Plane Geometry - Similarity and Congruence | AI tutor The No.1 Homework Finishing Free App

'In triangle ABC, where 43AB=6, AC=4, and cosB=3/4, answer the following questions: (1) Find the length of side BC. (2) When angle C is acute, find the area of triangle ABC. (3) For triangle ABC from (2), determine the radii of its circumcircle and incircle.'

'(3) In order for triangle ABC to be an acute triangle, angles A, B, and C all need to be acute. Here, when t is a real number, t^{2}-(t-2)=t^{2}-t+2=\\left(t-\\frac{1}{2}\\right)^{2}+\\frac{7}{4}>0, so it always holds that t^{2}>t-2. Therefore, the condition for angles A and B to both be acute is that for the y-coordinates of points A, B, and C, t-2<t^{2}-t-1<t^{2}, and for the y-coordinate difference of the 42 points A, B to be t^{2}-(t-2) =\\left(t-\\frac{1}{2}\\right)^{2}+\\frac{7}{4}>0, therefore t^{2} \\neq t-2 can also be considered valid.'

'Illustrate the regions represented by the following inequalities.'

'Let the radius of the inscribed circle be r, the radius of the circumscribed circle be R, and h=r/R. Also, let ∠A=2α, ∠B=2β, ∠C=2γ.'

'Internal and External Division Points\nFind the coordinates of the points that internally and externally divide the line segment AB in the ratio of m to n.\nInternal division point $\\left(\\frac{n x_{1}+m x_{2}}{m+n}, \\frac{n y_{1}+m y_{2}}{m+n}\\right)$\nExternal division point $\\left(\\frac{-n x_{1}+m x_{2}}{m-n}, \\frac{-n y_{1}+m y_{2}}{m-n}\\right)$'

'Practice book 57, p.127 Let A(x1, y1), B(x2, y2), and P(a, t). The equations of the tangents at points A and B are x1x + y1y = 1 and x2x + y2y = 1, respectively.'

'Quadrilateral ABCD is inscribed in a circle, ∠ABC=∠DAB, so ∠ADC=∠BCD. As CD is common, AD=BC. Arcs of equal length have equal central angles, so ∠ACD=∠BAC. Let ∠ACD=∠BAC=θ(0<θ<α), then ∠ACB=π-(α+θ), ∠CAD=α-θ. In triangle ABC, by the Law of Sines, AB/sin{π-(α+θ)}=2*1, BC/sinθ=2*1. Therefore, AB=2sin(α+θ), BC=2sinθ. Also, in triangle ACD, by the Law of Sines'

'Practice (1) When the triangle PAB with points A(0, -2), B(0, 6), and point P as vertices moves such that AP:BP=1:3, find the locus of point P.'

'Understand the coordinates of the internal and external division points between two points.'

"By reducing Figure A to 1/4, a similar figure is obtained, which when placed in (1) to (3) of Figure A, results in Figure B. Next, by reducing Figure B to 1/4, a similar figure is obtained, which when placed back in (1) to (3) of Figure A, results in a self-similar shape. Apply this self-similar shape to the pattern of Pascal's triangle."

'Find the acute angle formed by the two lines y=5x(1) and y=\\frac{2}{3}x(2).'

'Convert the following angles from degrees to radians and from radians to degrees.'

'TR 132\nFind the angle formed by lines 2 \ y=-\\frac{2}{5} x \ (1) and \ y=\\frac{3}{7}x \ (2).\nAssuming the angle formed by the two lines is acute.\nLet the angle formed by lines (1) and (2) with the positive direction of the \ x \ axis be denoted by \ \\alpha, \eta \'

'Find the locus of a point P equidistant from points A(-1,-2) and B(-3,2).'

'STEP forward'

'Use the addition formula to find the values of sin 75° and tan 15°. Since 75° is not a standard angle on the protractor, it cannot be directly calculated using trigonometric definitions. By expressing 75° in terms of the sum or difference of angles like 30°, 45°, 60°, etc., you can use the addition formula to determine the trigonometric functions of 75°.'

'Are the following two lines parallel or perpendicular?'

'(1) The x-coordinate of the intersection points of parabolas P₁ and P₂ is given by the equation x² - 2tx + 2t = -x² + 2x, which simplifies to x² - (t+1)x + t = 0. Solving this, we get (x-1)(x-t) = 0, which leads to x=1, t. When 0<t<1, S is the area of the red region in the diagram, given by'

'Condition for two lines to be perpendicular\nFor two lines y=m_{1} x+n_{1} and y=m_{2} x+n_{2}, the lines are perpendicular when the product of their slopes is -1.'

'When the size of an angle represented by a radius is specified, the position of the radius is determined, but conversely, even if the position of the radius is determined, there are countless angles it can represent, not just one. This is because the radius returns to its original position after a full rotation.\\n\\nThe angle formed by radius OP and the initial line OX is denoted by $\\alpha$, then the angle represented by radius OP is $\\quad \\alpha+360^{\\circ} \\times n \\quad (n$ is an integer $) \\) \\( \\cdots,-300^{\\circ}, 60^{\\circ}, 60^{\\circ}+360^{\\circ}=420^{\\circ}, \\cdots$ which is the same for radii'

'The point of division and external division point m, n are positive numbers. When a point P on the line segment AB satisfies AP: PB = m: n, the point P is said to divide the line segment AB in the ratio m: n, and the point P is called the inner point of the line segment AB. Also, when a point Q on the extension of the line segment AB satisfies AQ: QB = m: n (m≠n), the point Q is said to divide the line segment AB in the ratio m: n, and the point Q is called the external point of the line segment AB. Generally, the following holds:\nInner division\nOuter division when m>n'

'Find the locus of point P such that the ratio of its distances from points O(0,0) and A(3,6) is 1:2.'

'Find the coordinates of the following points.'

"My house is closer to Mr. A's house."

'Choose a layer from points A to F at Y point that is same as the e layer at point X in Figure 2, and provide the symbol.'

'The pyramid P-ABCD is similar, with a similarity ratio of square to triangle plus square equals 3 to 8 plus 3, which simplifies to 3 to 11. Therefore, the ratio of their volumes is 27 to 1331.'

'The shaded area of the triangle in the figure on the right (4) is denoted as K. Assuming the area of quadrilateral ABCD is 1, the area of rectangle BCQP is also 1, so the area of rectangle RPQS is also 1. The area of triangle RPQ is 1/2. Furthermore, triangles RBU and QSU are similar, with a similarity ratio of RB: QS = 2:1, hence RU: UQ = 2:1. Additionally, triangles PBT and QST are congruent, so it is known that PT = TQ. Therefore, K is 1/2 times the area of triangle RPQ, which is 1/6 times, so K = 1/2 * 1/6 = 1/12. Therefore, the area of quadrilateral ABCD is 12 times K.'

'4 Planar Figures - Ratios of Sides and Areas (1) As shown in the diagram on the right, mark the center of the circle as O and join O with points E, F, G, and H on the circumference. Furthermore, since triangle ABD is an equilateral triangle, the angles marked with symbols are 60 degrees, and the angles marked with • are 60 ÷ 2 = 30 degrees. Thus, all right-angled triangles with ○ and are half of equilateral triangles. Therefore, focusing on triangle ODH, HD:OD = 1:2, and focusing on triangle AOD, OD:AD = 1:2, so if the length of HD is taken as 1, the length of OD is 1 × 2/1 = 2, and the length of AD is 2 × 2/1 = 4. Hence, AH:HD = (4-1) : 1 = 3 : 1.'

'Answer the following questions.'

'Problem on measurement of length and accuracy (1) The scale divides 39mm into 20 equal parts and has the finest graduated line drawn on it, so the interval of one graduation is 39 ÷ 20=1.95 (mm).'

'In Figure 10, the distance between XY is 65-30=35 (increments) as measured by the eyepiece micrometer, and it is 50 increments as measured by the objective micrometer. One increment on the objective micrometer is 10 micrometers, so with 50 increments, it becomes 10 x 50 = 500 micrometers. Therefore, the length visible for each increment on the eyepiece micrometer is 500 ÷ 35 = 14.28..., which is equal to 14.3 micrometers.'

'Regarding the underlined part in line 4, sentences A to C describe either Rausudake, Iwakisan, or Choukaisan. Choose the correct sentence and mountain combination from the following options and answer with the number.'

'Upon investigating this land, it was found that the length of AC is 15 meters, the length of BC is 18 meters, and the size of angle B is exactly twice the size of angle C. In this case, how far is T from B?'

'When cutting this solid with a plane passing through points P, Q, and F, the plane intersected edge AE at point R.'

'Next, draw a line perpendicular to the line connecting the center O and point B, passing through point B. The point where the two lines intersect is point C. Since the lengths of CA and CB are always equal, a circle can be drawn with point C as the center and passing through points A and B. The arc of this circle is the path that the Poan alien walked.'

'Considering the same approach as in (2)(1), we have OI:ID=3:1, which means that if the area of triangle HID is taken as 1, then the area of triangle HOI is 1 × 3/1 = 3. Therefore, the area of quadrilateral EFGH is 3 × 8 = 24. Also, the area of triangle HOD is 1 + 3 = 4, so the area of triangle AOH is 4 × 3/1 = 12, and the area of triangle AOD is 4 + 12 = 16. Hence, the area of quadrilateral ABCD is 16 × 4 = 64, and the ratio of the areas of quadrilateral EFGH to quadrilateral ABCD is 24:64 = 3:8.'

'(2) If triangle CDB is congruent to triangle FDE in the figure above (3), find the measure of angle FED.'

'The area of triangle AFC is equal to the area of triangle AEC. Furthermore, when both triangles are added to triangle ADC, the areas of triangle CDF and triangle AED are also equal. Therefore, the area of triangle AED is 3 × 1 ÷ 2 = 1.5 (cm^2), so the area of triangle CDF is also 1.5 cm^2 and the area of the square with CD as one side is twice the area of triangle CDF, which is 1.5 × 2 = 3 (cm^2).'

'Regarding two microscopes, choose one of the options for the visibility when the magnification of the eyepiece lens is increased from 10x to 40x, and provide the symbol.'

'In triangle ABC, angle B is a right angle and in triangle ACD, angle C is a right angle, and the angles marked with dots are equal. Point E is the intersection of the extensions of sides BC and AD. The length of side AB is 2 cm, and the length of side BC is 1 cm. (2) What is the length of CE in cm?'

'(1) As the distance value increases in Figure 2, the illuminance value decreases. In other words, as the distance between the light bulb and the illuminance meter increases, the illuminance decreases.'

"In figure (2), Mark moves along the circumference with X as the center initially, while Harry moves along the circumference with Y as the center. In figure (2), triangle OFX and triangle YOX are both triangles that are half of an equilateral triangle, so if we set XF=1, then OX=1×2=2, and XY=2×2=4. Therefore, the ratio of the radii of the circumferences that Mark and Harry move on is XF:YF=1:(4-1)=1:3. Next, the central angle of the part Mark moves on is 120 degrees, and there are a total of 6 such parts. Also, the central angle of the part Harry moves on is 60 degrees, and there are a total of 3 such parts. Therefore, the ratio of the distances Mark and Harry move is {1×2×π×120/360×6}:{3×2×π×60/360×3}=4:3, so we can determine that Mark's speed is 4/3=1 1/3 times Harry's speed."

"In another method of solving, in figure 4, triangles DBG and DCG are congruent, let the angle BDG be θ and angle CDG be φ, then θ + φ = 90 degrees, so the sum of 2θ and 2φ is 180 degrees. Therefore, the size of angle ADB is 2θ. Additionally, taking point H on BD such that AD = AH, triangles ATH and ATD being congruent implies the size of angle AHT is also 2θ. Consequently, from the exterior angles of triangle ABH, we find that the size of angle HAB is θ, as shown in figure 5. In figure 5, AC has a length of 15 meters, and the lengths of DB and DC are equal, making the length of the bold line segment 15 meters. Furthermore, given that AD = BH and DT = HT, the length of BT is half of the bold line segment's length, which is 15 ÷ 2 = 7.5 meters. It's worth noting that the length of BT is independent of the length of BC."

'(2)As shown in figure 3, extending BA and PQ intersect at point M, where MF and AE intersect at point R. In triangle MAQ and triangle MBP in the figure, AQ is 4 cm (8-4) in length, and BP is 6 cm (8-2) in length. Hence, the similarity ratio is AQ:BP = 4:6 = 2:3. Therefore, the length of MA is calculated as 6*2/3-2=12 cm. Furthermore, triangles MRA and FRE are also similar, with a similarity ratio of MA:FE = 12:9 = 4:3, which means AR:RE = 4:3.'

'Since triangle ADC and triangle CDB are similar, CD=cm, which can be expressed as 1:=:3. Also, when P:Q=R:S, Q × R=P × S, so × =1 × 3=3. Therefore, the area of the square with CD as one side can also be calculated as 3 cm^2.'

'Determine the value of p, so that the two vectors m=(1, p) and n=(p+3, 4) become parallel.'

'Prove: In triangle ABC, let the sides BC, CA, and AB be divided internally by points P, Q, and R in the ratio m:n (m>0, n>0). If 24R, then the centroids of triangles ABC and PQR coincide.'

'State the conditions for line segment AB and CD to be parallel, as well as the conditions for them to be perpendicular, for different points A(α), B(β), C(γ), D(δ).'

'For the existence range of point P in the triangle OAB on the plane, if OP = sOA + tOB, then the existence range of point P is as follows: (1) Line AB if and only if s + t = 1; in particular, line segment AB if and only if s + t = 1, s ≥ 0, t ≥ 0. (2) The perimeter and interior of triangle OAB if and only if 0 ≤ s + t ≤ 1, s ≥ 0, t ≥ 0. (3) The perimeter and interior of parallelogram OACB if and only if 0 ≤ s ≤ 1, and 0 ≤ t ≤ 1.'

'In the xy-plane, consider the points \\( \\mathrm{F}_1(a, a), \\mathrm{F}_2(-a,-a) \\) and let \ \\mathrm{P} \ be a point whose distance product from these points is a constant value of \ 2 a^2 \. Denote the locus of point \ \\mathrm{P} \ as \ C \. It is given that \ a>0 \.\n(1) Find the equation of \ C \ in terms of the Cartesian coordinates \\( (x, y) \\).\n(2) Find the polar equation of \ C \ with the origin as the pole and the positive x-axis as the initial line, in polar coordinates \\( (r, \\theta) \\).\n(3) Prove that the portion of \ C \ excluding the origin lies in the combined range of the first and third quadrant in the plane.'

'Convert Cartesian coordinates (x, y) to polar coordinates (r, θ).'

'In a practice exercise, 3 points A, B, C are located on a circle with center O and radius 1, such that (3) 3213OA + 12OB + 5OC = 0. Let angle AOB be α and angle AOC be β. Determine: (1) Prove that OB is perpendicular to OC. (2) Find cosα and cosβ.'

'In an equilateral triangle ABC with side length a, let P₁ be the foot of the perpendicular from vertex A to side BC. Let Q₁ be the foot of the perpendicular from P₁ to side AB; R₁ be the foot of the perpendicular from Q₁ to side CA; and P₂ be the foot of the perpendicular from R₁ to side BC. By repeating this process, points P₁, P₂, ..., Pn, ... will be located on side BC. Determine the limiting position of point Pn. Angle is basically 26°.'

'(4) Orthocenter (in the case of an acute triangle \ \\triangle \\mathrm{ABC} \) the intersection point of three altitudes \\( \\mathrm{H}(\\vec{h}) \\)\nLet \ \\mathrm{D}, \\mathrm{E} \ be the intersection points of the line \ \\mathrm{AH} \ with side \ \\mathrm{BC} \ and the line \ \\mathrm{CH} \ with side \ \\mathrm{AB} \ respectively, then \ \\mathrm{BD}=\\frac{\\mathrm{AD}}{\\tan B}, \\mathrm{DC}=\\frac{\\mathrm{AD}}{\\tan C} \ gives\n\\\mathrm{BD}: \\mathrm{DC}=\\tan C: \\tan B\\nSimilarly, \ \\mathrm{AE}: \\mathrm{EB}=\\tan B: \\tan A \\nTherefore, from (*) we get \ \\triangle \\mathrm{BCH}: \\triangle \\mathrm{CAH}: \\triangle \\mathrm{ABH}=\\tan A: \\tan B: \\tan C \\nThus, from \\left( ** \\) we have \\( \\quad \\vec{h}=\\frac{(\\tan A) \\vec{a}+(\\tan B) \\vec{b}+(\\tan C) \\vec{c}}{\\tan A+\\tan B+\\tan C} \\)'

'Find the polar equations of the following circle and line in polar coordinates. Assume a>0.'

'Let s be not equal to 0. For distinct 3 points O(0,0), P(s, t), Q(s+6t, s+2t), where points P, Q are in the same quadrant and OP // OQ, let α be the angle between the line OP and the positive direction of the x-axis. Find the value of tan α.'

'Problem 107: Applications of trigonometry\nTo measure the height of a building, the angle of elevation to the top point P of the building was measured from a point 10 meters away and at a height of 1.5 meters, and it was found to be 65 degrees.\nUsing the trigonometric table at the end of the book, answer the following questions:\n(1) Determine the height of this building. Round off to the nearest meter.\n(2) From a point 15 meters away from the building, determine the angle of elevation to point P, following the same procedure.'

'In triangle ABC, ∠C=90°, AB:AC=5:4. On the extension of side BC beyond point C, take CD=376. Let E be the midpoint of side AB, and let BF be the perpendicular dropped from point B to line AD. Answer the following questions: (1) Prove that EF=EC. (2) Find the ratio of the areas of triangle ABC to triangle CEF.'

'Prove that in a non-equilateral triangle ABC, when O is the circumcenter, G is the centroid, and H is the orthocenter, G lies on segment OH and OG:GH=1:2.'

'Using the sine rule and cosine rule: Find the sides and angles of the triangle.'

'Basic Example 131 Area of a Triangle'

'Circumferential angle: The size of the circumferential angle for an arc is constant, half the size of the central angle for that arc.'

'There is a circular pool in a nearby park. One day, I and my friend decided to measure the area of this pool, so we went out with a tape measure and chalk. We marked points A, B, and C at three places on the edge of the pool. When we measured the horizontal distances AB, BC, CA, they were 9m, 6m, 12m respectively. 1. Find the sine, cosine, and tangent of angle ABC. 2. Find the area of this pool.'

'Let θ be an acute angle. When one of sin θ, cos θ, tan θ takes a specific value, find the values of the other 2 trigonometric ratios in each case.'

'In triangle ABC, where AB = 3, AC = 2, and ∠BAC = 60°, let D be the intersection point between the angle bisector of ∠A and BC. Find the length of segment AD.'

'Mathematics I'

'The proof is as follows: Connect EP, FQ, EF, and the perpendicular bisector of EF. Let O be the center of the circle. Connect OE, OF, the extensions of AD, and the extensions of BC intersect at P. Connect OF, the extension of ED intersect at Q. By the corollary from example 1, EF bisects ∠BOF and ∠EOQ, and ∠EOF=∠EOQ. Also, triangle EOF is congruent to triangle POQ (by side-side-angle), so EP^2 + FQ^2 = EO^2 + OF^2 = EF^2.'

'Prove that in triangle ABC, when the sizes of angles A, B, C are represented by A, B, C respectively, the equation cos((A+B)/2) = sin(C/2) holds true.'

"Cheva's Theorem\nWhen a line connecting the 3 vertices A, B, and C of triangle ABC or points on or extended from the sides BC, CA, AB intersects with the sides or their extensions, and the intersection points are P, Q, R, then\n\ \\frac{BP}{PC} \\cdot \\frac{CQ}{QA} \\cdot \\frac{AR}{RB} = 1 \"

'In triangle ABC, sinA:sinB:sinC=5:7:8. Therefore, cosC is (fill in the blank). Furthermore, if the length of side BC is 1, the area of triangle ABC is (fill in the blank).'

'365'

'Relationship between sides and angles of a triangle\nTheorem\n14\n1. In a triangle\n 1. In a triangle, the angle opposite the larger side is greater than the angle opposite the smaller side.\n 2. In a triangle, the side opposite the larger angle is longer than the side opposite the smaller angle.\nThat is, \ \\mathrm{AB}<\\mathrm{AC} \\Leftrightarrow \\angle \\mathrm{C}<\\angle \\mathrm{B} \'

"When circles O and O' with radii 5 and 8 respectively are externally tangent at point A, and the common external tangent line of these two circles intersects circles O and O' at points B and C, respectively, let the intersection of BA extended and circle O' be D. Prove: (1) AB is perpendicular to AC. (2) Prove that points C, O', D are collinear. (3) Find the ratio of AB:AC:BC."

'In triangle ABC, let R be the radius of the circumcircle. If A=30°, B=105°, a=5, find the values of R and c.'

'Number of rectangles (including squares) formed by the intersection of 7 lines x=k(k=0,1,2,⋯6) and 5 lines y=l(l=0,1,2,3,4) on the coordinate plane. Also, number of rectangles with an area of 4.'

'In triangle ABC, by the cosine rule\n\n\\[\n\egin{aligned}\n\\mathrm{AC}^{2}= & 2^{2}+(\\sqrt{3}+1)^{2} \\\\\n & -2 \\cdot 2(\\sqrt{3}+1) \\cos 60^{\\circ} \\\\\n = & 6\n\\end{aligned}\n\\]\n'

'Basic Example 70 Ratio of areas of centroid and triangle'

'Law of Cosines'

'Example 124 Maximum angle of a triangle In triangle ABC, find the measure of the largest angle of this triangle under the following conditions. (1) a/13=b/8=c/7 (2) sinA: sinB: sinC=1: √2: √5'

'Using the sine rule, find the lengths of the other sides of the following triangle: A is 45° and the length of the opposite side a is 2.'

'(1) In \ \\triangle \\mathrm{ABC} \, find the following.\n3106\n(A) When \ A=60^{\\circ}, c=1+\\sqrt{6}, a+b=5 \, find \ a \.\n(1) When \ A=60^{\\circ}, a=1, \\sin A=2 \\sin B-\\sin C \, find \ b, c \.\n(2) In \ \\triangle \\mathrm{ABC} \, if \ b=2, c=\\sqrt{5}+1, A=60^{\\circ} \, find whether \ C \ is acute, right, or obtuse angle.'

'There is a piece of origami in the shape of an equilateral triangle ABC with a side length of 10 cm. Points D on side AB and E on side AC are taken such that segment DE is parallel to side BC. When folding the paper along segment DE, let S be the area of the overlap between triangle ADE and quadrilateral BCED. The maximum value of S occurs when the length of segment DE is x cm, and at this point, S = y cm².'

'Find the length of segment DE. Given: BC = 60, AM = 20. In triangle ABC, let M be the midpoint of BC, and D, E be the points where the angle bisectors of angles AMB and AMC intersect sides AB and AC.'

'(2) In triangle ABC, where AB=4, BC=3, and CA=2, let D and E be the points where the angle A and its exterior angle bisectors intersect line BC. Determine the length of segment DE.'

'Inside acute angle PR XOY, 2 points A and B are given as shown in the figure on the right. On the half-lines 480 X and OY, points P and Q are taken respectively to minimize AP + PQ + QB, where should P and Q be placed respectively.'

'In triangle ABC on the right, G is the centroid of triangle ABC, and segment GD is parallel to side BC. Find the ratio of the areas of triangles DBC and ABC.'

'Length of angle bisector in a triangle'

'In triangle ABC, by the cosine rule\n\n\\[\n\egin{array}{l} \\cos \\angle \\mathrm{ACB}=\\frac{(\\sqrt{3}+1)^{2}+(\\sqrt{6})^{2}-2^{2}}{2(\\sqrt{3}+1) \\cdot \\sqrt{6}} \\\\\n=\\frac{2 \\sqrt{3}+6}{2 \\sqrt{6}(\\sqrt{3}+1)} \\\\\n=\\frac{2 \\sqrt{3}(1+\\sqrt{3})}{2 \\sqrt{6}(\\sqrt{3}+1)} \\\\\n=\\frac{1}{\\sqrt{2}} \\\\\n\\text { Therefore } \\quad \\angle \\mathrm{ACB}=45^{\\circ} \\\\\n\\text { Hence } \\quad \\angle \\mathrm{ACD}=75^{\\circ}-45^{\\circ}=30^{\\circ} \\\\n\\text { Therefore }\n\\end{array}\n\\]\n'

'From the given conditions, when determining the other three elements of a triangle, the methods for using the theorems based on the conditions are as follows: 1. 1 side and its adjacent angles (obtaining b, c, A from the conditions of a, B, C) A = 180° - (B + C); Sine theorem: a / sinA = b / sinB = c / sinC; 2. 2 sides and the included angle (obtaining a, B, C from the conditions of b, c, A) Cosine theorem a² = b² + c² - 2bc cosA to find a; Cosine theorem cosB = (c² + a² - b²) / (2ca) to find B; C = 180° - (A+B); 3. 3 sides (obtaining A, B, C from the conditions of a, b, c) Cosine theorem cosA = (b² + c² - a²) / (2bc) to find A; Cosine theorem cosB = (c² + a² - b²) / (2ca) to find B; C = 180° - (A + B).'

'In triangle ABC, when sin A: sin B: sin C = 5: 16: 19, find the measure of the largest angle in this triangle.'

'In triangle ABC with AB=13, BC=15, and CA=8, a perpendicular AD is drawn from point A to side BC. Find the following values: (1) Length of BD (2) sin angle B (3) tan angle C'

'Prove the following in a acute-angled triangle ABC (AB > AC) with the angle A bisector AD, median AM, perpendicular AH:'

'There is a quadrilateral ABCD inscribed in a circle. If AB=8, BC=3, BD=7, and AD=5, find the length of A and side CD. Also, find the area S of quadrilateral ABCD.'

'In triangle ABC, if C is 45 degrees, b is sqrt(3), and c is sqrt(2), find A, B, and a.'

'(3) Since $\\mathrm{EQ} / / \\mathrm{BD}$, we have $\\mathrm{EQ}: \\mathrm{BD} = \\mathrm{AE}: \\mathrm{AB} = 1:(1+k)$, therefore $\\mathrm{EQ} = \\frac{\\mathrm{BD}}{1+k}$. Also, $\\mathrm{QF} / / \\mathrm{DC}$, so $\\mathrm{QF}: \\mathrm{DC} = \\mathrm{AF}: \\mathrm{AC} = 1:(1+k)$, hence $\\mathrm{QF} = \\frac{\\mathrm{DC}}{1+k}$. Therefore, $\\frac{\\mathrm{EQ}}{\\mathrm{QF}} = \\frac{\\mathrm{BD}}{\\mathrm{DC}} = 1$.'

'Quadrilateral ABCD is circumscribed around circle O. Let the points where sides AB, BC, CD, DA intersect circle O be P, Q, R, S respectively, and let the lengths of line segments AP, BQ, CR, DS be a, b, c, d. When none of the three lines AC, PQ, RS are parallel to each other:\n(1) Let the intersection point of AC and PQ be X, prove that AX: XC = a: c.\n(2) Let the intersection point of AC and RS be Y, prove that AY: YC = AX: XC.'

'Describe the properties of triangle A: a^2 = 64, b^2 + c^2 = 61'

'In an isosceles triangle, the two base angles are equal. Furthermore, the angle bisector of the vertex angle of an isosceles triangle bisects the base perpendicularly. Using this information, solve the following problem: In isosceles triangle ABC, if the vertex angle ∠A = 100 degrees, what is the measure of the base angle ∠B?'

'In triangle ABC, a point O inside the triangle is connected to the three vertices, intersecting with sides BC, CA, and AB at points D, E, and F, and the extension of FE passes point E to intersect with the extension of side BC at point G.'

'Problem 381. Comparison of perimeter of a triangle'

'In triangle ABC, each side is tangent to a circle at points P, Q, R as shown in the figure below. Find the lengths of segment AQ and BC.'

'Basic example 133: Length of the angle bisector in a triangle (2)'

"Inverse of Ceva's Theorem"

"In triangle ABC, if point P divides side BC in the ratio m:n, point Q divides side CA in the ratio l:m, and point R divides side AB in the ratio n:l, then the lines AP, BQ, and CR intersect at one point. Prove this using the converse of Ceva's theorem."

'Prove that the following equality holds in triangle ABC: \\[ \\left(b^{2}+c^{2}-a^{2}\\right) \\\\tan A=\\left(c^{2}+a^{2}-b^{2}\\right) \\\\tan B \\]'

'In triangle ABC, let D be the point where the angle bisector of angle B intersects side AC. Find the length of segment BD.'

'Key points to solve angle problems'

'Basic Example 68 Using Circumcenter and Orthocenter\nLet H be the orthocenter of acute triangle ABC, O be the circumcenter, and OM be the perpendicular line from O to side BC. Also, take point K on the circumcircle of triangle ABC such that segment CK becomes a diameter of the circle. Prove the following:\n1. BK = 2OM\n2. Quadrilateral AKBH is a parallelogram\n3. AH = 2OM'

'Quadrilateral inscribed in a circle'

"Using the converse of Ceva's theorem, prove that the three medians of a triangle intersect at one point."

"Prove the inverse of basic figure 90's theorem"

"Math A\nrefers to the situation when the 4 points A', P, Q, B' are collinear. Therefore, by finding the point A symmetrical to point A' with respect to the half ray OX, and the point B symmetrical to point B' with respect to the half ray OY, we obtain the intersection point P of line A'B' and half ray OX, and the intersection point Q of line A'B' and half ray OY."

'In triangle ABC, with AB=8, BC=3, CA=6, let D be the point where the angle bisector of angle A intersects line BC. Find the length of segment CD.'

'In an acute-angled triangle ABC, let BD and CE be the altitudes dropped from vertices B and C to their respective opposite sides. If BC=a, express angle A in terms of angles. You may use the property that if in a line segment PQ the angle PRQ=90°, the point R lies on the circle with PQ as a diameter.'

'PRACTICE 6° AC=BC, AB=6 In the right-angled isosceles triangle ABC where AC=BC and AB=6, two rectangles with equal heights are created as shown in the figure on the right. Find the maximum value of the sum of the areas of the two rectangles when they are made to be maximum.'

'Prove that in acute triangle ABC, with orthocenter H and circumcenter O, midpoint of side BC as M, and midpoint of segment AH as N, the length of segment MN is equal to the radius of the circumcircle of triangle ABC, using the fact that AH=2OM.'

'In the diagram, if AR:RB=3:4 and BP=PC, find AQ:QC.'

'Given line segment AB of length a and two line segments of length b and c, draw a line segment of length \ \\frac{a c}{b} \.'

'When to apply the sine rule and cosine rule? Both the sine rule and cosine rule can be used to find lengths of sides and sizes of angles, and sometimes it can be confusing which one to use. Is there a method to determine this?'

'Find the length of line segment BF.'

'Prove that when the length of the diagonal AC of quadrilateral ABCD is smaller than any of its sides, the length of diagonal BD is greater than any of its sides.'

'At point H on the level ground PR, a pole stands perpendicular to the ground. When seeing the top of the pole from points A and B, the elevation angles are 30 degrees and 60 degrees respectively. Also, in the ground survey, it is known that the distance between A and B is 20 meters, and ∠AHB=60 degrees. Determine the height of the pole. Assume eye height is not considered.'

'Chapter 4: Geometry and Measurement EX In the quadrilateral ABCD inscribed in a circle, where DA = 2AB and ∠BAD = 120°, and E is the intersection of diagonals BD and AC, E divides the segment BD into 3:4.\n(1) BD = ?AB, AE = 1 ?AB.\n(2) CE = ? ?AB, BC = I? ?AB.\n(3) AB:BC:CD:DA = 1: ? : power : 2.\n(4) If the radius of the circle is 1, then AB = ?, and the area of the quadrilateral ABCD is S = ?.'

'In the quadrilateral ABCD inscribed in a circle, with AB=2, BC=1, CD=3, and cos∠BCD=-1/6. Find the length of AD and the area of the quadrilateral ABCD.'

'For the given line segment AB, plot the following points. (1) Point E dividing the line segment AB internally in the ratio 3:2 (2) Point F dividing the line segment AB externally in the ratio 3:1'

"In triangle ABC, where AB=3, BC=4, and CA=6, let D be the point where the bisector of angle A's exterior angle intersects line BC. Find the length of segment BD."

'Find the areas of triangle ABC and parallelogram ABCD in the given figure.'

'Basic Exercise 122 Solution for Triangles (1) For each case, find the remaining side lengths and angles of triangle ABC. (1) a=√3, B=45°, C=15° (2) b=2, c=√3+1, A=30°'

'Sine rule'

'Hanako and Taro decided to work on the following [problem] together and try to think using graphic drawing software.'

'When a point P lies on the line segment AB and the ratio of AP to PB is m: n, let the length of the side be denoted by k. In a quadrilateral inscribed in a circle, the similarity of the triangles formed by the diagonals is used.'

'Prove that in triangle ABC, if the angles ∠A, ∠B, ∠C are denoted by A, B, C respectively, then the equation (1+tan^2(A/2))sin^2((B+C)/2)=1 holds true.'

'(4) Find the triangle with the smallest circumcircle radius.'

"The term gradient is used to describe the slope of roads and railways. Using the trigonometric ratios, answer the following questions. (1) The gradient of a road is often expressed in percentages (%). The percentage indicates how many meters the elevation increases when moving 100 meters horizontally. On a certain road, there is a sign indicating 23%. What is the approximate angle of this road's slope? (2) The gradient of a railway is often expressed in permilles (‰). The permille indicates how many meters the elevation increases when moving 1000 meters horizontally. On a certain railway line, there is a sign indicating 18‰. What is the approximate angle of this railway line's slope?"

'Relationship between sides and angles of a triangle'

'In \ \\triangle ABC \, where \ \\angle C=90^\\circ \ and \ AB:AC=5:4 \, a point \ D \ is constructed on the extension of side \ BC \ such that \ CA=CD \. Let \ E \ be the midpoint of side \ AB \, and let \ BF \ be the perpendicular from point \ B \ to line \ AD \. Answer the following questions: ［Miyazaki University］\n(1) Prove that \ EF=EC \.\n(2) Find the ratio of the areas of \ \\triangle ABC \ and \ \\triangle CEF \.'

'From (5) BC = 9, BD: DC = 4: 5, we have BD=\\frac{4}{9} BC=\\frac{4}{9} \\cdot 9=4. Therefore, BD \\cdot BC =4 \\cdot 9 = 36'

'(2) By the law of cosines'

'In triangle ABC, show the relationship between a², b², and c² based on the range of angle A.'

'Please explain the meanings of the following terms: corresponding angles, vertical angles, acute angles, obtuse angles, interior angles, exterior angles, congruent, similar, perpendicular bisector, angle bisector, acute triangle, right triangle, obtuse triangle, chord, arc, central angle, inscribed angle, tangent to a circle, opposite side, diagonal, parallelogram.'

'Prove that triangle ABC is similar to triangle AEF.'

'In triangle ABC, with BC=5, CA=3, AB=7. Let D and E be the points where angle A and its exterior angle bisector intersect line BC, the length of segment DE is to be found.'

'Please use the properties and definitions of circumcenter, incenter, orthocenter, and centroid to answer the following triangle problems.'

'Alternative solution (Same until step 11)\n (1) From \ \\triangle \\mathrm{AQC}=\\frac{3}{7} \\triangle \\mathrm{ADC}=\\frac{3}{7} \\cdot \\frac{2}{3} \\triangle \\mathrm{ABC}=\\frac{2}{7} \\triangle \\mathrm{ABC} \ likewise\n\n \\triangle \\mathrm{BRA}=\\frac{3}{7} \\triangle \\mathrm{BEA}=\\frac{3}{7} \\cdot \\frac{2}{3} \\triangle \\mathrm{BCA}=\\frac{2}{7} \\triangle \\mathrm{ABC} \\ \\triangle \\mathrm{CPB}=\\frac{3}{7} \\triangle \\mathrm{CFB}=\\frac{3}{7} \\cdot \\frac{2}{3} \\triangle \\mathrm{CAB}=\\frac{2}{7} \\triangle \\mathrm{ABC} \n \n Thus, \\triangle \\mathrm{PQR}=\\triangle \\mathrm{ABC}-(\\triangle \\mathrm{AQC}+\\triangle \\mathrm{BRA}+\\triangle \\mathrm{CPB}) \\=\n \\triangle \\mathrm{ABC}-3 \\cdot \\frac{2}{7} \\triangle \\mathrm{ABC}=\\frac{1}{7} \\triangle \\mathrm{ABC} \n \\triangle \\mathrm{ABC}=\\frac{1}{2} \\cdot 1 \\cdot \\frac{\\sqrt{3}}{2}=\\frac{\\sqrt{3}}{4} therefore \\ \n \\triangle \\mathrm{PQR}=\\frac{1}{7} \\cdot \\frac{\\sqrt{3}}{4}=\\frac{\\sqrt{3}}{28}'

'Basic Example 802 Application of Equations\nIn the triangle ABC as shown on the right, with BC=20 cm, AB=AC, and ∠A=90°. Points D and E are taken on sides AB and AC respectively such that AD=AE. Perpendiculars are drawn from D and E to the side BC, intersecting at F and G. Find the length of side FG when the area of rectangle DFGE is 20 cm².'

"Converse of Menelaus' Theorem"

'Given a quadrilateral ABCD inscribed in a circle, where AB=8, BC=3, BD=7, and AD=5, find the length of CD. Also, calculate the area S of the quadrilateral ABCD.'

'Basic Example 1142 Angles Formed by Lines\n(1) Find the angle α formed by the line y=-1/√3x and the positive direction of the x-axis, and the angle β formed by the line y=1/√3x and the positive direction of the x-axis. Also, find the acute angle formed by the two lines. Assume that 0° < α < 180°, 0° < β < 180°.\n(2) Find the acute angle θ formed by the two lines y=-√3x and y=x+1.'

'In the right-angled isosceles triangle ABC, where AC = BC and AB = 6, two rectangles with equal vertical lengths are constructed as shown in the diagram to the right. What is the maximum value of the sum of the areas of the two rectangles when constructed for maximum sum?\nFrom the given conditions, AC = BC = 6 / √2 = 3√2.\n\nAs shown in the diagram, let D, E, F, G be the points, and let the vertical length of the rectangles be denoted as x:\n\nDE = AE = AC - CE = 3√2 - 2x\nFG = AG = AC - GC = 3√2 - x\n\nAlso, since 0 < CE < AC\n0 < 2x < 3√2, which implies 0 < x < 3√2 / 2\n\nLet y be the sum of the areas of the two rectangles:\n\ny = x(3√2 - 2x) + x(3√2 - x)\n = -3x^2 + 6√2x\n = -3(x - √2)^2 + 6\n\nIn (1), y attains its maximum value of 6 at x = √2.'

"Using Menelaus' theorem in triangle ABC and line DF"

'Similarity condition of triangles: Two triangles are similar if one of the following conditions holds. [1] The ratio of three sides is equal. [2] Two pairs of sides are proportional and the included angles are equal. [3] Two pairs of angles are equal.'

'What is the shape of ∆ABC that satisfies the following equations: (1) b sin^2 A + a cos^2 B = a (2) a/cos A = b/cos B = c/cos C?'

"Prove the following using Ceva's theorem when a straight line connecting the vertices A, B, C of triangle ABC to a point O not lying on the sides or their extensions intersects the opposite sides or their extensions at points P, Q, and R respectively:"

'Prove that in triangle ABC, if M is the midpoint of side BC and the angle bisectors of ∠AMB and ∠AMC intersect sides AB and AC at points D and E, respectively, then DE // BC.'

'Prove that triangle ABC, satisfying the following conditions, is an equilateral triangle:\n(1) The centroid and circumcenter coincide.\n(2) The circumcenter and incenter coincide.'

'Regarding a rhombus with a sum of diagonal lengths of 10 cm:\n(1) Find the maximum area.\n(2) Find the minimum perimeter.'

'Theorem 2: In triangle ABC with AB ≠ AC, the intersection of the exterior angle bisector of ∠A and the extension of side BC divides side BC in the ratio of AB:AC.'

'In triangle ABC, let D be the midpoint of side AB, E be the midpoint of segment CD, and F be the intersection point of AE and BC. Find the ratio of AE to EF.'

'Solve the following problem about a triangle.'

'Master the use of Venn diagrams and conquer example 49!'

''

'Quadrilateral ABCD is inscribed in a circle, with AB=4, BC=2, and DA=DC.'

'Example 378\nIn triangle ABC with AB=10, BC=5, CA=6, let ∠A and its external angle bisectors intersect side BC or its extension at points D and E. Find the length of segment DE.'

'Master the Sine Rule and conquer Example 126!'

'Find cosA using the cosine rule, and then calculate the area and height of the triangle using the result.'

'What distance did you move horizontally when walking 80 meters on a slope with an 8° inclination from the horizontal? Also, how many meters did you descend vertically?'

'When two sides and the angle between them are given, we can use the cosine rule.'

"Given an equilateral triangle ABC with side length 1. Point P is taken on arc BC that does not include vertex A, such that PA=a, PB=b, PC=c (b>c). Let's calculate the value of a²+b²+c². Since ∠APB=∠APC=α degrees, cosine rule can be applied in triangle ABP."

'From two points A and B, which are 1 km away from each other on the sea, both points see the same mountain top C. From point A, the elevation angle towards the east is 30 degrees, while from point B, the angle towards the northeast is 45 degrees. Find the height CD of the mountain. Assume point D is directly below C, and points A, B, D are on the same horizontal plane. Also, assume sqrt(6)=2.45.'

'Basic Example 1\nFind the values of trigonometric ratios for 30°, 45°, 60°\nFind the sine, cosine, and tangent values for 30°, 45°, 60°\nGUIDE Trigonometric Ratios for 30°, 45°, 60°\nTry drawing the triangle ruler diagram'

'If 90° < A < 180°, in the diagram on the right, segment BD is the diameter of the circumcircle of triangle ABC. In this case, \ \\angle BAC + \\angle BDC = 180° \ which means \ \\angle BDC = 180° - A \, hence \ a = \\mathrm{BD} \\sin \\angle \\mathrm{BDC} \ \\( = \\mathrm{BD} \\sin (180° - A) \\) \ = \\mathrm{BD} \\sin A \ \ \\mathrm{BD} = 2 R \, therefore \ \\quad a = 2 R \\sin A \'

'There are three cases for the positional relationship between line `ℓ` and plane `α`.'

'Determining the shape of a triangle from the equality of sides and angles'

'Point H is the incenter of triangle DEF because it is the intersection of angle bisectors of angle DFE and angle FDE.'

'Please explain the properties of angle bisectors and ratios in a triangle.'

'In triangle ABC, if a²cosA sinB=b²cosB sinA holds, what is the shape of triangle ABC?'

'In triangle ABC, if a²cosA sinB = b²cosB sinA holds, what shape is triangle ABC?'

'Let O be the circumcenter of acute triangle ABC. If the angle BAO bisector intersects the circumcircle of triangle ABC at point D, prove that AB is parallel to OD.'

'Draw a perpendicular from point D to side AB, let the intersection be H, then AH=BH=\\frac{1}{2}. Therefore, using (2), \\cos 36^{\\circ} =\\frac{AH}{AD}=\\frac{\\frac{1}{2}}{\\frac{\\sqrt{5}-1}{2}}=\\frac{1}{\\sqrt{5}-1} \\ =\\frac{\\sqrt{5}+1}{(\\sqrt{5}-1)(\\sqrt{5}+1)}=\\frac{\\sqrt{5}+1}{4}. Focus on triangle DAH.\n\nReferring to it, draw a perpendicular from vertex A to side BC, let the intersection be E, then BE=\\frac{1}{2} BC=\\frac{\\sqrt{5}-1}{4}.\n\nTherefore, \\cos 72^{\\circ}=\\frac{BE}{AB}=\\frac{\\sqrt{5}-1}{4}. The angle bisector of an isosceles triangle bisects the base perpendicularly.'

'There are two circles tangent at point P. As shown in the figure to the right, two lines passing through point P intersect the outer circle at points A and B, and the inner circle at points C and D. Prove that AB is parallel to CD.'

'In the diagram on the right, L, M, N are the points of tangency of the sides of △ABC with the inscribed circle, ∠C=90°, AL=3, BM=10. (1) Let r be the radius of the inscribed circle, express the lengths of AC and BC as r. (2) Find the value of r.'

'In triangle ABC, when b=2√6, c=3√2+√6, and A=60°, find the length of the remaining side and the size of the other angle.'

'In triangle ABC, where AB=AC=5, BC=√5. Let D be a point on side AC such that AD=3. Extend side BC on the side of B to a point E, which is different from B, at the intersection of the circumcircle of triangle ABD. (3) Show that DE=EP.'

'In trapezoid ABCD, where AD // BC, AB=5, BC=7, CD=6, DA=3. Let E be the intersection of the line passing through D parallel to AB and side BC, and let ∠DEC=θ. Find the following values.'

'Chapter 7: Applications to Triangles\n135\nIn triangle ABC, where AB = 7, BC = 4√2, and ∠ABC = 45°, with the center of the circumscribed circle of triangle ABC denoted as O.\n(1) CA = $is a square, and the radius of the circumscribed circle O is$.\n(2) On arc BC of circumscribed circle O, excluding point A, a point D is taken such that CD = √10. In this case, given that ∠ADC = $, let AD = x, then x = √.'

'In triangle ABC, let point D divide side AB in the ratio 3:2, and point E divide side AC in the ratio 4:3. Let the intersection of BE and CD be point P, and the intersection of line AF and BC be point F. Find the ratio BF:FC.'

'Given that L, M, N are the points of tangency of the sides of triangle ABC with the incircle, ∠C=90°, AL=3, BM=10. (1) Express the lengths of AC and BC in terms of r, the radius of the incircle. (2) Find the value of r.'

'From points A and B, the points C and D on the opposite side of the waterway were observed as shown in the map on the right. It is assumed that points A, B, C, and D are at the same height.\n(1) Find the lengths of BD and BC (meters).\n(2) Find the length of CD (meters).\n\nThe answers can remain in square root form.\n& 〜GUIDE Use the sine theorem and cosine theorem to find applicable triangles.\n(1) In triangle ABD, one side and two angles are known, enabling the use of the sine theorem.\n(2) In triangle BDC, two sides and the angle between them are known, enabling the use of the cosine theorem.'

'Example 123: Right triangle and trigonometric values'

'Prove that in triangle ABC, if ∠B > ∠C, then b > c.'

'Chapter 3 Properties of Figures\nFurthermore, in ∆AFE and ∆ABC\n∠A is common, ∠AFE=∠ABC\nTherefore, since two sets of angles are equal, ∆AFE ∝ ∆ABC\nAF:AB=1:2\n∆AFE:∆ABC=1²:2²=1:4\nHence, ∆AFE=1/4 ∆ABC=1/4⋅12S=3S\n(2) From (1), let the area of quadrilateral AFGE be T\nT=∆EFG+∆AFE=S+3S=4S\nTherefore, from (1) and (2), ∆ABC/T=12S/4S=3\nTherefore 3 times'

'In the figure to the right, calculate the sine, cosine, and tangent values of angle θ.'

'Using the right-angled triangle ABC on the right, find the values of sine, cosine, and tangent of 15 degrees.'

'In the figure to the right, point I is the incenter of triangle ABC. Calculate the following: (1) α (2) CI: ID'

'(1) Given α=90°, AB=2, BC=3, find the sizes of the three angles of △ABC.\n(2) Given α=70°, β=γ, find the lengths of the three sides of △ABC.'

'In the given figure, find the value of α. Here, (1) states that BC = DC and (3) mentions that point O is the center of the circle.'

'Express in terms of trigonometric ratios of acute angles'

'Theorem 1: The intersection of the internal bisector of angle A of triangle ABC with side BC divides side BC in the ratio of AB:AC.'

'Let quadrilateral ABCD be inscribed in circle O, where AB=2, BC=3, CD=1, and ∠ABC=60°. Find:\n(1) The length of segment AC\n(2) The length of side AD\n(3) The radius R of circle O'

'In triangle ABC, with AB = AC = 5, BC = √5. Let point D be on side AC such that AD = 3, and let E be the intersection point (other than B) of the extension of side BC and the circumcircle of triangle ABD. Let P be the intersection point of AB and DE, then DP/EP = .'

"In mathematics, prove the following: In the diagram above, line AB touches the circles O and O' at points A and B respectively. If the radii are r and r' (r < r'), and the distance between the centers of the two circles is d, then prove that AB is equal to sqrt(d^2 - (r'-r)^2)."

"The area of a triangle can be calculated as half of the base multiplied by the height. Let's express this formula using trigonometry."

'There is a straight slope with a length of 125 meters. Climbing up this slope, the height increases by 21.7 meters. What is the approximate angle of inclination of this slope? Also, what is the horizontal distance of this slope in meters? Consider using trigonometric ratios.'

'Radius and Area of the Incircle of a Triangle'

'Answer the following question. Find the other elements of the triangle when a=√3+1, A=75°, C=60°, or when a=√3-1, A=15°, C=120°.'

'(2) When the circumcenter and incenter of triangle ABC coincide, let that point be O. Since O is the circumcenter, OB=OC. Therefore ∠OBC=∠OCB. Also, point O is the incenter of triangle ABC. \n\n[\nStarting equation set\n\\angle B=2\\angle OBC\n\n\\angle C=2\\angle OCB\nEnding equation set\n\\]\n\nLikewise, we can derive that\nincenter\n\n\\angle A=\\angle C\n\nTherefore, \\\angle A=\\angle B=\\angle C\\nThus, triangle ABC is an equilateral triangle.'

"From the edge of the roof of a 20m high building, looking down at a certain point, the angle formed with the horizontal plane is 32°. Find the distance between that point and the building. Also, find the distance between that point and the edge of the building's roof. Round to two decimal places."

'Use the theorem of angles in a circle to derive the interior angles of each triangle, and utilize the law of cosines and the law of sines.'

'In triangle ABC, given that a=√3, B=45°, and C=15°, find the following:\n(1) b\n(2) c\n(3) The value of cos 15°'

'Quadrilateral ABCD is inscribed in a circle with AB=4, BC=2, and DA=DC. Let E be the intersection point of the diagonals AC and BD, F be the point that divides segment AD in the ratio 2:3, and G be the intersection point of line FE and DC. (1) Find the ratios of AE:EC and GC:GD.'

'In triangle ABC, where AB=4, BC=5, and CA=6, let D and E be the points where angle A and its exterior angle bisector intersect line BC. Find the length of segment DE.'

'65 \\\\mathrm{AB}=2 r \\\\sin \\theta, \\\\mathrm{OH}=r \\\\cos \\theta'

'In triangle ABC, AB=AC=1, ∠ABC=72°. Point D is taken on side AC such that ∠ABD=∠CBD.\n(1) Find the measure of ∠BDC.\n(2) Find the length of side BC.\n(3) Find the value of cos 36°.'

'In triangle ABC, points D and E are the midpoints of sides BC and AC, respectively. Also, let AD and BE intersect at F, the midpoint of segment AF at G, and the intersection of CG and BE at H. (1) If BE=6, find the lengths of segments FE and FH. (2) Find the ratio of the areas of triangle EHC to triangle ABC.'

"■ Incenter ...... Intersection of triangle's internal angle bisectors\nAngle bisectors\nPoint P is on the angle bisector of ∠ ABC P is on the angle bisector of two lines ⇔ It is equidistant from BA and BC - In other words, The bisector of ∠ ABC is a set of points that are equidistant from the two lines BA and BC."

"Given two circles O and O' externally tangent at point A. If the tangent to circle O' at point B intersects circle O at two points C and D as shown in the diagram, prove that AB bisects the exterior angle of ∠CAD."

'In the figure on the right, assume both hypotenuses have a length of 1. Find the lengths of the remaining sides and fill in the blanks. Then, verify the values of sine, cosine, and tangent for 30, 45, 60 degrees.'

'Finding the length of the remaining side with 2 sides and 1 diagonal given'

'In △ABC, with the radius of the circumcircle being R, find the following: (1) When a=10, A=30°, B=45°, find C, b, R. (2) When b=3, B=60°, C=75°, find A, a, R. (3) When c=2, R=√2, find C.'

'In △ABC, where the radius of the circumcircle is R, find the following.'

"In triangle ABC where AB=5, BC=6, and CA=4, let's denote the intersection of the external bisectors of angles B and C as P. (1) Let PD be the perpendicular from P to line AB. Calculate the length of segment AD."

'In triangle ABC, let D be the point that divides side BC in the ratio 3:2, and let E be the point that divides side AB in the ratio 4:1. Let P be the intersection of line segments AD and CE, and let F be the intersection of line BF and side CA.'

'Cyclic Quadrilateral Theorem\nThe size of the cyclic angle corresponding to an arc is constant, being half of the central angle corresponding to that arc. In other words, in the figure on the right, $\\angle \\mathrm{APB}=\\frac{1}{2} \\angle \\mathrm{AOB}$ In particular, when $\\mathrm{AB}$ is a diameter, $\\angle \\mathrm{APB}=90^{\\circ}$\n\nConverse of the Cyclic Quadrilateral Theorem\nFor 4 points $\\mathrm{A}, \\mathrm{B}, \\mathrm{P}, \\mathrm{Q}$, if points $\\mathrm{P}, \\mathrm{Q}$ lie on the same side of line $\\mathrm{AB}$\n\n\\n\\angle \\mathrm{APB}=\\angle \\mathrm{AQB}\n\\n\nthen the 4 points $\\mathrm{A}, \\mathrm{B}, \\mathrm{P}, \\mathrm{Q}$ are on the same circle.'

'Chapter 6 Trigonometric Ratios - 115 TR 121'

'On the same horizontal plane, there are 3 points A, B, and C, with tower PC standing at C. Given that AB=80 m, ∠PAC=30°, ∠PAB=75°, and ∠PBA=60°. Find the height PC of the tower. It is okay to leave the answer in square root form.'

"There are three similar right-angled triangles ABC and A'B'C'. Since the ratios of the corresponding sides are equal, there are the following three equations regarding the ratios. Let's consider these three ratios: (1) BC/AB = B'C'/A'B', (2) AC/AB = A'C'/A'B', (3) BC/AC = B'C'/A'C'"

'Chapter 3 Properties of Geometric Figures - 195\n(2) Point E is the midpoint of side AC, so triangle ABC = 2 triangle EBC\nAlso, since BF:FE = 2:1, BE:FE = 3:1\n\n\triangle EBC = 3 triangle EFC\\]\nFurthermore, FH:HE = 2:1, so FE:HE = 3:1, thus triangle EFC = 3 triangle EHC\nTherefore\n\\[\egin{aligned}\ntriangle ABC & = 2 triangle EBC = 2 \\cdot 3 triangle EFC \\\\\n& = 6 triangle EFC = 6 \\cdot 3 triangle EHC \\\\\n& = 18 triangle EHC\n\\end{aligned}\\nConsequently triangle EHC: triangle ABC = 1:18\n— due to having a common height\n\ triangle ABC: triangle EBC = AC:EC \\n\ triangle EBC: triangle EFC = BE:FE \\n\ triangle EFC: triangle EHC = FE:HE \'

'In the diagram on the right, let ∠A = α, ∠B = β. Find the sine, cosine, and tangent values of α and β.'

'(1) As shown in the figure, for a regular pentagon and points A, B, H, when ∠AOB = 360° / 5 = 72°, r = 10, and θ = 1/2 × 72° = 36°, using the result from the previous question, the length of one side is\nAB = 2 × 10 × sin 36°\n= 20 × 0.5878\n= 11.756, rounding to AB = 11.8. The length of the perpendicular is OH = 10 × cos 36° = 10 × 0.8090 = 8.090, rounding to OH = 8.1.'

'In triangle ABC, find the following. Where the area of triangle ABC is denoted as S. 76 (1) When A=120°, c=8, S=14√3, find a, b (2) When b=3, c=2.0°<A<90°, S=√5, find sinA, a (3) When a=13, b=14, c=15, and the length of the perpendicular line from vertex A to side BC is denoted as h, find S, h'

'Prove that when three different lines x+y=1 (1), 4x+5y=1 (2), ax+by=1 intersect at one point, the three points (1,1), (4,5), (a,b) lie on the same line.'

'Find the locus of point P such that the ratio of its distances from points A(0,0) and B(5,0) is 2:3.'

'Find the coordinates of the point that divides the line segment AB externally in the ratio m:n, where A(x1, y1) and B(x2, y2) are two points on the coordinate plane.'

'Find the value of a when triangle ABC is an isosceles triangle.'

'(1) Since the slopes of the two lines are equal, the two lines are parallel.\n(2) From y=2x+4, y=-\\frac{1}{2}x+3, we can determine that the slopes of the two lines are 2 \\cdot\\left(-\\frac{1}{2}\\right)=-1, hence, the two lines are perpendicular.'

'What is a radian?'

'Find the locus of the point P that satisfies the following conditions: (1) The sum of the squares of the distances from points A(-4,0) and B(4,0) to point P is 36. (2) The ratio of distances from points A(0,0) and B(9,0) to point P is PA:PB=2:1. (3) The point P varies such that the triangle PAB with points A(3,0) and B(-1,0) as vertices satisfies PA:PB=3:1.'

'When point P is on the line x+y=5, find the coordinates of point P that minimizes the length of the broken line AP + PB connecting points A(2,5) and B(9,0).'

"The coordinates of the intersection point of two lines given by equations (1) ax + by + c = 0 and (2) a'x + b'y + c' = 0 are obtained as the solutions of the simultaneous equations (1) and (2)"

'When the point P satisfies the condition AP:BP = 2:3 and the line segment AB connects A(0,0) and B(5,0), find the locus of point P.'

'For shape A_{n+1}, focus on the rightmost column. Placing a tile horizontally in the bottom right corner results in three possible configurations as shown in Figure 3, where the remaining part matches A_{n}, and two possible configurations as shown in Figure 4, where the remaining part matches B_{n}.'

'Find the angle formed by 2 lines (1) Find the angle θ (0<θ<π/2) formed by the lines y=3x+1 and y=1/2x+2. (2) Find the slope of the line that forms an angle with y=2x-1 of π/4.'

'In (2) 0 < α < π/2, the radius representing angle α is equal to the radius representing 6α. Find the magnitude of angle α.'

'Given three points A(6,1), B(2,3), and C(a,b), find the values of a and b when triangle ABC is an equilateral triangle.'

'Prove that the centroid of triangle DEF coincides with the centroid of triangle ABC when points D, E, and F are taken on the sides BC, CA, and AB of triangle ABC respectively, such that BD:DC = CE:EA = AF:FB = 37. [Kinki University]'

'In triangle ABC, let D be a point on side BC such that it divides BC in the ratio 3:2. Prove that 3(2AB^2 + 3AC^2) = 5(3AD^2 + 2BD^2).'

'Symmetric points on a line'

'Prove that in triangle ABC, points P and Q divide side BC into three equal parts, such that BP=PQ=QC. Prove the following relationship holds: $$ 2AB^{2}+AC^{2}=3(AP^{2}+2BP^{2}) $$'

'In triangle ABC, with AB=15, BC=18, AC=12, find the intersection point D of the angle bisector of angle A and side BC. Determine the lengths of segments BD and AD.'

'Explain the Sine Rule and Cosine Rule, and solve an example problem.'

'Using the trigonometric table, answer the following questions: (1) In figure (A), find the values of x and y. Round to the nearest hundredth. (2) In figure (B), find the approximate size of acute angle θ.'

'By the Sine Rule and Cosine Rule'

'Please explain the relationship of corresponding angles when two lines are parallel.'

'When c=√6, find the angles of the triangle. The results obtained using the cosine rule are A=75°, C=60°.'

'Dropping perpendicular OI from vertex O to triangle DEG, we find that I is the center of the circumscribed circle of triangle DEG. Since GI is the radius of the circumscribed circle of triangle DEG, by the law of sines, we have GI=\ \\frac{1}{2 \\sin 60^\\circ} = \\frac{1}{\\sqrt{3}} \. Therefore, OG=\ \\frac{1}{2} \\mathrm{BG} = \\frac{\\sqrt{10+2 \\sqrt{5}}}{4} \. Please perform the following calculations.'

'Explain how to divide a polygon into triangles in order to calculate its area.'

'Practice At point A, which is at the same altitude as a certain tower, the angle of elevation to the top of the tower was measured to be 30 degrees. Furthermore, at point A, at a distance of 114m, there is point B where angle KAB is 75 degrees and angle KBA is 60 degrees. At this time, the distance between A and K is x meters, and the height of the tower is y meters.'

'In the quadrilateral ABCD inscribed in a circle, with AD // BC, AB=3, BC=5, and ∠ABC=60 degrees, find the following:\n(1) The length of AC\n(2) The length of CD\n(3) The length of AD\n(4) The area of the quadrilateral ABCD'

'By the sine rule, \ \\frac{a}{\\sin A}=2R \, hence \ \\frac{\\sqrt{2}}{\\sin A}=2 \\cdot 1 \, so \ \\sin A=\\frac{\\sqrt{2}}{2} \'

'In triangle ABC, when a=√2, b=2, and A=30°, find c, B, and C. Similar to basic example 154, when two sides and one angle of a triangle are given, the triangle may not be uniquely determined. First, formulate an equation for c using the cosine rule. As c will have two possible values, find B and C for each case. Refer to the discussion on the right page for an alternative solution using the sine rule.'

'In △ABC, let M be the midpoint of side BC.'

'Sine theorem\nIn \ \\triangle \\mathrm{ABC} \, let the radius of the circumscribed circle be \ R \, then\n\\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}=2 R\'

'On line segment AB of length 6, two points C and D are taken such that AC=BD. It is given that 0<AC<3. Find the minimum value of the sum S of the areas of three circles with diameters AC, CD, and DB, and the length of segment AC at that time.'

'From a location on the sea to a lighthouse standing on top of a cliff with a height of 30 meters, the zenith angle is 60 degrees, and from the same location to the zenith angle of the bottom of the lighthouse is 30 degrees, find the height of the cliff.'

'The area of triangle ABC is 12√6, and the ratio of its side lengths is AB:BC:CA=5:6:7. In this case, what is the value of sin∠ABC, denoted as , and what is the radius of the inscribed circle of triangle ABC, denoted as .'

'As shown in the figure, observing the points P and Q on the opposite bank of the river from points A and B 100 meters apart, the following values were obtained: ∠PAB=75°, ∠QAB=45°, ∠PBA=60°, ∠QBA=90°. Answer the following questions in this case.'

'Vertices B, E, G are all on the surface of the sphere S, and BG is the diameter of the sphere S, so triangle EBG is a right triangle with ∠BEG = 90°. Starting from EG = 1, perform the following calculations.'

'In triangle ABC, if ∠A = 60 degrees, AB = 7, AC = 5, then let D be the point where the bisector of ∠A intersects side BC. Find the length of AD.'

'From a location on the sea surface to the top of a lighthouse with a height of 30 meters, the angle of elevation to the tip is 60 degrees, and the angle of elevation to the bottom of the lighthouse is 30 degrees. Find the height of the cliff.'

'Please list three conditions for the congruence of triangles.'

'Determine the conditions for the triangle to exist.'

'In triangle ABC, what kind of triangle is it when the following equations hold true?'

'Point P starts from vertex B and moves towards vertex A at a speed of 1 unit per minute along edge AB. Also, point Q starts at the same time from vertex C and moves towards vertex B at a speed of 2 units per minute along edge BC. Find the minimum distance between points P and Q when the distance between them is minimized.'

'(1) c=\\sqrt{2}, A=105^{\\circ}, C=30^{\\circ} or c=\\sqrt{6}, A=75^{\\circ}, C=60^{\\circ}'

'A person with a height of 1.5 meters standing on flat ground wanted to know the height of a tree. The angle of elevation from point A to the top of the tree was 30°, and the angle of elevation from point B, which was 10 meters closer to the tree, was 45°. Calculate the height of the tree.'

'In triangle ABC, let R be the radius of the circumscribed circle. Find the following:\n(1) If a = 2, cos B = -1/3, and c = 4 cos B, then find b and cos A'

'Ratio of Sun and Moon Distance'

'In triangle ABC, when a=1+√3, b=2, C=60°, find:\n(1) The length of side AB\n(2) The measure of ∠B\n(3) The area of △ABC\n(4) The radius of the circumcircle\n(5) The radius of the incircle'

'In ancient Greece, the study of trigonometry advanced along with astronomy. The ancient Greek astronomer Aristarchus used the following relationship to seek the approximate distance ratio between the Sun and the Moon.'

'In triangle ABC with ∠B=60° and AB+BC=1, let M be the midpoint of side BC. The length of segment AM is minimized when BC=.'

'Consider a scalene triangle ABC, with the longest side being BC and the shortest side being AB, where AB=c, BC=a, CA=b (a≥b≥c). Let the area of triangle ABC be denoted as S.'

'Translate the given question into multiple languages.'

'Sine Rule and Cosine Rule'

'When m>0, n>0, point P lies on segment AB, and AP: PB=m: n, point P is said to internally divide segment AB in the ratio m: n [For more details, refer to Mathematics A]. Let AB=k, representing the length of the other side as k. Utilize the similarity of triangles formed by the diagonals of an inscribed quadrilateral.'

'In triangle ABC, let S represent the area. Find the following. It is assumed that triangle (2) is not an obtuse triangle.'

'Let the areas of triangles AID, BEF, and CGH be denoted as T1, T2, and T3, respectively. In this case, which of the following options fits in place of S?'

'In triangle ABC, let R be the radius of the circumscribed circle. When A=30°, B=105°, and a=5, find the values of R and c.'

'Prove the following equations in triangle ABC'

'In the quadrilateral ABCD inscribed in a circle with DA=2AB, ∠BAD=120°, (1) BD = square root of 3 times AB, AE = AB, (3) AB:BC:CD:DA=1:square root of 3:2, (4) If the radius of the circle is 1, then AB = square root of 3 and the area of the quadrilateral ABCD is S=3.'

'As shown in the figure on the right, draw squares ADEB, BFGC, and CHIA with sides AB, BC, and CA as one side of each square, and then connect points E and F, G and H, I and D.'

'Calculate the gradient of this railway line. The gradient of the railway line is 18%, and when you move 1000m horizontally, the elevation increases by 18m. Calculate the slope angle θ using trigonometry.'

'angle between two lines'

'Basic question 124 Maximum angle of a triangle'

'Please explain the difference between the sine rule and cosine rule.'

'Given \2 \\sin \\theta = \\sqrt{2}\, we can find that \\\sin \\theta = \\frac{1}{\\sqrt{2}}\. On the circumference of radius 1, the points where the \y\ coordinate is \\\frac{1}{\\sqrt{2}}\ are \\\mathrm{P}\ and \\\mathrm{Q}\. Therefore, the desired \\\theta\ corresponds to \\\angle \\mathrm{AOP}\ and \\\angle \\mathrm{AOQ}\.'

'Geometry and Measurement\n157\nEX 394\n(1) Using the diagram on the right, find the value of \ \\sin 18^{\\circ} \. (2) Using the diagram on the right, find the values of \ \\sin 22.5^{\\circ}, \\cos 22.5^{\\circ} \, and \ \\tan 22.5^{\\circ} \.\nHINT: To find the trigonometric ratios of special angles, you can create a right triangle that includes that angle.'

'Problem 218 Basic Example 136 Radius of Circumcircle and Incircle of a Triangle\nIn △ABC, where AB=6, BC=7, CA=5, find the radius R of the circumcircle and the radius r of the incircle.'

'In triangle ABD, according to the sine theorem, BD/sin120° = 2 × 1, hence BD = 2 sin120° = √3. On the other hand, BD = √7 × AB, therefore √7AB = √3, so AB = √3/√7 = √21/7. Therefore, S = triangle ABD + triangle CBD = 1/2 × k × 2k sin120° + 1/2 × 3k × 2k sin(180°-120°) = √3/2 × k² + 3√3/2 ×k² = 2√3 k² = 2√3 AB² = 2√3 (√3/√7)² = 7√3/7, the radius of the circumscribed circle R = 1, angle BCD = 180°-angle BAD = 180°-120° = 60°, and the sum of the diagonals of the inscribed quadrilateral is 180°.'

'Basics of Trigonometry'

'In a quadrilateral ABCD, let the lengths of the diagonals AC and BD be p and q, and let one of the angles formed by the diagonals be θ. Express the area S of the quadrilateral ABCD in terms of p, q, and θ.'

'Mathematics I'

'Prove that the equality cos (A+B)/2 = sin (C/2) holds true when the sizes of angles A, B, and C of triangle ABC are represented as A, B, and C respectively.'

'Which one to use, sine rule or cosine rule?'

'In triangle ABC, let R be the radius of the circumcircle. If A=30°, B=105°, and a=5, find the values of R and c.'

'In triangle ABC with AB=6, BC=4, CA=5, let D be the point where the angle bisector of angle B intersects side AC. Find the length of segment BD.'

'In triangle ABC, where AB = 3, AC = 2, and ∠BAC = 60°, let D be the intersection point between the angle bisector of ∠A and BC. Find the length of segment AD.'

'126 Surveying Problem (Plane) (1) (1) (0) From two points A and B 100 meters apart, measurements were taken to identify two points P and Q on the opposite bank of a river, with values obtained as shown in the figure. (1) Find the distance between A and P. (2) Find the distance between P and Q. Basics 107, 120, 121 Distances and direction (segments and angles) can be considered as sides and angles of triangles. Consider which triangle in the diagram to focus on, and think about when to apply the sine rule or cosine rule.'

'In triangle ABC, if sin A: sin B: sin C = 3: 5: 7, find the ratio of cos A: cos B: cos C.'

'Basic Example 133 Length of angle bisector in triangle (2)'

'Find the equations of the lines and parabolas obtained by moving the following line and parabola parallel to the x-axis by -3 units and the y-axis by 1 unit.\n(A) Line: y=2x-3\n(B) Parabola: y=-x^{2}+x-2'

'Basic Example 106 Right Triangle and Trigonometric Ratios\nIn triangle ABC as shown in the figure, find the following:\n(1) The values of sinθ, cosθ, tanθ\n(2) The lengths of segments AD and CD'

'(2) In triangle ABC, by the cosine rule'

'Measurements were taken from points A and B, which are 50 meters apart, to points P and Q on the opposite bank of the river, yielding the values as shown in the diagram. Calculate the distance between points P and Q.'

'In triangle ABC, if 7/sin A=5/sin B=3/sin C holds true, find the measure of the largest angle in triangle ABC.'

'Length of the angle bisector in a triangle'

'In an isosceles triangle ABC where angle A is 36 degrees and BC = 1, the intersection of the bisector of angle C and side AB is called D.\n(1) Find the lengths of segments DB and AC.\n(2) Find the lengths of segments DB and AC again. Using the result from (1), determine the value of cos 36 degrees.\n[Source: Kobe Gakuin University]\nBasic 106'

'Example Problem 140 Minimum Area of a Triangle\nA equilateral triangle ABC with side length 2 is given. Points D on side AB and E on side CA are such that AD = CE. Let the area of quadrilateral DBCE be S.\n(1) Find the minimum length of segment DE and the length of segment AD at that point.\n(2) Find the minimum value of S and the length of segment AD at that point.\nGiven basics 66, 121, 131'

'Take point E on side BC such that AB // DE, then quadrilateral ABED is a parallelogram.'

'What is the shape of triangle ABC that satisfies the following equations:\n(1) b*sin^2(A) + a*cos^2(B) = a\n(2) a/cos(A) = b/cos(B) = c/cos(C)?'

'Use the sine rule to find the largest angle of triangle ABC. The given conditions are as follows: sin A : sin B : sin C = 5 : 16 : 19.'

"The word used to describe the slope of roads and railways is gradient. Using the 'table of trigonometric ratios', answer the following question. (1) The gradient of a road is often expressed in percentage (%). Percentage represents how many meters the elevation increases when 100 meters are traveled horizontally. On a certain road, there is a sign indicating 23%. Approximately how many degrees is the slope of this road?"

'(1) In triangle ABC, if the angle A bisector intersects side BC at point D, prove that BD:DC = AB:AC.'

'(3) As shown in the figure, when a regular dodecagon is divided into 12 congruent triangles\n\\n\\mathrm{AB} = 1, \\quad \\angle \\mathrm{AOB} = 360^{\\circ} \\div 12 = 30^{\\circ}\n\'

'Find the lengths of the remaining sides and the sizes of the angles of \ \\triangle \\mathrm{ABC} \ in each of the following cases: (1) \ A=60^{\\circ}, B=45^{\\circ}, b=\\sqrt{2} \ (2) \ a=\\sqrt{2}, b=\\sqrt{3}-1, C=135^{\\circ} \'

'(2) In triangle ABC, with BC = 6, CA = 5, and AB = 7, let D be the intersection point of the angle bisector of ∠A and side BC. Using (1), find the length of segment AD.'

'Let D be the point that divides side AB of △ABC in the ratio 1:2 internally, E be the point that divides side AC in the ratio 2:1 internally, and F be the point that divides side BC in the ratio t:(1-t). Here, t is a real number satisfying 0<t<1.'

'In tetrahedron ABCD, let points P, Q, R, S be the points that internally divide edges AB, CB, CD, AD in the ratio t:(1-t) [0<t<1].'

'On the TR coordinate plane, when the endpoints A and B of a segment AB of length 6 move along the y-axis and x-axis respectively, the trajectory of point P which divides the segment AB in a 3:1 ratio is to be determined.'

'Explain the following curves:\n(1) Ellipse $\\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1$ shifted parallel to the $x$-axis by 2 units and the $y$-axis by -3 units; center at point (2, -3); foci at two points (2+√5, -3), (2-√5, -3)\n(2) Hyperbola $\\frac{x^{2}}{4}-\\frac{y^{2}}{25}=1$ shifted parallel to the $x$-axis by -2 units and the $y$-axis by -3 units; vertices at two points (0, -3), (-4, -3); foci at two points (√29-2, -3), (-√29-2, -3); asymptotes are two lines $5x+2y+16=0$, $5x-2y+4=0\n(3) Parabola$y^{2}=4x$shifted parallel to the$x$-axis by -2 units and the$y$-axis by 1 unit; vertex at point (-2, 1), focus at point (-1, 1); directrix is the line$x=-3$'

'Polar coordinates and polar equations'

'(4) In the coordinate plane, let the curve represented by the polar equation $r=2 \\cos \\theta$ be denoted as $C$ and let the points on $C$ with polar coordinates $\\left(\\sqrt{2}, \\frac{\\pi}{4}\\right)$ and $(2,0)$ be denoted as $\\mathrm{A}$ and $\\mathrm{B}$, respectively. Also, let $\\ell$ be the line passing through $\\mathrm{A}$ and $\\mathrm{B}$, and let $D$ be the circle centered at $A$ with radius equal to the length of segment $\\mathrm{AB}$.\n(1) Find the polar equation of the line $\\ell$.\n(2) Find the polar equation of the circle $D$.'

'Prove that the midpoints of the diagonals AG and BH of parallelogram ABCD-EFGH coincide.'

'Trajectory of points with a constant ratio of distances from a point and a line'

'In triangle OAB, let point D internally divide side AB in the ratio 2:1, point E be the image of point D under symmetry about line OA, and point F be the intersection of the perpendicular from point B to line OA. Let vector OA be a and vector OB be b such that |a|=4 and a⋅b=6.'

'A circle with the midpoint of side BC as the center and passing through point A.'

'(2) Prove that \ \\overrightarrow{\\mathrm{GU}} \ is perpendicular to the plane QTV.'

'Mathematics C Practice 108 Point Q lies on the circumference of a circle with diameter OP, so ∠OQP=π/2 and PQ=1. Therefore, △OPQ=1/2 OQ・PQ=1/2 OQ, which implies that we only need to consider the maximum length of the segment OQ. However, point Q lies on an ellipse with the major axis on the y-axis, and the length of the segment OQ decreases monotonically with respect to s, where 0 ≤ a ≤ s. Therefore, when a=0, i.e., point P lies on the y-axis, the length of segment OQ is maximum.'

'Try to prove the following properties of the shape using the complex number plane.\nFor the quadrilateral ABCD\n(1) AB·CD+AD·BC≥AC·BD holds.\n(2) Equality holds in (1) when the quadrilateral ABCD is inscribed in a circle.'

'(1) Find the angle θ formed by the two planes α and β. Note that 0° ≤ θ ≤ 90°.'

'Using the complex plane, prove the following theorems: (1) In triangle ABC, let the midpoints of sides AB and AC be D and E, respectively. Then, BC // DE and BC=2DE (Midpoint Theorem). (2) In triangle ABC, let M be the midpoint of side BC. Then, the equation AB^2+AC^2=2(AM^2+BM^2) holds true (Midline Theorem).'

'In space, there are four points O, A, B, C not on the same plane. Let s and t be real numbers satisfying 0<s<1,0<t<1. Let A0 be the point that divides the line segment OA in a 1:1 ratio, B0 be the point that divides the line segment OB in a 1:2 ratio, P be the point that divides the line segment AC in an s:(1-s) ratio, and Q be the point that divides the line segment BC in a t:(1-t) ratio. Furthermore, assume that the four points A0, B0, P, Q lie on the same plane. (1) Express t in terms of s. (2) Given |OA|=1, |OB|=|OC|=2, ∠AOB=120°, ∠BOC=90°, ∠COA=60°, and ∠POQ=90°, find the value of s.'

'Example 36 Minimum length of a broken line (space)\nIn coordinate space, consider points A(1,0,2), B(0,1,1).\n(1) When point P moves on the xy plane, find the minimum value of AP+PB.\n(2) When point Q moves on the x-axis, find the minimum value of AQ+QB.'

'In the parallelogram ABCD, point E divides side AB in the ratio 3:2, point F divides side BC in the ratio 1:2, and the midpoint of side CD is M. Let P be the intersection of segments CE and FM, and let Q be the intersection of line AP and diagonal BD. If vector AB is represented as a and vector AD is represented as b, express vectors (1) AP and (2) AQ in terms of a and b.'

'Example 23 Positional relationship of centroid, circumcenter, and orthocenter of a triangle\nLet the centroid of triangle ABC be G and the circumcenter be E, prove the following:\n[Yamanashi University]\n1. Vector GA + Vector GB + Vector GC = Vector 0\n2. Vector EA + Vector EB + Vector EC = Vector EH, let H be the orthocenter of triangle ABC.\n3. The three points E, G, and H are collinear and EG:GH = 1:2'

'Prove that when a line passing through the centroid G of triangle ABC intersects the sides AB and AC at points 25D and E respectively, where point D is different from points A and B, and point E is different from points A and C, then DB/AD + EC/AE = 1.'

'Example 106 Dilations of Ellipse and Circle'

'Find the coordinates of point R, which is equidistant from points O(0,0,0), F(0,2,0), G(-1,1,2), and H(0,1,3).'

'Find the polar equation of a line with an angle α with the initial line.'

'Example 132: Using parametric representation to find the minimum area of the triangle formed by the tangent of ellipse and coordinate axes'

'Prove the conditions when triangle ABC is an isosceles triangle with AB=BC.'

'In triangle ABC, let points D, E, and F divide sides AB, BC, and CA internally in the ratios of m:n, respectively. For any pair of natural numbers (m, n), if AE is perpendicular to DF, what kind of triangle is ABC?'

'(3) The figure formed by the line segment AP passing through is the black region in the right figure, including the boundary line. Here, G and H are the points of intersection of the two tangent lines drawn from point A to circle K. cos ∠AEH = EH / AE = a / 2a = 1/2, 0 < ∠AEH < π, therefore ∠AEH = π / 3. Also, ∠AEH = ∠AEG, so ∠GEH = 2/3π. Thus, the area S of the figure formed by the line segment AP passing through is S = 2 * △AEH + (area of circle K) - (area of sector EGH) = 2 * (1/2) * a * sqrt(3)a + πa^2 - (1/2) a^2 * (2/3)π = sqrt(3)a^2 + (2/3)πa^2.'

'These two tangents pass through the point P(x_{0}, y_{0})'

'The equation of tangent at point P(x1, y1) is (x1 x)/a^2 - (y1 y)/b^2 = 1 (x1 > a), and x1^2/a^2 - y1^2/b^2 = 1. When x=a, with y1 ≠ 0, we get y = b^2(x1 - a)/(a y1). When x=-a, with y1 ≠ 0, we get y = -b^2(x1 + a)/(a y1). Thus, Q(a, b^2(x1 - a)/(a y1)), R(-a, -b^2(x1 + a)/(a y1)). Therefore, the center of the circle C1 with diameter QR is (0, -b^2/y1), and if the radius is r, then r^2 = a^2 + (b^2 x1/a y1)^2 = a^2 + (b^4 x1^2)/(a^2 y1^2) = a^2 + b^2 + b^4/(y1^2). Hence, the equation of circle C1 is x^2 + (y + b^2/y1)^2 = a^2 + b^2 + b^4/(y1^2).'

'Since $\\mathrm{FP}: \\mathrm{PH}=e: 1$, then $\\mathrm{FP}^{2}=e^{2} \\mathrm{PH}^{2}$, thus'

'In triangle ABC, let point L divide side AB in the ratio 2:1 and let M be the midpoint of side AC. Let P be the intersection of segments CL and BM, and let N be the intersection of line AP and side BC. Express vector AP and AN in terms of vectors AB and AC. Furthermore, find the ratio AP:AN.'

'Find the polar coordinates of point Q.'

'If quadrilateral ABDC is a parallelogram, find the values of a, b, c from vector AB = CD.'

'Find the acute angle formed by the following two lines.'

'(1) In triangle ABC where AB=8, BC=7, and CA=5, let I be the incenter. Express vector AI in terms of vectors AB and AC.'

'Point Q moves on the circumference with radius 5 centered at point O, and point P moves on the circumference with radius 1 centered at point Q. At time t, the angles that OQ and QP make with the positive direction of the x-axis are t and 15t respectively. If the angle that OP makes with the positive direction of the x-axis is ω, find dω/dt.'

'In a regular tetrahedron ABCD with edge length 2, find the dot product of vector AB and vector AC.'

'Find the equation of the line that bisects the area of triangle ABC with vertices A(20,24), B(-4,-3), and C(10,4) and passes through the point P that divides side BC in the ratio 2:5.'

'Internal division point and external division point\nThe coordinates of the point that divides the line segment AB in the ratio m:n are\nInternal division ... ((nx_{1}+mx_{2})/(m+n), (ny_{1}+my_{2})/(m+n))\nExternal division ... ((-nx_{1}+mx_{2})/(m-n), (-ny_{1}+my_{2})/(m-n))'

'On the coordinate plane, the parabolas C₁: y=-p(x-1)²+q and C₂: y=2x² are tangents to the same line at the point (t, 2t²). Here, p and q are positive real numbers, and t lies in the range 0 < t < 1.'

'Find the coordinates of a point P that is equidistant from points A(3,3), B(-4,4), and C(-1,5).'

'Translate the given text into multiple languages.'

'Find the coordinates of the point P on the y-axis equidistant from points A(3,-4) and B(8,6).'

'What is the meaning of the given problem?'

'(2) In triangle ABC, let D be the point that divides side BC in the ratio 1:3. Prove that the equation 3AB^{2}+AC^{2}=4AD^{2}+12BD^{2} holds.'

'For a point P(x, y) on the xy-plane other than the origin O, let point Q satisfy the following conditions: (A) Q lies on the half-line OP with O as the starting point. (B) The product of the lengths of line segments OP and OQ is 1. (1) Express the coordinates of Q in terms of x and y. (2) Determine the locus of Q as point P moves around the circle with equation (x-1)^{2}+(y-1)^{2}=2, excluding the origin. (3) Determine the locus of Q as point P moves around the circle with equation (x-1)^{2}+(y-1)^{2}=4.'

'Find the coordinates of a point P that is equidistant from points A(3,3), B(-4,4), and C(-1,5).'

'Given three distinct points A, B, and C on the circumference of a circle with center O and radius 1 in the plane. Prove that the radius r of the inscribed circle of triangle ABC is less than or equal to 1/2.'

'In the xy-plane of mathematics, there are two points P and Q on a semi-line with origin O as the starting point, satisfying OP · OQ = 4. When point P moves along the curve (x-2)² + (y-3)² = 13, (x, y)≠(0,0) excluding the origin, find the trajectory of point Q.'

'Find the locus of points that are at a distance ratio of 2:1 from points A(-4,0) and B(2,0).'

'Illustrate the radii of the following angles. Also, identify the quadrant in which they lie.'

'Angle α is such that 0<α<π/2 and the radius representing α coincides with the radius representing 6α. Find the magnitude of angle α.'

'(5) Find the locus of points where the angle subtended at fixed points A and B is constant angle α.'

'In triangle ABC, let the lengths of sides BC, CA, and AB be a, b, c respectively. If triangle ABC is inscribed in a circle of radius 1 and ∠A = π/3, find the maximum value of a + b + c.'

'Sine curve appearing in the shape formed by cutting a cylinder'

'Find the point dividing the line segment AB connecting two points A(x1, y1) and B(x2, y2) into m:n internally.'

'Translate the given text into multiple languages.'

'Prove that in triangle ABC, where the sizes of angles A and B are α and β respectively, and the lengths of their opposite sides are denoted as a and b, the inequality b^2/a^2 < (1-cos β)/(1-cos α) < β^2/α^2 holds when 0 < α < β < π.'

'For the polar coordinates centered at O, there are 2 points A(4, -π/3) and B(3, π/3). Find: (1) the length of segment AB, (2) the area of triangle OAB.'

'Consider Mathematics III\nAlso, consider the conic curve when 𝑡 value is the solution. Prove that it is a hyperbola or ellipse, and find the coordinates of the foci.'

'Based on the chart editing policy, please solve the following problem:\n2. Find the length of the hypotenuse of a right triangle. (Using the Pythagorean theorem)\nProblem: Find the length of the hypotenuse of a right triangle with side lengths of 3 cm and 4 cm.'

'Translate the given question into multiple languages.'

'Express X, Y in terms of x, y, and θ when point P(X, Y) is rotated around the origin O by angle θ to get point Q(x, y).'

'Regarding polar coordinates, find the polar equations of the following circle and line. Assume that a>0.'

'Let the polar coordinates of points A, B, C, and D be (r₁, θ+π/6), (r₂, θ), (r₃, θ), and (r₄, θ+π/3) respectively. Triangle ABC is an isosceles triangle with AB=AC, and triangle DBC is an isosceles triangle with DB=DC.'

'For a triangle $\\triangle ABC$ with side length 2 and $\\cos A=\\frac{7}{9}$, let the length of side $AB$ be $x(0<x<1)$ and the area of $\\triangle ABC$ be $S$. [Similar to Aichi University of Education] (1) Express $S$ in terms of $x$. (2) Find the maximum value of $S$. Also, determine the lengths of the three sides of $\\triangle ABC$.'

'Please indicate the condition for points A(α), B(β), C(γ), D(δ) to be such that AB and CD are perpendicular.'

'In the isosceles trapezoid ABCD with AD // BC, where AB=2 cm, BC=4 cm, and ∠B=60°. If ∠B is increased by 1°, by how much will the area S of trapezoid ABCD increase? Assume π=3.14.'

'In polar coordinates, find the polar equation of the locus of points P where the distance ratio from the pole O and the line g is constant, passing through point A(3, π) and perpendicular to the initial line.'

'Through the point O, find the polar equation of a line that forms an angle with the initial line and α.'

'Conditions for a quadrilateral to be inscribed in a circle'

'In the complex plane, let three points O(0), A(α), B(β) form a triangle OAB, where ∠AOB = π/6, and OA/OB = 1/√3. Then, α^(2)-1 α β+β^(2)=0 holds true.'

'Given that triangle ABC with vertices A(-1), B(1), C(√3i) forms an equilateral triangle and triangle PQR with vertices P(α), Q(β), R(γ) forms an equilateral triangle. Prove that the equation α²+β²+γ²-αβ-βγ-γα=0 holds true.'

'Determine the value of a so that lines AB and AC are perpendicular.'

'As shown on the right, when OP1=1, and P1P2=½OP1, P2P3=½P1P2, ... continue indefinitely, what point do points P1, P2, P3, ... approach infinitely close to?'

'In triangle OAB, let point D divide side AB in the ratio 2:1, point E be the symmetric point of point D with respect to line OA, and point F be the intersection of the perpendicular from point B to line OA and line OA. Let vector OA=a, vector OB=b, with |a|=4 and a∙b=6. (1) Express vector OF using vector a. (2) Express vector OE using vectors a and b.'

'On a plane, there is a triangle OAB with OA=8, OB=7, AB=9 and a point P, where OP=wOA+tOB is expressed (w, t are real numbers).'

'Prove that points A, B, and C satisfy AB ⊥ AC.'

'For the existence range of points on the plane within triangle OAB, if \ \\overrightarrow{OP} = s\\overrightarrow{OA} + t\\overrightarrow{OB} \, then the range for the point P is'

'When the length of line segment AB is 8, point A lies on the x-axis, and point B moves along the y-axis, find the locus of point P that divides line segment AB in a ratio of 3:5.'

'(2) In the regular hexagon ABCDEF, express vector FB in terms of vector AB and vector AC.'

'There is a regular pentagon with side length 1 on the plane, and its vertices are sequentially A, B, C, D, E. Answer the following questions:\n(1) Prove that edge BC is parallel to line segment AD.\n(2) Let the intersection of line segments AC and BD be F. Describe the shape of quadrilateral AFDE and provide its name and reasoning.\n(3) Find the ratio of the lengths of line segments AF and CF.\n(4) If vector AB=a and vector BC=b, express vector CD in terms of vectors a and b.'

'Find the distance between points A and B when the coordinates of point A are (3, π/4) and the coordinates of point B are (4, 3π/4) in polar coordinates.'

'In triangle ABC, let D, E, and F be the points that internally divide the sides AB, BC, and CA in the ratios of m : n, respectively. What kind of triangle is triangle ABC if for any pair of natural numbers (m, n), AE is perpendicular to DF?'

'In an equilateral triangle ABC with side length a, let P1 be the foot of the perpendicular from vertex A to side BC, Q1 be the foot of the perpendicular from P1 to side AB, R1 be the foot of the perpendicular from Q1 to side CA, and P2 be the foot of the perpendicular from R1 to side BC. By repeating this procedure, points P1, P2, ..., Pn are determined on side BC. Find the point towards which Pn approaches.'

'What kind of figure does the point w represent for the following expressions:\n(1) When the point z moves on a circle with radius 1 centered at the origin O, w=3-iz\n(2) When the point z moves on a circle with radius 1 centered at 1-sqrt{3}i, w=(2+2√3i)z'

'In triangle ABC, there is a point P inside. Let Q be the intersection of AP and side BC, such that BQ:QC=1:2, and 24AP:PQ=3:4. Prove that the equation 4PA+2PB+PC=0 holds.'

'Assume the perimeter of triangle ABC is 36 and the radius of the inscribed circle in triangle ABC is 3. Find the area of triangle QBC when point Q satisfies the condition 6→AQ+3→BQ+2→CQ=→0.'

'In triangle OAB, let the point dividing side AB in the ratio 2:1 be D, let the point symmetric to point D about line OA be E, and let the intersection of the perpendicular from point B to line OA with line OA be F. Let →OA=a, →OB=b, |a|=4, a⋅b=6. (1) Express →OF in terms of vector a. (2) Express →OE in terms of vectors a and b.'

'In the quadrilateral ABCD with AD // BC and BC=2AD, prove (1) that points P and Q lie on the line AB. (2) Show that points P, Q, and D are collinear.'

'Point Q divides side AC of triangle ABC internally in the ratio 1:2, and point P divides side BC in the ratio m:n (m>0, n>0). Let R be the intersection point of segments AP and BQ. A line passing through point R intersects sides AB and AC at points D and E respectively. Also, let vec{b}=→AB and vec{c}=→AC.\n(1) Express vector AR in terms of m, n, vec{b}, and vec{c}.\n(2) Let k=AB/AD+AC/AE. Show the relationship between m and n such that k is constant regardless of the position of point D on segment AB, and find the value of k at that time.'

'Prove that the incenter P(z) of triangle OAB with vertices O(0), A(α), and B(β) satisfies the equation z=|β|α+|α|β/|α|+|β|+|β-α|.'

'Pass through the pole O and find the polar equation of the line that forms an angle α with the initial line.'

'When the point z moves on the following figure, what kind of figure does the point w represented by w=(-√3+i) z+1+i draw? (1) A circle with a radius of 1/2 centered at -1+√3i (2) The perpendicular bisector of the line segment connecting the 2 points 2,1+√3i'

'Prove that the product of the lengths of the segments PQ and PR is constant when perpendiculars PQ and PR are drawn from any point P on a hyperbola to the two asymptotes.'

'In quadrilateral ABCD, where AD // BC and BC = 2AD. Answer the following questions when points P and Q satisfy the conditions.'

'Prove that the incenter of the triangle OAB with different points O(0), A(α), B(β) as vertices is P(z), where z satisfies the equation z = (|β|α + |α|β) / (|α| + |β| + |β-α|).'

'In triangle OAB, let point C be the point which divides the side OA in the ratio 2:1, and point D be the point which divides segment BC in the ratio 1:2. Let E be the intersection point of line OD and side AB. Express the following vectors in terms of vector OA and vector OB.'

'In the parallelogram ABCD, let E be the point dividing side AB in the ratio 3:2, F be the point dividing side BC in the ratio 1:2, and M be the midpoint of side CD. Let P be the intersection of line CE and line FM, and Q be the intersection of line AP and diagonal BD. If vector AB=a, and vector AD=b, express vector (1) AP and (2) AQ in terms of a and b.'

'(4) When point E and point F are equal, since point E is the centroid of triangle ABC, line AE passes through the midpoint of side BC. Also, from (2), AE is perpendicular to BC. Therefore, line AE is the perpendicular bisector of side BC. Hence, triangle ABC is an isosceles triangle with AB = AC. Therefore, AB: AC = 1:1'

'(3) Since $\\ sqrt {3} \\ cos \\ frac {\\ pi} {6} = \\ frac {3} {2}, \\ sqrt {3} \\ sin \\ frac {\\ pi} {6} = \\ frac {\\ sqrt {3}} {2}$, the coordinates of point A are $\\ left (\\ frac {3} {2}, \\ frac {\\ sqrt {3}} {2} \\ right)$, therefore, the slope of line OA is $\\ frac {1} {\\ sqrt {3}}$, so the slope of the required line is $- \\ sqrt {3}$. Therefore, the equation is\n$[y- \\ frac {\\ sqrt {3}} {2} = - \\ sqrt {3} \\ left (x- \\ frac {3} {2} \\ right)]$\nthat is $\\ sqrt {3} x +y = 2 \\ sqrt {3}$\nSubstituting $x = r \\ cos \\ theta, y = r \\ sin \\ theta$\n\n$[\\ sqrt {3} r \\ cos \\ theta + r \\ sin \\ theta = 2 \\ sqrt {3}]$'

'In triangle OAB, let point C divide side OA in the ratio 2:3, and point D divide side OB in the ratio 4:5. The intersection of segments AD and BC is point P, and the intersection of line OP and side AB is point Q. If \\\overrightarrow{OA}=\\vec{a}\ and \\\overrightarrow{OB}=\\vec{b}\, express \\\overrightarrow{OP}\ and \\\overrightarrow{OQ}\ in terms of \\\vec{a}\ and \\\vec{b}\. [Sim. Kinki Univ.]'

'Let \ a>0 \. Consider the curve \ K \ represented by the polar equation \\( r=a(1+\\cos \\theta) (0 \\leqq \\theta<2 \\pi) \\) (cardioid). Answer the following questions.'

'When a line passing through the centroid G of triangle ABC intersects sides AB and AC at points 25D and E respectively, where D is different from points A and B, and E is different from points A and C, prove that DB/AD + EC/AE = 1.'

'(4) For plane PQR and edge OD, the following holds. When q=1/4, plane PQR is? When q=1/5, plane PQR is? When q=1/6, plane PQR is?'

'(1) Prove: In tetrahedron OABC, for t satisfying 0<t<1, let points K, L, M, N be the points where edges OB, OC, AB, AC are internally divided in the ratio 43t:(1-t). Prove that quadrilateral KLNM is a parallelogram.'

'Component representation of dot product'

'Solving for y in (1), we get y = ±(b/a)√(x² - a²), so y = ±(b/a)x√(1 - a²/x²). As x approaches infinity, y approaches ±(b/a)x. The same is true when x is negative and its absolute value approaches infinity. Therefore, the two lines y = (b/a)x and y = -(b/a)x are the asymptotes of hyperbola (1) (the lines that a curve approaches as it gets close). These asymptotes are also the two lines represented by (x/a - y/b)(x/a + y/b) = 0 with 1 on the right side of (1) replaced by 0.'

'Find the condition to show that AB is parallel to CD.'

'Similarities and Differences between Plane and Space 2'

'Find the conditions for the quadrilateral PQSR to be a parallelogram.'

'Find the vertex, focus, and asymptotes of the following hyperbolas. Also, sketch their general shape.\n(1) \ \\frac{x^{2}}{4} - \\frac{y^{2}}{4} = 1 \\n(2) \ 25 x^{2} - 9 y^{2} = -225 \'

'Consider an equilateral triangle ABC with side length 1 on the plane. For a point P, let the vector v(P) be given by v(P)=→PA−3→PB+2→PC.'

'The following 4-7 are basic facts about an ellipse.'

"(2) Let's denote ∆ABC as S. Then"

'In the complex plane, let A(0), B(β), C(γ). Find the complex numbers representing points E and G.'

'For the points A(1,2), B(2,3), C(-1,2), find the equation of the line passing through point A and perpendicular to BC. Find the acute angle α formed by the lines x-2y+3=0 and 6x-2y-5=0.'

"As k varies from 1 to 2, the line segment A'B' moves parallel to the line segment AB to CD as shown in the diagram."

'In the trapezoid ABCD shown to the right, where AD=a and BC=b. Let E be a point dividing AB in the ratio m, and let F be the intersection of CD with the line through E parallel to AD, then EF=(na+mb)/(m+n) holds.'

'What curve will result from shrinking or enlarging the circle x^2 + y^2 = 4 in the following ways?\n(1) Shrinking by a factor of 1/2 along the y-axis\n(2) Enlarging by a factor of 3 along the x-axis'

'Let O be the centroid of triangle ABC. A line l passing through point O but not through vertex A intersects sides AB and AC at points P and Q respectively. Let S be the area of triangle ABC and T be the area of triangle APQ. Determine the equation of line l that minimizes T/S, and find the minimum value of T/S.'

'(1) Circle with radius 1 centered at the point dividing segment AB in 2:3 ratio\n(2) Circle with diameter AD where point D divides side BC in 3:2 ratio'

'Find the acute angle \ \\alpha \ formed by the two lines \ x-2 y+3=0,6 x-2 y-5=0 \.'

'Inside the right triangle ABC0 with an interior angle of 90 degrees, an infinite series of squares B0B1C1D1, B1C2D2, and so on are being constructed. Let the length of one side of the nth square Bn-1BnCnDn be an, and let its area be Sn. For each natural number k greater than 1, ak=ralpha(k-1) holds true, where a0=1.'

'Prove that in trapezoid ABCD, AD // BC and AD:BC = 1:2.'

'Points P and Q are on sides OA and OB of an equilateral triangle OAB with a side length of 1. When the area of triangle OPQ is exactly half of the area of triangle OAB, find the range of possible values for the length of PQ.'

'Find the coordinates of point Q, which divides the line segment AB in the ratio m:n.'

'In the regular hexagon $\\mathrm{ABCDEF}$, express $\\overrightarrow{\\mathrm{FB}}$ in terms of $\\overrightarrow{\\mathrm{AB}}$ and $\\overrightarrow{\\mathrm{AC}}$.'

'Question 62\n(1) Determine the points D that internally divide the line segment BC in the ratio 5:4, and the points E that internally divide the line segment AD in the ratio 2:1.\n(2) Find the ratio of V_{1} : V_{2}.'

'(1) Right isosceles triangle with BA=BC\n(2) Equilateral triangle\n(3) Right triangle with ∠A=π/3, ∠B=π/6, ∠C=π/2'

'In the complex plane, let the points representing z1, z2, z3, z4, z5 be denoted as A, B, C, D, E respectively. Of the following (0) to (5), the correct ones are (E) and (G). (0) △ABC is an equilateral triangle. (1) △BCD is an equilateral triangle. (2) △OCE is a right-angled triangle. (3) △BCE is a right-angled triangle. (4) The quadrilateral ABDC is a parallelogram. (5) The quadrilateral AOEC is a parallelogram.'

'Prove the following theorems using the complex plane: (1) In triangle ABC, with the midpoints of AB and AC as D and E respectively, it holds that BC // DE and BC = 2DE (Midpoint Theorem). (2) In triangle ABC, when the midpoint of BC is M, the equation AB^2+AC^2=2(AM^2+BM^2) is true (Median Theorem).'

'Prove that in the isosceles triangle ABC, taking point D on the base BC and drawing the chord ADE of the circumcircle of triangle ABC. At this point, AB squared equals AD times AE.'

'238'

'Drawing Similarity with 64 similar methods'

'(2) In triangle $ABC$, point $D$ lies on the extension of side $BC$, points $E$ and $F$ lie on sides $AC$ and $AB$ respectively, satisfying the following conditions: $\\frac{BD}{DC}=\\frac{AB}{AC} \\cdots \\cdots \\\\ \\frac{CE}{EA}=\\frac{BC}{BA} \\cdots \\cdots \\\\ \\frac{AF}{FB}=\\frac{AC}{BC} \\cdots \\cdots$'

'Question 51 | Size of sides and angles in a triangle\nIn triangle ABC, let M be the midpoint of side BC and D be the intersection point of the angle bisector of angle A and side BC.\nProve the following (1), (2):\n(1) AB > BD\n(2) If AB > AC, then ∠BAM < ∠CAM'

"Practice 44: Let the intersection of AB and PQ be R, and the intersection of PQ and CD be R'. Since AD//BC, we have PR:RQ=AP:BQ, PR':R'Q=PD:QC, AP:PD=BQ:QC=m:n. AP:BQ=½AD:½BC=AD:BC, PD:QC=¼AD:¼BC=AD:BC. Therefore, AP:BQ=PD:QC."

'In triangle ABC with circumradius R, find the following when (1) angle B = 120° and R = 6: b'

'Pythagorean theorem and its converse: In triangle ABC, if BC=a, CA=b, AB=c, then ∠C=90° if and only if a²+b²=c².'

'Important Example 102 Shape of a Triangle'

'In triangle ABC, according to the cosine rule, cos B = \\frac{144+121-100}{2 \\cdot 12 \\cdot 11} = \\frac{165}{2 \\cdot 12 \\cdot 11} = \\frac{5}{8}. From (1), AD^2 = 144+36-144 \\cdot 6 \\cdot \\frac{5}{8} = 90, and since AD > 0, AD = 3 \\sqrt{10}. Let ∠ADB = θ. In triangle ABD, according to the cosine rule, AB^2 = AD^2 + BD^2-2AD \\times BD \\cos θ. Also, BD:CD = 2:3, so CD = \\frac{3}{2}BD. In triangle ADC, according to the cosine rule, AC^2 = AD^2 + CD^2-2AD \\times CD \\cos(180^{\\circ}-θ) = AD^2+\\left(\\frac{3}{2}BD\\right)^2+2AD \\times \\frac{3}{2}BD \\cosθ = AD^2+\\frac{9}{4}BD^2+3AD \\times BD \\cosθ. Hence, 6AB^2 + 4AC^2 = 6(AD^2+BD^2-2AD \\times BD \\cosθ) + 4(AD^2+\\frac{9}{4}BD^2+3AD \\times BD \\cosθ) = 10AD^2+15BD^2.'

'Explain and prove the conditions for triangle similarity.'

'In \ \\triangle ABC \, if points \ P \ and \ Q \ lie on or extended from sides \ AB \ and \ AC \ respectively, then the following properties hold: \n[1] \ PQ // BC \\Leftrightarrow AP: AB=AQ: AC \\n[2] \ PQ // BC \\Leftrightarrow AP: PB=AQ: QC \\n[3] \ PQ // BC \\Longrightarrow AP: AB=PQ: BC \'

'In △ABC, since OA=OC, angle OCA=angle OAC=40°, thus α=180°-2×40°=100°. Also, since OA=OB, OB=OC, angle OAB=angle OBA=β, and angle OBC=angle OCB=25°. Therefore, in △ABC, 2×40°+2×25°+2β=180°, hence 2β=50°, so β=25°. Another solution: First find β, then according to the inscribed angle theorem, α=2(β+25°)=2(25°+25°)=100°.'

'Let the radius of the circumcircle be R, according to the law of sines, 6/sin C = 2R, thus R = 8/√7 = 8√7/7. Let the radius of the incircle be r, then △ABC = r/2(6+4+5), △ABC = 15√7/4, therefore r = √7/2.'