Plane Geometry - Properties of Basic Shapes (Points, Lines, Angles, Triangles, Quadrilaterals, Circles) | AI tutor The No.1 Homework Finishing Free App

'Find the value of k representing a circle passing through the origin (0,0).'

'Understand the explanation of distance between 1 points. Find the formulas for the distance between the origin O and point P(a), and between points A(a) and B(b).'

'(1) What kind of shape does the equation $x^{2}+y^{2}+4x-6y+4=0$ represent?\n(2) In order for the equation $x^{2}+y^{2}+2px+3py+13=0$ to represent a circle, determine the range of values for the constant $p$.'

'(1) Find the coordinates of the midpoint of the chord formed by the intersection of the line x+y=1 and the circle x^{2}+y^{2}=4, and determine the length of the chord.'

'(1) Passing through both the x-axis and y-axis, and point A(-4,2). (2) Passing through point (3,4), touching the x-axis, and having its center on the line y=x-1.'

'Plot the region that satisfies the inequalities $x^{2}+y^{2}>1$ and $|x| + |y| > 1$, and explain their relationship.'

'Example 29 | Shape of a triangle For 4 points A(4,0), B(0,2), C(3,3), D, answer the following questions.'

'Find the coordinates of the internal division points, external division points, and centroid in Example 30'

'Practice (63=>This Book p.137) (2) Let the coordinates of point P be (x, y), then from AP^2+BP^2=18 we get {(x-1)^2+(y-4)^2}+{(x+1)^2+y^2}=18, simplifying gives x^2+y^2-4y=0, which implies x^2+(y-2)^2=2^2. Therefore, the points that satisfy the condition lie on the circle (1). Conversely, any point on the circle (1) satisfies the condition. Hence, the desired locus is a circle with center at (0,2) and radius 2.'

'For the line segment connecting A (-3) and B (6), find the coordinates of the following points: (1) Point dividing internally in the ratio 2:1 (2) Point dividing externally in the ratio 2:1 (3) Point dividing externally in the ratio 1:2 (4) Midpoint'

'Taking line BC as the x-axis and point P as the origin, the coordinates of the vertices of triangle ABC can be expressed as: \nA(a, b), B(-c, 0), C(2c, 0)\nwhere b ≠ 0, c > 0. Verify the equation 2AB² + AC² = 3(AP² + 2BP²).'

'What is a diameter?'

'Find the equation of a circle that touches both the x-axis and the y-axis.'

'Find the trajectories of the following points Q and R with respect to the point P moving on the parabola y=x^2 and the two points A(3,-1), B(0,2).'

'Since the point (3,4) lies on the line 3x-2y-1=0, the desired line passes through the points (-7,-11) and (-1,6).'

'Practice (2) Parabola y=x^2 and line y=m(x+2) intersect at different points A and B. Find the locus of the midpoint of segment AB as the value of m varies.'

'Find the standard equation of a circle.'

'The equation of the line passing through the points (-1,1) and (3,-1) is $y-1=\\frac{-1-1}{3-(-1)}(x+1)$, which simplifies to $x+2y=1$.'

'When the chapter 3 (28 t) takes all real values, for the three points A(t, t^{2}), B(t, t-2), C(t+√3, t^{2}-t-1), answer the following questions:\n(1) Prove that for each real number t, A and B are distinct points.\n(2) Find all t values that make triangle ABC a right triangle.\n(3) Determine the range of t values that make triangle ABC an acute triangle.'

'The equation of the perpendicular bisector of line segment BC is y-0=-2(x-5), which simplifies to y=-2x+10. By solving equations (4) and (5) simultaneously, we get x=4, y=2. Therefore, the center of the circumscribed circle is at point (4,2) and the radius is sqrt{(8-4)^{2}+(5-2)^{2}}=5. Thus, the equation we are looking for is (x-4)^{2}+(y-2)^{2}=25.'

'Circle with center (-2, 3) and radius 3'

'When the intersection points of 22 circles, a circle passing through the intersection points of a circle and a line, and the equations of lines in terms of x, y are written as f(x, y), the curve represented by the equation f(x, y) = 0 (including cases where it represents a line) is called the curve f(x, y) = 0 and the equation is called the equation of the curve.'

'Find the equation of the tangent line at point P(4,6) on the circle.'

'Find the values of the constant k for which the lines do not form a triangle.'

'The results from (1) to (3) indicate that the coordinates of the centroid G of △PQR change from $\\left(\\frac{3-8+2}{3}, \\frac{3+1+1}{3}\\right)$ to $\\left(-1, \\frac{5}{3}\\right)$.'

'(2) Find the locus of the point P for which the sum of the squares of the distances from the points A(1,4) and B(-1,0), i.e., AP^2 + BP^2 is 18. [(2) Hokkai Gakuen University]'

'Given that the length of the perpendicular dropped from point (2,1) to the line kx + y + 1 = 0 is √3, find the value of the constant k.'

'Find the equations of the lines parallel and perpendicular to the line 4x+3y-6=0 passing through the intersection point of the lines 2x-y-1=0 and x+5y-17=0.'

'Intersect at points (-1, 3) and (3, -1)'

'Find the distance between the following two points.'

'(2) (1) From (1), in triangle △AOB where ∠AOB = 90°, the circle passing through points A, B, O has AB as its diameter.'

'Practice Example 17 Area of a Sector'

'Find the arc length and area of a sector with a radius of 4 and a central angle of 150°.'

'Problem (1) In the coordinate plane, when the points A(a, 2), B(5, 1), C(-4, 2a) are collinear, find the value of the constant a.'

'(1) It touches both the x-axis and the y-axis, passing through point A(-4,2). (2) Passing through point (3,4), touching the x-axis, with the center on the line y=x-1.'

'Problem to find the coordinates of point P. Find the coordinates of point P (x, y) lying on the line connecting two points A (6, -3) and B (1, 7).'

'Outside of the circle (x-2)^{2}+(y-1)^{2}=5. Including the boundary.'

'The coordinates of the centroid G of triangle ABC with vertices A(x1, y1), B(x2, y2), and C(x3, y3) are (\\frac{x1+x2+x3}{3}, \\frac{y1+y2+y3}{3})'

'Maths II river 36 books p.119\n(1) The radius r is the distance between the center (-5,4) and the origin, so r^2=(-5)^2+4^2=41\nTherefore, the equation of the circle we seek is (x+5)^2+(y-4)^2=41\n(2) The center is the midpoint of the diameter, so its coordinates are (-3+3)/2, (6+(-2))/2 which is (0,2)\nThe radius r is the distance between the center (0,2) and the point A(-3,6), so r^2=(-3-0)^2+(6-2)^2=25\nTherefore, the equation of the circle we seek is x^2+(y-2)^2=25\nAnother solution (2) On the circumference, let P(x, y) be a point different from A, B, then AP ⊥ BP so, when x ≠ -3, x ≠ 3, (y-6) / (x-(-3)) * (y-(-2)) / (x-3) = -1\nTherefore, (x+3)(x-3)+(y-6)(y+2)=0 which is x^2+(y-2)^2=25\nThis equation holds when x=-3, x=3, i.e., the points (-3,6), (-3,-2), (3,6), (3,-2) satisfy it, so this is the equation of the circle we seek.'

'For a standard angle, determine whether the following proposition is true and explain why.\n"There are no angles greater than 360 degrees"'

'Taking OP as the radius.'

'(5) A line parallel to the y-axis is perpendicular to the x-axis. Since the x-coordinate of the point it passes through is 5, we have x=5'

'If a parabola and a circle are tangent at one point, as shown in the figure, if the parabola and circle are tangent at the point (0,3) or the point (0,-3), then a = ±3, so the desired value of a is a = -37/4, ±3.'

'Important Example 58: Intersection Points of Parabola and Circle\nLet r be a positive constant. Consider the parabola y=x^{2} and circle x^{2}+(y-2)^{2}=r^{2}, and answer the following questions:\n(1) When r=2, find all coordinates of intersection points between the parabola and the circle.\n(2) Investigate how the number of intersection points between the parabola and the circle changes as r varies over all positive real values.'

'Find the values of a when the lines (a-2)x+ay+2=0 and x+(a-2)y+1=0 are parallel, coincident, or perpendicular.'

'Understand the formula for the distance between 2 points in a plane. Find the formula for the distance between points O(0; 0), A(x_{1}, y_{1}), B(x_{2}, y_{2}).'

'For the two circles \ x^{2}+y^{2}-2 x-4 y+1=0, x^{2}+y^{2}=5 \:\n(1) Find the equation of the line passing through the two intersection points of the two circles.\n(2) Find the center and radius of the circle passing through the two intersection points of the two circles and the point (1,3).'

'Which of the following lines are parallel to each other, and which are perpendicular?\n(1) y=2x+3\n(2) y=√2x-1\n(3) y=-2x+1\n(4) 2x-√2y+1=0\n(5) x+2y-5=0'

'(1) Find the equation of a circle with center (-5,4) passing through the origin.\n(2) Find the equation of a circle with diameter AB where A(-3,6) and B(3,-2).'

'Given circles, denoted as C_1 and C_2, respectively. (1) Let the coordinates of the point of tangency on circle C_1 be (x_1, y_1) such that x_1^2 + y_1^2 = 9'

'Important Example 49 Distance between a point on a parabola and a line\nGiven two points A(0,1) and B(2,5) and the parabola y=x^{2}+4x+7. Let there be a moving point P on the parabola.\n\nFind the minimum value of the area S of triangle PAB.'

'Plot the range of existence of point (a, b) on the ab plane when the line segment connecting points A(1,-2) and B(-2,1) intersects the parabola y=x^{2}+ax+b at only one point other than A and B.'

'Calculate the distance between two points on a plane.'

'The equation of the line AB is x/a + y/b = 1. Let point P(a, b) and the distance between point P and the line AB be d. Find the maximum value of d.'

'Number of lattice points'

'Let the equation of the required circle be (x-1)^(2)+(y+√3)^(2)=r^(2) (r>0). The condition for the circle (2) to be tangent to circle C is 0<r<5 and √((1-0)^(2)+(-√3-0)^(2))=5-r, therefore r=5-√4=3. Hence, the required equation is (x-1)^(2)+(y+√3)^(2)=9'

'Find the coordinates of point Q:\n\nLet the coordinates of point Q be (x, y).\n(1) Let OP=r, and let the angle between OP and the positive direction of the x-axis be α, then r cosα=-2, r sinα=3.\nTherefore, x=r cos(α+5/6π)=r cosα cos5/6π-r sinα sin5/6π.'

'Coordinates of points\nLet the points A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) be given.\nFind the distance between two points.\nAB=√((x₂-x₁)² + (y₂-y₁)²)\nIn particular, the distance between the origin O and A is OA=√(x₁² + y₁²)'

'Example 55 Tangent Conditions and Equations of Circles and Lines'

'Taking point B as the origin and edge BC as the x-axis, the coordinates of each vertex can be represented as A(0, a), B(0, 0), C(b, 0), D(b, a). Prove that PA² + PC² = PB² + PD².'

'On a plane, there are n circles such that any two circles intersect each other and no three or more circles intersect at the same point. How many parts does the plane get divided into by these circles?'

'Passing range of points and curves'

'Similar question Taking a point P(1/2, 1/4) on the coordinate plane. When two points Q(α, α^2) and R(β, β^2) on the parabola y=x^2 move such that the three points P, Q, R form an isosceles triangle with QR as the base, find the locus of the centroid G(X, Y) of triangle PQR. [University of Tokyo]'

'The parabola and the circle have 4 shared points when the vertex of the parabola is on the line segment connecting point (0, -37/4) and point (0, -3) (excluding the endpoints) as shown in the figure. Therefore, -37/4 < a < -3.'

'Two circles tangent to the coordinate axes and a line'

'Taking the line BC as the x-axis and the perpendicular bisector of side BC as the y-axis, the midpoint L of side BC becomes the origin O, and the coordinates of each vertex can be represented as A(a, b), B(-c, 0), C(c, 0). In this case, L(0,0), M((a+c)/2, b/2), N((a-c)/2, b/2), therefore, the coordinates of the intersection points dividing the three medians AL, BM, CN in a 2:1 ratio are ((a/3), (b/3)), ((-c+(a+c))/(2+1), (0+b)/(2+1)), ((c+(a-c))/(2+1), (0+b)/(2+1)), all of which are ((a/3), (b/3)), so the three medians AL, BM, CN intersect at this point.'

'Practice 1: Find the radius and total area of the incircles of equilateral triangles'

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

'(1)\n{% raw %}\\(\\mathrm{AB}^{2}=(0-4)^{2}+(2-0)^{2}=20\\)\\(\\mathrm{BC}^{2}=(3-0)^{2}+(3-2)^{2}=10\\)\\(\\mathrm{CA}^{2}=(4-3)^{2}+(0-3)^{2}=10\\)\\{% endraw %}\nTherefore, BC=CA, BC^2 + CA^2 = AB^2, so ΔABC is a right-angled isosceles triangle with ∠C=90∘.'

'Find the equation of a line'

'Practice Let real number t satisfy 0<t<1, consider the 4 points O(0,0), A(0,1), B(1,0), C(t,0) on the coordinate plane. Also, define point D on segment AB such that ∠ACO=∠BCD. Find the maximum area of triangle ACD. [University of Tokyo]'

'When the point (x, y) moves inside a circle with radius 1 centered at the origin, depict the range of motion of the point (x+y, x y).'

'For the circle $C: x^{2}+y^{2}=25$, answer the following questions:\n1. Find the equation of a circle with center at $(12,5)$ that is tangent to circle $C$ externally.\n2. Find the equation of a circle with center at $(1,-\\sqrt{3})$ that is tangent to circle $C$ internally.'

'Find the arc length and area of a sector with radius 4 and central angle 150 degrees.'

'Let a and b be positive real numbers. The parabolas C1: y = x^2 - a and C2: y = -b(x - 2)^2 are both tangent to the line ℓ at the point P(x0, y0). Define S1 as the area enclosed by the line x = 0, the parabola C1, and the tangent line ℓ, and S2 as the area enclosed by the line x = 2, the parabola C2, and the tangent line ℓ. Answer the following questions:\n(1) Express a, x0, y0.\n(2) Express the ratio of areas S1 : S2 in terms of b.'

'Distance between a point and a line'

'Represent the set of points (x, y) that satisfy y=x+1 in a figure with a straight line as the boundary. Also, represent the regions of points that satisfy y>x+1 and y<x+1 in a figure.'

'Find the coordinates of a point P on the x-axis equidistant from points A(-1,2) and B(3,4).'

'Find the equation of a circle passing through the point (2,1) and tangent to the x-axis and y-axis.'

'Basic Example 70: Given A(-2,1), B(6,-3), C(1,7), find the coordinates of the following points.'

'Consider the following problem.'

'Investigate how the number of intersection points between the circle (x-1)^2+(y-1)^2=r^2 and the line y=2x-3 changes depending on the radius r.'

'Standard Example 103'

'Given the three vertices A(5,-2), B(1,5), C(-1,2), find the lengths of the three sides of triangle ABC and determine what type of triangle it is.'

'Plot the region represented by the following inequalities.'

'Find the equation of the tangent line drawn from point A(3,1) to the circle x^2+y^2=2 and the coordinates of the point of tangency.'

'Let A and B be the intersection points of the parabola y=9-x^{2} and the x-axis. When a trapezoid is inscribed in the area enclosed by this parabola and the x-axis, with segment AB as the base, determine the maximum area of this trapezoid.'

'Find the number of lattice points within the area enclosed by y = -x^2 + 8x and y = x (including the boundary).'

"A circle with center C(a, b) and a constant distance r(>0) from C is a collection of points with C as the center and radius r. The circle with center C is simply called circle C, and the equation satisfied by any point (x, y) on the circle is called its equation. Let's try to find the equation of this circle. The condition for a point P(x, y) to be on circle C is CP = r, expressed in coordinates as √((x-a)^2 + (y-b)^2) = r, squaring both sides gives (x-a)^2 + (y-b)^2 = r^2. Since both sides of (2) are positive, (1)⇔(2)⇔(3), hence (3) is the equation of the desired circle. The form of equation (3) with knowledge of the center (a, b) and radius r is called the basic form of the circle equation. The equation of a circle with radius r and center (a, b) is (x-a)^2 + (y-b)^2 = r^2. The equation of a circle with radius r and center at the origin is x^2 + y^2 = r^2. Note that setting a=b=0 in 1 results in 2. When r=1, it is called a unit circle. Additionally, 1 can be considered as a translation of 2 parallel to a along the x-axis and b along the y-axis."

'(1) Let the circle (x-1)^2 + (y+2)^2 = 9 be C. When the circle (x+1)^2 + (y-1)^2 = 4 is C₁, determine the position relationship between C and C₁.'

'Point D is in the fourth quadrant, Circle D is tangent to the x-axis and y-axis, so the coordinates of point D can be assumed as (d, -d) and the radius is d. Since point D is below the line l, we have 3d - 4d - 12 < 0. The distance between point D and line l is |3d - 4d - 12| /√(3^2+4^2) = (d + 12) / 5. Since Circle D is tangent to line l, the distance between point D and line l is (d + 12) / 5 = d. Therefore, d = 3.'

'Find the equation of the tangent line at point A on the circle with A(0,3) and B(8,9) as the diameter.'

'There are points A(-1,3) and B(5,11).'

'Find the value of a when the two circles touch each other.'

'Find the equation of a line that is tangent to the circle x^2 + y^2 = 9 and parallel to the line 4x + 3y - 5 = 0.'

'For points A(0,1) and B(4,-1): (1) Find the equation of a circle C1 with center on the line y=x-1 passing through points A and B. (2) Find the equation of a circle C2 which is symmetric to the circle C1 found in (1) with respect to the line AB. (3) Let P and Q be points on circles C1 and C2, respectively. Find the maximum length of the line segment PQ. [Gunma University]'

'There is a parallelogram ABCD with vertices A(-2,3), B(5,4), and C(3,-1). Find the coordinates of vertex D and the intersection point P of the diagonals.'

'Find the equation of the following circle:\n(1) A circle with center at (1,1) that is tangent to the line 2x-y-11=0'

'Given points A(6,0) and B(3,3), when point P moves on the circle x^2+y^2=9, find the locus of the centroid G of triangle ABP.'

"Let C and C' be the two intersection points A, B, and the midpoint of line segment AB be M. Therefore, the length of segment OM is equal to the distance between the origin O and the line ℓ."

'Using the coordinates (p, q) of point B, determine the condition for the line AB to be perpendicular to the line ℓ when the slope of ℓ is 2.'

'When exactly 2 tangent lines can be drawn from point P(1, ) to curve C, answer the following questions. (i) Find the equations of the 2 tangent lines. (ii) Let Q and R be the points of tangency between the lines found in (i) and curve C. Assume that the x-coordinate of Q is less than the x-coordinate of R. Find the area S of the figure enclosed by line segment PQ, line segment PR, and curve C.'

'Investigate the positional relationship between the following circles and lines, and find the coordinates of the intersection points if they exist.'

'What kind of shapes do the following equations represent?'

'Assume there are four circles on the coordinate plane that touch the x-axis, y-axis, and the line 3x + 4y - 12 = 0. Arrange the radii of these circles in ascending order and explain the relationship between the center of each circle and the line.'

'Find the slope of the line that makes an angle of \\frac{\\pi}{4} with the line x - \\sqrt{3} y = 0.'

'For the point A(-2, -3), find the coordinates of point Q which is symmetrical to point P(3, 7).'

'Find the equation of the tangent line at point P on the following circle.'

'Let A be the intersection of the two lines \ 3 x+2 y-4=0 \ (1) and \ x+y+2=0 \ (2). Determine the equation of the line passing through point A and B(3,-2) for (1). Determine the equation of the line passing through point A and parallel to the line \ x-2 y+3=0 \ for (2).'

'Given A(-2,-3), B(3,7), C(5,2), find the coordinates of the following points.'

"Let's find the equation of the tangent line at point (a, b) on the circle x^2 + y^2 = r^2."

'Let line 3x+2y-4=0 be (1) and x+y+2=0 be (2), with A as the intersection point of the two lines. Find the equation of the line passing through A and point B(3,-2).'

'Find the equation of the line that intersects the x-axis at (2, 0) and the y-axis at (0, -3).'

'What is the shape of the triangle ABC formed by the following 3 points?'

'Find the equations of the following circles:\n1. Circle with center at (2, -3) and radius 1\n2. Circle with center at (3, 4) passing through the origin\n3. Circle with diameter defined by the points (3, 1) and (-5, 7)\n4. Circle with center at (5, 2) tangent to the y-axis'

'Find the center and radius of the circle that passes through the two intersection points of the two circles \\( x^{2}+y^{2}=2,(x-1)^{2}+(y+1)^{2}=1 \\) and is tangent to the line \ y=x \.'

'Point A is in the first quadrant, Circle A is tangent to the x-axis and y-axis, so the coordinates of point A can be assumed as (a, a), with the radius being a. Since point A is below line l, we have 3a + 4a - 12 < 0. The distance between point A and line l is |3a + 4a - 12| / √(3^2 + 4^2) = (-7a + 12) / 5. Since circle A is tangent to line l, the distance between point A and line l is a, hence (-7a + 12) / 5 = a, leading to a = 1.'

'Master the formulas for internal and external division points coordinates and conquer example 74!'

'Circle and line'

'For which values of the constant k does the circle C: x^2+y^2+(k-2)x-ky+2k-16=0 pass through the points A(x, y) and B(x, y)? Here, . The line segment AB will be a diameter of the circle C only when k=.'

'Explain what the third quadrant is.'

'Show the region boundaries represented by the inequalities.'

'Problem to find the position relationship between a line and a circle, along with the coordinates of their intersection points.'

'Find the equations of the following circles.'

"The equation of the line passing through the two intersection points C and C' is \\square x+\\square y=15. Also, the area S of the triangle with the two intersection points and the origin O as vertices is S=\\square."

'What kind of shapes do these equations represent?'

'Are the following two lines parallel or perpendicular?'

'Master the formula for distance between a point and a line, conquer example 83!'

'Illustrate the radius of angle θ and specify in which quadrant the angle lies'

'When the triangle ABC is a right triangle with vertices A(1,1), B(2,4), and C(a,0), find the value of the constant a.'

'Find the coordinates of the point P on the circumference of the circle with equation x^2-2x+y^2-4y+4=0 that is closest to the point A(-1,1). Also, find the distance between the points A and P.'

'Point on a plane'

'(2) Right isosceles triangle where \ \\angle \\mathrm{A}=90^{\\circ} \'

'Regarding points A(0,1) and B(4,-1)'

'Chapter 4: Figures and Equations'

'Problem of finding the distance between two points on a plane.'

'Find the coordinates of the following points.'

'Given the circle TR: x^{2}+y^{2}=1, denoted as C_{0}, and let C_{1} be the circle obtained by translating C_{0} 2a units in the positive direction of the x-axis, where a is 0<a<1. Also, let A and B be the two intersection points of C_{0} and C_{1} in the first quadrant, and let P(s, t) be a point on C_{0} different from points A and B. Find the locus of the centroid G of triangle PAB as P moves on the part of C_{0} excluding the two points A and B.'

'73 (1) $\\mathrm{AB}=\\mathrm{CA}$ is an isosceles triangle'

"The position of point P on a plane is represented by a pair of real numbers, for example, (a, b). This pair (a, b) is called the coordinates of point P, where a is the x-coordinate and b is the y-coordinate. The point P with coordinates (a, b) is denoted as P(a, b). In this section, let's learn about points on a plane. The plane with coordinates is divided into 4 parts by coordinate axes. These parts are called quadrants, and they are named as the first quadrant, the second quadrant, the third quadrant, and the fourth quadrant in a counterclockwise manner. Note that the coordinate axes are not included in any quadrant. In the diagram, (+, +) indicates the signs of x and y coordinates in each quadrant."

'In this case, the distance between the center (0,0) of circle (1) and the line (2) is equal to the radius of the circle √k, so'

'Find the number of tangents that can be drawn from point P(1, ) to the curve C: y=x^3-x.'

''

'Given the center and radius of a circle, find the equation of the circle.'

'Find the equation of a circle with center (a, b) and radius r.'

'Find the circle passing through the intersection of 2 circles'

'Prove that for an acute triangle ABC, the following equation holds: tan A + tan B + tan C = tan A tan B tan C.'

'Find the locus of the midpoint P of the line segment connecting point A(2,0) and point Q as point Q moves along the circle x^2 + y^2 = 1.'

'Examine the position relationships between the following circles and lines, and if there are common points, find their coordinates.'

'Since the center of circle C3 is the origin O, the distance between circle C and circle C3 is PO=√(1^2+(-2)^2)=√5\nLet r3 be the radius of circle C3, as circle C3 is inscribed in circle C, we have r3 < 3 and √5 = 3 - r3\nTherefore r3 = 3 - √5\nHence, the equation of circle C3 is x^2 + y^2 = (3 - √5)^2'

'Plot the regions represented by the following inequalities.'

'There is a parallelogram ABCD with vertices A(-2,3), B(5,4), and C(3,-1). Find the coordinates of vertex D and the intersection point P of the diagonals.'

'Find the coordinates of the midpoint and the length of the segment cut by the circle with center (2, 1) and radius 2 from the line y=-2x+3.'

'Find the equation of the tangent line drawn from point A(7,1) to the circle x^2+y^2=25.'

'Find the equation of a circle passing through the points (0,2) and (-1,1) with its center on the line y=2x-8.'

'When a parabola (1) and a circle (2) have 4 common points, find the range of r.'

'Find the center and radius of the circle passing through the two intersection points of the two circles x^2+y^2=2 and (x-1)^2+(y+1)^2=1, which is tangent to the line y=x.'

'In Figure 6, what is the length that can be read with the vernier calipers in increments of how many millimeters?'

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'What are the values on the vertical axis for points (2) and (3)?'

'2021 Shibuya Academy Makuhari Middle School (1st time) (4)\nAs shown in Figure 5-1, there is a rectangular prism $ABCD-EFGH$ with a rhombus base and all rectangular side faces. Points $K, L, M, N$ are located on the edges $AB, BC, CD, DA$ respectively, with $AK: KB = AN: ND = 1: 1$, $BL: LC = DM: MC = 2: 1$.\nAdditionally, point O is located on the diagonal $EG$ of the rhombus $EFGH$, with $EO: OG = 1: 5$.\nConnecting each vertex of quadrilateral $KLMN$ to point O creates pyramid O-KLMN. Answer the following questions. The volume of the pyramid can be calculated as (base area) x (height ÷ 3).'

'Explain the difference between a bright red star and a dark red star.'

'There is a right triangle ABC as shown in the figure 2, and squares with sides of AD, BD, and CD respectively. In this case, what is the area of the square with CD as one side in square centimeters?'

'What is the size of cell A in figure 12 (length between PQ) in micrometers? Use the value obtained in (5) and answer in integers.'

'peninsula'

'(3) As shown in the graph on the right, cliff B is 48m above sea level and at a distance of 70m north from point A, cliff C is 53m above sea level and at a distance of 70m south from point A, simply note down their respective positions.'

'(2) The line drawn by point O becomes like a bold line. Firstly, the central angle of a semi-circle with a radius of 6 cm is between (2) and (3). Adding the part between 8 and 9 (which is equal to the length of a 60-degree arc) gives a total of 180×3+90+60×2 = 750 degrees. Moreover, the arc between 3 and 4 has a radius of 12+6=18 cm and a central angle of 30 degrees. Therefore, the length of the line drawn by point O is calculated as 6×2×3.14×750/360+18×2×3.14×30/360=(25+3)×3.14=87.92 cm.'

'In the figure 5-1, there is a right-angled triangle with angle A, where AB=3 cm and AC=6 cm, and a right-angled isosceles triangle with angle D, where DE and DF are both 6 cm. Answer the following questions about the geometric shape formed by combining these right-angled triangles. Take the value of Pi as 3.14. Also, note that the volume of a cone can be calculated by base area * height / 3.'

'The height above sea level of volcanic ash layer X is 53 meters at cliff A and 44 meters at cliff B. When marked with circles on a graph, it looks as shown in the right graph.'

'(4) A set of points where the difference in distance from A to the sound source and B to the sound source is constant at 350m forms a line, indicating the presence of the sound source. This is represented by (I). A curve where the difference in distance from two points to the sound source is constant is called a hyperbola.'

'Does not fit in A3 paper'

"Comets are celestial bodies in the solar system that revolve around the sun like planets. Comets exhibit a distinctive feature of suddenly brightening as they approach the sun from far away in the solar system and darkening abruptly and disappearing as they move away. Additionally, as shown in Figure 6, comets present a different appearance with a long tail waving unlike other celestial bodies. The tail of a comet extends in the opposite direction of the sun. (5) Suppose a new comet is discovered, and it is visible immediately after the sunset of that day. Describe the appearance of the comet's tail as a straight line."

'(3) The sound source A is at the position reached within 1 second, and B is at the position reached within 2 seconds. Therefore, drawing a circle with a radius of 350 × 1 = 350 meters centered at A, and a circle with a radius of 350 × 2 = 700 meters centered at B, the two intersection points of these two circles will indicate the position of the sound source.'

'Choose one correct statement that can be inferred from the graph in Figure 4.'

'Find the length of each side of the following triangle.'

'Equation of a plane perpendicular to two coordinate axes'

'Right triangle with angle C equal to 90 degrees'

'Let the polar coordinates of point A be (r₁, θ₁) and those of point B be (r₂, θ₂). Find the area of triangle OAB, denoted by S.'

'In parallelogram ABCD, let M be the midpoint of side AB, E be the point that divides side BC into 1:2, and F be the point that divides side CD into 3:1. If →AB=b and →AD=d'

'A regular hexagon ABCDEF with side length 1 is given. When point P moves along side AB and point Q moves along side CD independently, find the area where point R, which divides segment PQ in the ratio 2:1, can pass through.'

'Although it is possible to directly substitute z=x+yi into the equation (3)(2) and calculate it, the computation becomes very complicated (see the first consideration after the answer). Therefore, we first consider the equation of an ellipse with congruence to shape C and having the focus on the x-axis, then rotate it to find the equation of C. (1) K: \\frac{x^{2}}{2^{2}}+\\frac{y^{2}}{1^{2}}=1 The coordinates of the foci are, \\sqrt{2^{2}-1^{2}}=\\sqrt{3}, so it is (\\sqrt{3}, 0),(-\\sqrt{3}, 0). The length of the major axis is 2\\cdot2 and the length of the minor axis is 2\\cdot1, therefore, the area to find is \\pi\\cdot2\\cdot1=2\\pi.'

'The tangent at any point on the curve C in the first quadrant always intersects the positive parts of the x-axis and y-axis, and the intersection points are denoted as Q and R, respectively. The point P divides the segment QR internally in the ratio 2:1.'

'For polar coordinates, find the equations of the following circle and line: (1) A circle with center at point A(3, π/3) and radius 2. (2) A line passing through point A(2, π/4) and perpendicular to OA (O is the pole).'

'(2) 128\nIn the isosceles trapezoid \ \\mathrm{ABCD} \ where \ \\mathrm{AB}=2 \\mathrm{~cm}, \\mathrm{BC}=4 \\mathrm{~cm}, \\angle \\mathrm{B}=60^{\\circ} \, when \ \\angle \\mathrm{B} \ increases by \ 1^{\\circ} \, by how much does the area \ S \ of the trapezoid \ \\mathrm{ABCD} \ increase? Assume \ \\pi=3.14 \.'

'Find the distance between two points.'

'Find the position vector of the orthocenter of a triangle.'

'In the coordinate space, let A(1,0,2) and B(0,1,1). When point P moves along the x-axis, find the minimum value of AP+PB.'

'On the coordinate plane, the circle C passes through the point (0,0), its center lies on the line x+y=0, and it is tangent to the hyperbola xy=1. Find the equation of circle C. Here, the circles and hyperbolas are said to touch at a point if the tangents of the circle and the hyperbola coincide at that point.'

'For the ellipse $x^{2}+4 y^{2}=4$, find the locus of the point $P(a, b)$ outside the ellipse from which two tangents drawn to the ellipse intersect at right angle.\n[Type University of Tokyo]\nBasic 155'

'Curve represented by parametric variables'

'Find the geometric figure represented by all points P(z) that satisfy the equation $|z-\\alpha|=r(r>0)$.'

'On the curve \\sqrt[3]{x}+\\sqrt[3]{y}=1, let \\mathrm{P} be the point in the first quadrant where the tangent intersects the x-axis and y-axis at points \\mathrm{A}, \\mathrm{B} respectively. If the origin is \\mathrm{O}, find the minimum value of \\mathrm{OA}+\\mathrm{OB}.'

'Find the coordinates and length of the chord formed by the intersection of the following line and curve.'

'Basic Concepts 1 Polar and Straight-line Equations (1) Polar equation of a circle with center at the pole O and radius a r=a r=2a cos θ r^2-2r r₀ cos(θ-θ₀)+r₀^2=a^2 θ=α r cos (θ-α)=a (a>0) (2) Circle with center at (a, 0) and radius a r=2a cos θ (3) Circle with center at (r₀, θ₀) and radius a r^2-2r r₀ cos(θ-θ₀)+r₀^2=a^2 (4) Line passing through the pole O and forming an angle α with the initial line θ=α (5) Line passing through point A(a, α) and perpendicular to OA'

'Let the equation $2 x^{2}-8 x+y^{2}-6 y+11=0$ represent a conic section $C_{1}$. Also, let $a, b, c$ $(c>0)$ be constants, and the equation $(x-a)^{2}-\\frac{(y-b)^{2}}{c^{2}}=1$ represent a hyperbola $C_{2}$. Determine the values of $a, b, c$ when the two foci of $C_{1}$ and the two foci of $C_{2}$ form the 4 vertices of a square.'

'Using the 4 points given in question (1) A(2,4), B(-3,2), C(-1,-7), D(4,-5), form a quadrilateral\n(2) Using the 3 points A(0,2), B(-1,-1), C(3,0) as vertices, connect another point D to form a parallelogram. Find the coordinates of the fourth vertex D.'

'(4) For the plane PQR and edge OD, the situations are as follows. When q = 1/4, plane PQR is . When q = 1/5, plane PQR is 又. When q = 1/6, plane PQR is ネ. Choose the one that fits two 〜 ネ, one from 0 to 5, you can choose the same option repeatedly.'

'Find the locus of the center P of a circle that is tangent to both the circle $(x-4)^{2}+y^{2}=1$ and the line $x=-3$.'

'For the ellipse \\( \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0) \\), the coordinates of the foci are \\(\\left(\\sqrt{a^{2}-b^{2}}, 0\\right),\\left(-\\sqrt{a^{2}-b^{2}}, 0\\right)\\). The foci are on the x-axis, with the length of the major axis being 2a and the length of the minor axis being 2b.'

'Prove that the difference in distance from any point on the hyperbola to its two foci is constant.'

'State the condition for three distinct points A(α), B(β), C(γ) to be collinear.'

'Let the two endpoints A and B of a line segment of length 2 move along the x-axis and y-axis, respectively. When $\\mathrm{BP}=1$, find the trajectory of point P.'

'Passing through the point A(3, -4), find the line parallel to the line l: 2x-3y+6=0 and name it g. Determine the equation of the line g.'

'In the regular hexagon ABCDEF, with center O, point P divides side CD internally in the ratio 2:1, and point Q is the midpoint of side EF. If vector AB is a and vector AF is b, express vectors BC, EF, CE, AC, BD, QP in terms of vectors a and b.'

''

'When a point P(z) moves along the perimeter of a circle centered at -i with a radius of 1 (excluding the origin), the point Q(w) represented by (3) 114 w=1/z, what kind of shape does it trace?'

"Topic: Investigation of quadratics represented by complex number equations and rotation movement Mathematics In Chapter 3 of mathematics C, we learned about geometric shapes in the complex plane, and in Chapter 4 we learned about the properties of quadratics. Here, we will investigate cases where the shape represented by the complex number z's equation is a quadratic. First, let's confirm the basic concepts of quadratics with the following problem. CHECK 3-A The equation of the trajectory of point P with a total distance of 6 from points F(√5, 0) and F'(-√5, 0) is to be found."

'Find the vector equation of a circle.'

'Plot the region E on the ab-plane consisting of all points (a, b) where the hyperbola $\\frac{x^{2}}{4}-\\frac{y^{2}}{9}=1$ and the line $y=ax+b$ have common points.'

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

'Prove the equation of the tangent line at the point (x0, y0) on the circle using vectors.'

'In triangle △OAB with vertices O(0), A(1), B(ι), where ∠O is the right angle vertex, prove that when triangle △PQR is formed by points P(α), Q(β), R(γ) with ∠P as the right angle vertex, the equation 2α² + β² + γ² - 2αβ - 2αγ = 0 holds.'

'(2) The sum of the distances from point z to the 2 points (√3+3i)/2 and -(√3+3i)/2 is constant at 4, therefore, the figure C is an ellipse with the 2 foci at (√3+3i)/2 and -(√3+3i)/2. Let c be the distance from the origin, which is the center of this ellipse, to the foci. The coordinates of the foci on the xy-plane are (c, 0) and (-c, 0). This ellipse is congruent to an ellipse where the sum of the distances from points on the ellipse to the 2 foci is also 4.'

'Consider a circle C with radius a in the first quadrant of the xy-plane, which is tangent to both the line l: y=mx(m>0) and the x-axis. Also, consider circles tangent to the line l, the x-axis, and the circle C at one point each with a radius of b, where b>a. (1) Express t in terms of m. (2) Express b/a in terms of t. (3) Find the limit lim_{m \to +0} 1/m(b/a-1).'

'(1) Let the lengths of the three sides of triangle ABC be AB=8, BC=7, CA=9. Let vector AB=b and vector AC=c, and let P be the incenter of triangle ABC. Express vector AP in terms of b and c.'

'Midpoint theorem: In triangle ABC, let the midpoints of sides AB and AC be M and N respectively. Then MN // BC and MN = 1/2 BC'

'There is a quadrilateral ABCD inscribed in a circle. When AB=4, BC=5, CD=7, DA=10, find the area S of quadrilateral ABCD.'

'Circumcenter of a triangle'

'Determine the range of values for x so that a triangle with side lengths 3, 5, and x becomes an acute triangle.'

'A regular tetrahedron OABC with edge length of 6 is given. Let L be the midpoint of edge OA, M be the point that divides edge OB into 2:1, and N be the point that divides edge OC into 1:2. Find the area of triangle LMN.'

'In triangle ABC, if B=30°, b=√2, and c=2, find the values of A, C, and a.'

'Please provide the terms related to the circumcircle of a triangle and their definitions.'

'The center of the circle passing through points A, P, and Q is the point of intersection of the perpendicular bisectors of two chords.'

'(2) acute angle'

'Using the power of a point theorem, show the properties of the two tangents drawn from point P to a circle.'

'For a regular octagon, find the following numbers.\n(1) The number of quadrilaterals that can be formed by connecting 4 vertices\n(2) The number of triangles formed by connecting 3 vertices that share an edge with the regular octagon'

'On the line x=1, the point T is located where the y-coordinate is √3. The point P is the intersection of the line OT and a semicircle with radius 1. The angle we are looking for is ∠AOP.'

'On a rectangular floor with dimensions of 240 cm by 396 cm, we want to cover it with square tiles of side length a cm without any gaps. Find the maximum value of a in this case. Also, determine the number of tiles that can be laid out.'

'Explain and prove the properties of the circumcenter, incenter, and centroid of a triangle.'

'Determine whether the four points A, B, C, D on the right diagram lie on the same circle.'

'The radius of circumcircle is \ \\frac{85}{8} \, and the radius of incircle is 2'

'In a quadrilateral ABCD inscribed in a circle, with AB = 8, BC = 10, and CD = DA = 3. Find the area S of the quadrilateral ABCD.'

'79. In triangle ABC, AB=2, BC=4, CA=2√3. Let AD be the altitude from vertex A to side BC, and let E and F be the points of intersection of the circle with diameter AD and sides AB and CA, respectively. E and F are different from A. [Tokyo Jikeikai Medical University]\n(1) Prove that points E, B, C, and F lie on the same circle.\n(2) Find the area of triangle EBF.'

'As shown in the diagram, label all vertices of an equilateral triangle with side length 2 and each midpoint of the edges from 1 to 6. Match the outcome of the first dice roll with this number. Connect the points corresponding to the numbers rolled on the dice three times to create a shape. Find the expected value of the area of the resulting shape.'

'Theorem 25 Power of a Point Theorem II'

'In \ \\triangle ABC \, with \ \\angle A=90^{\\circ}, \\angle B=60^{\\circ}, \\angle C=30^{\\circ} \ and noting that \ AD \ is the diameter of a circle, draw the auxiliary lines \ AD, ED, EF, DF \.'

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

'Please explain about points inside and outside a circle and the sizes of angles.'

'In triangle ABC, O is the circumcenter. Find the angles α and β in the figure on the right.'

'Conditions for a parallelogram: A quadrilateral is a parallelogram if any of the following conditions are met. [1] Two pairs of opposite sides are parallel. [2] Two pairs of opposite sides are equal. [3] Two pairs of opposite angles are equal. [4] One pair of opposite sides is parallel and equal in length. [5] The diagonals intersect at their respective midpoints.'

'In the quadrilateral ABCD inscribed in a circle, where AB = BC = 1, BD = √7, and DA = 2, find:\n1. The position of point A\n2. The length of side CD\n3. The area of quadrilateral ABCD S'

'Find the circumcenter O of triangle ABC.'

'Given a quadrilateral ABCD inscribed in circle PR with AB = 4, BC = 5, CD = 7, DA = 10, find the area S of quadrilateral ABCD.'

'Determine the minimum length of the diagonal in a rectangle with a length of 40 cm. Also, describe the shape of the rectangle at this minimum. Let the vertical length of the rectangle be x cm, then the horizontal length is (20-x) cm. Since x>0 and 20-x>0, we have 0<x<20. Denote the length of the diagonal as l cm, l^2 =x^2+(20-x)^2 =2 x^2-40 x+400 =2(x-10)^2+200 (1) where l^2 reaches the minimum value of 200 at x=10. Since l>0, when l^2 is minimized, l is also minimized. Thus, the minimum value of the diagonal length l is sqrt(200)=10 sqrt(2)(cm). At this point, the horizontal length is also 10 cm, making the rectangle a square.'

'(2) Consider cases based on the length of one side. 11) A square formed by two vertically adjacent lines and two horizontally adjacent lines.'

'Take point O on the plane and define two perpendicular lines at point O, as shown in the diagram to the right. These are called the x-axis and y-axis, respectively. Point O is called the origin. In this case, if point A is located at coordinates (3, 2), please provide its x-coordinate and y-coordinate.'

'In the figure to the right, label all the vertices of an equilateral triangle of side length 2 and the midpoints of each side with numbers 1 to 6, corresponding to the rolls of a die. Roll the die 3 times and connect the numbers obtained to form a geometric shape. Find the expected value of the area of the resulting shape.'

'There are 10 non-intersecting lines on a plane, with no three lines passing through the same point. If two of the 10 lines are parallel, determine the number of intersection points and triangles formed by these 10 lines.'

'Example 4: Parabolic Antenna\nA parabola is called a parabola in English. The surface of a parabolic antenna used for satellite broadcasting reception is in the shape of the surface that is formed by rotating a parabola around its axis.'

"Circles O and O' with radii 5 and 8, respectively, are externally tangent at point A. Let B and C be the points where the common external tangent of these two circles touches circles O and O'. Extend BA to intersect circle O' at point D.\n(2) Prove that points C, O', and D are collinear.\n(3) Find the ratios of AB:AC:BC."

'Calculate the number of parallelograms formed by 3 parallel lines and 5 lines intersecting them.'

'Distance between two points\n(1) The distance between two points A(x1, y1) and B(x2, y2) on the coordinate plane is\nAB=√((x2-x1)^2+(y2-y1)^2)\nIn particular, the distance between the origin O and point A(x1, y1) is OA=√(x1^2+y1^2)\n(2) The distance between two points A(x1, y1, z1) and B(x2, y2, z2) in coordinate space is\nAB=√((x2-x1)^2+(y2-y1)^2+(z2-z1)^2)\nIn particular, the distance between the origin O and point A(x1, y1, z1) is OA=√(x1^2+y1^2+z1^2)'

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'For a regular decagon, find the following:\n(1) The number of diagonals\n(2) The number of triangles with 3 of the vertices of the decagon as vertices\n(3) The number of triangles from (2) that share only one side with the decagon'

'Find the distance between A and P.'

'Prove that the centroids of triangles ABC and DEF coincide.'

'The minimum value is 10√2 cm, square'

'For the given line segment AB, draw the following points.'

'Mathematics I\nTherefore, the area of triangle (ABC) is\n\nEX 1 A piece of paper in the shape of an equilateral triangle with a side length of (10 cm) is given. Let the vertices of this equilateral triangle be (A, B, C), and let point (P) on side (BC) be at a distance of (2 cm) from point (B). When folding this equilateral triangle paper so that point (A) coincides with point (P), let the intersection points of the folds with sides (AB, AC) be (D, E) respectively. At this time, if (AD=) is A(cm), (AE=) is B(cm), and the area of △ADE is C(cm^{2}).\n[From Kyoto Makie University]'

'In triangle ABC, where BC=17, CA=10, AB=9. Find the value of sinA, the area of triangle ABC, the radius of the circumcircle, and the radius of the incircle.'

'Basic example 85 The length of the segment cut by the parabola from the x-axis\n(1) Find the length of the segment cut by the graph of the quadratic function y=-x^{2}+3x+3 from the x-axis.\n(2) Prove that the length of the segment cut by the graph of the quadratic function y=x^{2}-2ax+a^{2}-3 from the x-axis is constant regardless of the value of the constant a.'

'Given AB=2√2, BC=4√2, CA=2√6, find the lengths of the sides of the triangle.'

'(2) In triangle ABC, if BC = 5, CA = 3, AB = 7. Let D and E be the points where angle A and its exterior bisector intersect line BC, respectively. Find the length of segment DE.'

'On a plane, there are 10 lines such that no three of them intersect at the same point. When exactly 2 of the 10 lines are parallel, determine the number of intersection points formed by these 10 lines and the number of triangles formed.'

'Plot the following points:\n(1) Point P equidistant from sides AB, BC, CA\n(2) Point Q equidistant from points A, B, C'

'Draw the following circles based on the circle O and chord AB shown on the right. Note that points P and Q are different from A and B, and are not on the perpendicular bisector of chord AB.'

'For a right triangle with side lengths a, b, c, where the circumradius is 3/2 and the inradius is 1/2, answer the following questions. Assume a ≥ b ≥ c:'

'Find the minimum length of the diagonal in a rectangle with a length of 40 cm. Also, determine what kind of rectangle it will be at that time.'

'Write the following mathematical terms and their corresponding definitions in Japanese.'

'The length of the shorter side is at least 1 meter and at most 3 meters'

'Find the equation of the parabola obtained by symmetrically moving the parabola y=-2x^2+3x-5 with respect to the following lines or points.'

'Find the angle θ formed by the following two lines. Assume 0° ≤ θ ≤ 90°.(1) AB and FG (2) AE and BG (3) AF and CD'

'In a equilateral triangle ABC with side length 1, let D divide BC in the ratio 1:2, let E divide CA in the ratio 1:2, let F divide AB in the ratio 1:2. Let P be the intersection of BE and CF, Q be the intersection of CF and AD, and R be the intersection of AD and BE. Find the area of triangle PQR.'

'In triangle ABC, if AB = 6, BC = 7, CA = 5, find the radii of the circumcircle R and the incircle r.'

'Please provide the terms and the pages that have shapes circumscribed by two circles.'

'On the semicircle with a radius of 1, the point whose x-coordinate is 1/2 is the point P. The angle we are looking for is ∠AOP.'

'Find the coordinates of the point Q, which is symmetrical to the point P(3, 4) with respect to the line y = 2x + 1.'

'Let the lengths of the sides of triangle ABC be a, b, c. If (a+b) : (b+c) : (c+a) = 4 : 5 : 6 and the area is 15√3, then find the circumradius R and inradius r of triangle ABC.'

'Incenter of a triangle'

'In triangle ABC, with the radius of the circumcircle denoted as R. When A=30 degrees, B=105 degrees, and a=5, find R and c.'

'When symmetrically moved about the origin, the vertex is at the point \\( \\left(-\\frac{3}{4}, \\frac{31}{8}\\right) \\), forming a concave parabola,\n\\[ y=2\\left(x+\\frac{3}{4}\\right)^{2}+\\frac{31}{8} \\quad\\left(y=2 x^{2}+3 x+5 \\text { also valid }\\right) \\]'

'Circle O intersects with circle B at points P and Q, and further intersects with the diameter FG of circle B at points A and center B. In addition, E is the intersection point of lines PQ and FG. If EA = x, AB = a, and BG = b, express x in terms of a and b.'

'Find the distance between the following two points.'

'There is an equilateral triangle paper with a side length of 10 cm. Let A, B, and C be the vertices of this equilateral triangle, and point P be a point on edge BC such that BP=2 cm. When folding this equilateral triangle paper so that vertex A coincides with point P, let the intersections of edges AB, AC, and the fold be denoted as D and E, respectively. At this point, AD= cm, AE= cm, and the area of triangle ADE is cm².'

'For a triangle ABC that is not an equilateral triangle, with circumcenter O, centroid G, and orthocenter H, prove the following: (1) Let L be the midpoint of side BC, and M, N be the midpoints of segments GH and AG respectively. Prove that quadrilateral OLMN is a parallelogram. You may use the fact that AH=2OL. (2) Prove that point G lies on segment OH. (3) Prove that OG:GH=1:2.'

'Find the length of the line segment cut from the x-axis by the graph of the following two quadratic functions:\n(A) y = 2x^2 - 8x - 15\n(B) y = x^2 - (2a + 1)x + a(a + 1) (where a is a constant)'

'Positional relationship between a parabola and the x-axis'

'Summary of Determining the Shape of a Triangle'

"There are two circles P and Q that intersect with two circles O and O'. As shown in the figure on the right, draw a line from point A, which is beyond P of segment QP, tangent to circle O and intersecting with circle O', with the points of tangency being C, and the points of intersection being B and D. If AB=a, BC=b, CD=c, express c in terms of a and b."

"Prove the following using the converse of Ceva's theorem:\n1. The three medians of a triangle intersect at one point.\n2. The three angle bisectors of a triangle intersect at one point."

'67 diagrams; in order of vertices and axes: (1) point (2, -1), line x=2 (2) point (-2, -3), line x=-2 (3) point (1, 1), line x=1'

'Quadrilateral ABCD is inscribed in a circle O, with AB=3, BC=CD=√3, and cos ∠ABC=√3/6. Find: (1) The length of segment AC (2) The length of side AD (3) The radius R of circle O'

'Find the area of the following shapes.\n1. Parallelogram ABCD with AB=2, BC=3, and ∠ABC=60 degrees\n2. Regular octagon circumscribed around a circle with radius 10'

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'In triangle ABC, when a=13, b=7, and c=15, find A.'

'For a right triangle where the sum of the lengths of the two legs is 16, what shape maximizes the area? Also, calculate the maximum value.'

"There are three cases for the position relationship between a circle and a line. Here, r is the radius of the circle, and d is the distance between the center of the circle and the line. [1] Intersection at 2 points (2 shared points) 0 ≤ d < r [2] Tangent (1 shared point) 0 ≤ d < r [3] Disjoint (no shared points) 0 ≤ d < r When there is only one shared point, the circle and the line are tangent, and this line is called the tangent, with the shared point being the point of tangency. Let's first investigate the properties of the tangents of a circle."

'Determine the conditions for the triangles to exist'

'In triangle ABC, where AB=6, BC=a, and CA=4, let M and N be the midpoints of BC and CA, respectively. (1) Find the value of a when AM=√10. (2) When a is the value from (1), find the length of segment BN.'

'(2) The longest side is CA, so AB + BC = 18, CA < AB + BC, therefore triangle ABC exists.'

"Let's review Example 65! Let's use the fact that the lengths of the two tangents drawn from a point outside the circle are equal. Once again, let's try eliminating unnecessary shapes."

'TRAINING 112 (1)\nLet the point O on the flat square be the origin, consider the coordinate plane with the east direction as the positive direction of the x-axis and the north direction as the positive direction of the y-axis.\nPoint A is located 28 units east of point O. Additionally, point P is south of the line connecting points O and A.\nPoint P is at a distance of 25 from O and 17 from A.\n(1) Find the coordinates of point A.\n(2) Find the coordinates of point P.'

'Practice 3: Incenter, Circumcenter, Centroid of a Triangle'

'Solve the following problems for triangle ABC:\n(1) Find the value of cos A.\n(2) Find the area S of triangle ABC.\n(3) Find the radius r of the incircle of triangle ABC.'

'Calculate the values of trigonometric functions in a right triangle'

'Proof that there are four points on a circle'

'One of the intersection points of two graphs is the point (-1,0)'

'Prove that the points B, C, F, E lie on a single circle when a perpendicular line AD is drawn from vertex A of acute triangle ABC to side BC, and perpendicular lines DE, DF are drawn from D to sides AB and AC, respectively.'

'In right triangle ABC, with AB > AC and ∠A = 90°, draw the perpendicular AD from vertex A to edge BC.'

'In the pentagon ABCDE circumscribed about a circle, where AB = 7, BC = 3, CD = 5, DE = 6, ∠BCD = 120° and ∠A = 82°, find:\n(1) The length of segment BD\n(2) The length of segment AD\n(3) The length of side AE\n(4) The area of quadrilateral ABDE'

'When A=90^{\\circ}, \\sin A=\\sin 90^{\\circ}=1, so 2R \\sin A=2R \\cdot 1=2R. Also, side BC is the diameter of the circumcircle of triangle ABC, so a=2R, hence a=2R \\sin A.'

'(1) \\\\( \\theta=30^{\\circ}, \\\\ 150^{\\circ} \\\\\\\n(2) \\\\( \\theta=45^{\\circ} \\\\\\\n(3) \\\\( \\theta=120^{\\circ} \\\\\\\n'

'66 diagram; (1) point (-1,0) and line x=-1 in order of vertices and axis, (2) point (1,1) and line x=1'

'Chapter 7 Applications to Triangles'

'In △ABC, suppose AB = 7√3 and ∠ACB = 60°. What is the radius of the circumcircle O of △ABC? Let point P move on the arc AB containing point C of the circumcircle O.'

'(1) Find the measures of the three angles of triangle ABC where ∠A=90°, AB=2, and BC=3.\n(2) Find the lengths of the three sides of triangle ABC where ∠A=70° and ∠B=∠C.'

'(1) ∠C < ∠B < ∠A\n(2) CA = AB < BC'

'In the diagram on the right, point I is the incenter of triangle ABC. Find the following: (1) 𝛼 (2) AI:ID'

'Consider a coordinate plane with point O as the origin, eastward as the positive direction of the x-axis, and northward as the positive direction of the y-axis.'

'The graph of the quadratic function y = ax^2 + 2ax + a + 6 (a≠0) intersects the x-axis at two points P and Q, and the length of line segment PQ is 2√6. Determine the value of the constant a.'

'(1) 60°\n(2) 50°\n(3) 140°'

'In the figure on the right, point O is the circumcenter of triangle ABC. Find the angles alpha and beta.'

"When thinking as described in the question, it is unclear where to apply the properties of the learned shapes. First, point A is the point of tangency of the two circles O and O', so let's draw the common tangents of the two circles passing through point A. By focusing on a part of the figure, the relevant properties will become apparent."

'Using the result from the previous question, find the length of a side of the next regular polygon that is inscribed in a circle with a radius of 10. Also, find the length of the perpendicular dropped from the center O of the circle to a side of the regular polygon. You may use trigonometric tables. Round the result to two decimal places.'

'In the given figure, find the value of x. Here, PT is the tangent to the circle, and T is the point of contact.'

'PRACTICE 4 (3) In △ABC, BC=a, CA=b, AB=c, the radius of the circumscribed circle is 3, and the area is S. In this case, S=ABc. The answer choices are (0) 1/2 (1) 1/3 (2) 1/6 (3) 1/8 (4) 1/12'

'In triangle ABC with AB=6, BC=a, and CA=4, let M and N be the midpoints of sides BC and CA, respectively. (1) Find the value of a when AM=√10. (2) When a has the value from (1), find the length of segment BN.'

'(1) In triangle ABC, if a=1, b=√3, and A=30°, find the lengths of the remaining side and the size of the angle.'

'Example 1 In the square ABCD with side length 8, points P, Q, and R are taken on the sides AB, BC, and CD respectively such that AP=x, BQ=2x, CR=x+4(0<x<4). Since the areas of triangles PBQ, QCR are represented by A, B respectively when expressed in terms of x, the area of triangle PQR is minimized when x=.'

'Find the area of the quadrilateral ABCD, where 77^{3} AB =5, BC=6, CD=5, DA=3, and ∠ADC=120^{\\circ}.'

'Inside angle XOY, there is a point A such that ∠XOA=30° and OA=3. When points P and Q are taken on OX and OY respectively, find the minimum value of AP+PQ+QA.'

'On a rectangular floor of 2m 40cm by 3m 72cm, you want to lay square tiles of side length a cm without any gaps. Find the maximum value of a. Also, determine the number of tiles that can be laid.'

'Explain problems related to the sides and angles of a triangle, and prove the following theorems:\n1. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.\n2. The difference between the lengths of any two sides of a triangle is less than the length of the third side.'

'angle between two lines'

"Prove that when a circle O passing through the common chord AB of two intersecting circles O and O', and chord CD of circle O and chord EF of circle O', the four points C, D, E, F are concyclic. It is given that the four points C, D, E, F are not collinear."

'In △ABC, where the radius of the circumcircle is 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'

'74 degrees isosceles triangle, the maximum value is 32'

'(1) Which of the following quadrilaterals is inscribed in a circle?\n(2) In acute triangle ABC, point D is taken on side BC (different from B and C), and perpendiculars DE and DF are drawn from point D to sides AB and AC respectively. Prove that quadrilateral AEDF is inscribed in a circle.'

'In a town where the roads are like a grid, find the shortest path from point A to point B.\n(1) How many possible routes are there?\n(2) How many of the routes pass through point C from (1)?\n(3) How many of the routes from (1) do not pass through point C?'

'Explanation using the condition of a quadrilateral inscribed in a circle\n(1) Which of the right quadrilaterals ABCD can be inscribed in a circle?\n(2) There is a quadrilateral ABCD inscribed in a circle, and a line parallel to side AD intersects sides AB, DC at points E, F respectively. Prove that the quadrilateral BCFE is also inscribed in the circle.'

''

'Find the equations of the parabola when the given parabola in the example (1) is moved symmetrically about (1) axis (2) with respect to the origin.'

'■Circumcenter…Intersection point of the perpendicular bisectors of the sides of a triangle\nMiddle school\nPerpendicular bisector of a line segment\nPoint P lies on the perpendicular bisector of line segment AB ⇔ PA=PB\nIn the same line\nIn other words\n“The perpendicular bisector of line segment AB is a collection of points equidistant from points A and B”'

'(1) How many triangles can be formed by connecting 3 vertices of a regular pentagon? How many of those triangles share 2 sides with the regular pentagon? (2) How many line segments can be formed by connecting 2 vertices of a regular pentagon?'

'Point P is on the semicircle with a radius of √5, so OP = √5\nIn the right triangle OPQ, OQ² + 2² = (√5)², hence OQ² = 1 and OQ = 1\nTherefore, the coordinates of point P are (-1,2)\n\nTherefore, sin θ = 2/√5, cos θ = -1/√5, tan θ = 2/-1 = -2'

'When the sum of the lengths of the two sides forming a right angle triangle is 16, what shape maximizes the area of the triangle? Also determine the maximum value.'

'Summary of determining the shape of a triangle'

'In the tetrahedron ABCD with side length 4, let M be the midpoint of side CD and let the angle AMB be θ'

'An angle ∠XOY=30° with a point A where OA=3 inside the angle. Points P and Q are taken on OX and OY respectively. Find the minimum value of AP+PQ+QA.'

"In the figure on the right, the two circles O and O' are tangent externally. A and B are the points where the common tangent of circles O and O' intersect the circles. If the radii of circles O and O' are 6 and 4 respectively, find the length of segment AB."

'In triangle 128 (3), when á=√6+√2, b=2, and C=45°, find the length of the remaining side and the size of the angle.'

'There is a quadrilateral ABCD inscribed in a circle with radius TR, where the lengths of the sides are AB=√7, BC=2√7, CD=√3, and 141DA=2√3. Find: (1) the value of cos B (2) the length of the diagonal AC (3) the area S of the quadrilateral ABCD'

'There are 4 roads running east to west and 4 roads running north to south. How many shortest paths are there for the following destinations: (1) From point A to point B. (2) From point A, passing through points C and D, to point B. (3) Of the shortest paths from point A to point B, those that pass through at least one of points C or D.'

'A parabola y = x² and a circle x² + (y - 5/4)² = 1 touch at two different points. Find the area S of the region enclosed by the shorter arc of the circle with the two tangent points as endpoints and the parabola.'

'(1) Distance between point A and line BC\n(2) Area of triangle ABC'

'Given the curve y=9-x^2 intersects the x-axis at points A and B, and a trapezoid ABCD is inscribed in the region enclosed by curve and line segment AB. Find the maximum area of this trapezoid. Also, determine the coordinates of point C at that time.'

'Find the arc length and area of the following sector:'

'Basic Example 69 Coordinates of the centroid of a triangle'

'On the plane, there is a parabola C: y=x^2-2 and a line l: y=4x. (1) Find the coordinates of the intersection points A and B of C and l. (2) Let moving point P on C move from A to B. Find the coordinates of point P where the area of triangle PAB is maximized.'

'Passing through point A(3,1), find the equation of the line PQ passing through the points of tangency P and Q of two tangents that are tangent to the circle x^2 + y^2 = 5.'

'Find the area of the triangle with vertices O(0,0), A(x1, y1), and B(x2, y2).'

'Find the equation of a circle passing through point (4,2) and tangent to the x-axis and y-axis.'

'Find the distance between two points A(a) and B(b) on a number line.'

'Find the coordinates of point Q after rotating point P(4, 2√3) around the origin by π/6.'

'Find the coordinates of the following points: (5, 1), (9, 5), (3, 9)'

'Illustrate the region represented by the system of inequalities x² + y² - 2x + 2y - 7 ≥ 0, x ≥ y.'

'Find the equation of the tangent to the circle x^2 + y^2 = r^2 at point (x1, y1).'

'Let a be a constant satisfying a > 1. There is a point M(2, -1) on the coordinate plane. For a point P(s, t) different from M, point Q is taken such that the three points M, P, Q are collinear in that order, and the length of segment MQ is a times the length of segment MP.'

'Consider a circle with radius r and center C, and a line ℓ with distance d from C. Determine the position relationship between the circle and the line based on the relationship between d and r.'

'What will be the shape of the intersection points P(x, y) of the two lines l: tx-y=t and m: x+ty=2t+1 as t varies over real values? Find their equations and plot them.'

'When the line (3) shares a point with region D, the slope m is maximized when the line is tangent to circle C. Calculate the maximum slope m at this point.'

'Find the equation of the following circles.'

'Distance between a point and a line'

'Find the coordinates of a point equidistant from the three points A(1,5), B(0,2), and C(-1,3).'

'What inequality does the shaded area in the figure represent? Exclude the boundary line.'

'Consider two points A(-1,2) and B(4,2) on the coordinate plane. Let t be a real number such that 0 < t < 1. Point P divides the line segment OA in the ratio t:(1-t) and point Q divides the line segment OB in the ratio (1-t):t. Find the minimum length of the line segment PQ and the corresponding value of t.'

'When point Q moves on the circle x^{2}+y^{2}=9, find the locus of point P that divides the line segment AQ connecting point A(1,2) and Q internally in the ratio 2:1.'

'Find the equation of a line that is a tangent to the circle x^2+y^2=8 and perpendicular to the line 7x+y=0.'

'When the line with slope -1 intersects region D, and the circle with center (3,2) has a distance to the line less than or equal to the radius 1. Find the maximum value of n in this case.'

'Passing through point A(3,1), let P and Q be the points of contact of the two tangent lines that touch the circle x^{2}+y^{2}=5. Find the equation of the line PQ.'

'For the circles \\( (x-5)^{2}+y^{2}=1 \\) and \ x^{2}+y^{2}=4 \, find: (1) How many common tangents do the two circles have? (2) Determine the equations of all common tangents for the two circles.'

'Find the range of constant k such that the circle x^2 + y^2 - 4x - 6y + 9 = 0 (1) and the line y = kx + 2 have a point in common.'

'Do the given circle and line have a point in common? If so, find the coordinates of that point.'

'Given the radius and the radius representing an angle, let P be the intersection point with a circle of radius r.'

'Reduce the given angle of 42140° to an angle on the unit circle in standard position.'

'The centroid of the triangle formed by the moving point of distance 2 from the origin with fixed points (5,0) and (0,3) lies on the curve x^{2}+y^{2}-Ax-By+C=0.'

'Find the following coordinates: (\x0crac{3}{14}, 0) and (0,-\x0crac{3}{4}).'

'For triangle ABC with vertices A(1,1), B(2,4), and C(a, 0) to be a right-angled triangle, find the value of a.'

'Shapes and Equations\nFor the three points A(0,0), B(2,5), C(6,0), find the coordinates of point P when PA² + PB² + PC² is minimized.'

'Find the equation of the tangent line to the circle at point A (4,6) on the circle x^2 + y^2 - 2x - 4y - 20 = 0.'

'In the coordinate plane, find the equations of the two lines that pass through the point (-2, -2) and are tangent to the parabola y=1/4 x^{2}.'

'When the 3 lines 2x-y-1=0, 3x+2y-2=0, y=1/2x+k intersect at point A, what is the value of k, and what are the coordinates of point A?'

'Find the situation of the three points A(-2, -2), B(2, 6), C(5, -3) on the coordinate plane.\n(1) Find the equation of the perpendicular bisector of segment AB.\n(2) Find the coordinates of the circumcenter of triangle ABC.'

'Find the equation of the line passing through point (7,1) and tangent to the circle x^2+y^2=25, and find the coordinates of the point of tangency.'

'For the circles given by (x-5)^2 + y^2 = 1 and x^2 + y^2 = 4, (1) how many common tangents do the two circles have in total? (2) Find all the equations of the common tangents for the two circles.'

'A circle with its center on the line y = -4x + 5 and tangent to both the x-axis and y-axis, find the equation of the circle.'

'Find the equation of the circle passing through the points (4, -1), (6, 3), and (-3, 0).'

'Investigate the shape of the triangle ABC with vertices A(2a, a+√3a), B(3a, a), C(4a, a+√3a). Here, a>0.'

'Find the coordinates of the point that divides the line segment AB between points A(x1, y1) and B(x2, y2) internally in the ratio m:n.'

'Find the equation of the tangent line at the given point on the given circle.'

'Find the coordinates of the remaining vertex D of the parallelogram with vertices A(4,5), B(6,7), C(7,3).'

'Determine the values of a and b so that triangle ABC with vertices A(1,0), B(0,3), and C(a, b) becomes an equilateral triangle.'

'On the number line, there are 3 points A(3), B(-3), C(5). Point D divides the line segment AB internally in the ratio 2:1, point E divides the line segment AC externally in the ratio 3:1, find the coordinates of the point that divides the line segment DE internally in the ratio 3:4.'

'Let n≥2. There are n circles on a plane, where no two circles intersect each other and three or more circles do not intersect at the same point. How many intersection points are formed by these circles?'

'Find the length of the line segment AB where A and B are the points of intersection of the parabola y=x^{2} (1) and the line y=x+3(2).'

'Conversely, the point P(x, y) on the line x+y=2 satisfies AP^2-BP^2=4. Therefore, the required trajectory is the line x+y=2.'

'Find the arc length and area.\n(1) Radius is 10 with angle π/5\n(2) Radius is 3 with angle 15°'

'Find the coordinates of the remaining vertex D of a parallelogram with vertices A(5,-1), B(3,3), and C(-1,-3).'

'When the point P (x, y) on the coordinate plane moves within the range of 3y ≤ x + 11, x + y - 5 ≥ 0, and y ≥ 3x - 7, find the maximum and minimum values of x² + y² - 4y.'

'Find the locus of points P that satisfy the following conditions:\n(1) Points P equidistant from points O(0,0) and A(3,2)\n(2) Points P such that angle OPA = 90 degrees, where O(0,0) and A(6,0)\n(3) Points P such that AP^2 - BP^2 = 4, where A(3,2) and B(1,0)'

'There are n circles on a plane, where any two circles intersect each other, and no three or more circles intersect at the same point. In how many parts do these circles divide the plane?'

'Find the distance between the following two points: (1) A(-3), B(2) (2) A(-2), B(-5)'

'Therefore, the coordinates of point C are $\\mathrm{C}\\left(-\\frac{2}{3}, \\frac{1}{3}\\right)$, and thus $\\mathrm{BC}=\\sqrt{\\left(-\\frac{2}{3}-\\frac{4}{3}\\right)^{2}+\\left\\{\\frac{1}{3}-\\left(-\\frac{2}{3}\\right)\\right\\}^{2}}=\\sqrt{5}$. Also, the distance between point A and line (3) is $\\frac{\\left|\\frac{1}{3}+2 \\cdot \\frac{4}{3}\\right|}{\\sqrt{1^{2}+2^{2}}} = \\frac{3}{\\sqrt{5}}$. Therefore, the area of the triangle we seek is $\\triangle \\mathrm{ABC}=\\frac{1}{2} \\cdot \\sqrt{5} \\cdot \\frac{3}{\\sqrt{5}} = \\frac{3}{2}$.'

'In the xy-plane, considering the lines l: x+t(y-3)=0 and m: tx-(y+3)=0 as t varies over all real numbers, what kind of shape do the intersection points of lines l and m form? [Gifu]'

'Regarding the triangle ABC with vertices A(1,1), B(2,4), and C(a,0)'

'Graph the following equations representing lines on the coordinate plane.'

'Find the equation of the tangent that satisfies the following conditions.'

'Chapter 3: Shapes and Equations'

'Given three vertices A(4,5), B(6,7), C(7,3) of a parallelogram, find the coordinates of the remaining vertex D.'

'Determine the shape of the triangle ABC with vertices A(2a, a+√3a), B(3a, a), and C(4a, a+√3a). Assume that a>0.'

'What kind of triangle is triangle ABC with vertices A(1, -1), B(4, 1), and C(-1, 2)?'

'Find the coordinates of the point $Q$ which is symmetric to point $P(3,2)$ with respect to the line $\\ell: x+y+1=0$.'

'Question 84'

'Find the equation of circle PR. (1) Passing through points (0,2), (-1,1) and with center on the line y=2x-8.'

'Find the conditions for the general form of a circle equation x^{2}+y^{2}+l x+m y+n=0 to represent a circle. Also find the center and radius.'

'Find the equations of the following circles'

'Find the coordinates of the following points. (1) Point dividing the line segment 3:1 internally (2) Point dividing the line segment 3:1 externally (3) Point dividing the line segment 1:3 externally (4) Midpoint'

'When the points A(a,-2), B(3,2), C(-1,4) are collinear, find the value of the constant a.'

'Find the tangent at a point on the circumference of a circle.'

'For points A(7,6), B(-3,1), and C(8,1) in the PR 3, let P be the midpoint of BC, Q be the point that divides CA externally in the ratio 3:2, and R be the point that divides AB internally in the ratio 3:2. Find the coordinates of the centroid of triangle PQR.'

'Given the parabola y = x^2 and points A(1,2), B(-1,-2), C(4,-1), find the locus of points Q and R as point P moves on the parabola y = x^2. (1) Point Q that divides the segment AP in the ratio 2:1 (2) The centroid R of triangle PBC.'

'Find the maximum value of k, such that the condition x^{2}+y^{2} ≤ 1 implies 3 x+y ≥ k.'

''

'Find the equations of 4 lines.'

'Find the locus of points P that satisfy the following conditions:'

'Find the scale of the midpoint of the line segment connecting point 4 on the scale and point 9 on the scale.'

'(1) In the coordinate plane, find all the equations of lines that are parallel to the line y=-2x and at a distance of √5 from the origin. [Tokyo Denki University] (2) Find the distance between the parallel lines 2x-3y=1 and 2x-3y=-6.'

'Plot the regions represented by the following inequalities.'

'Find the conditions of a and b such that the line y = ax + b has a point in common with the line segment connecting two points A(-1,5) and B(2,-1), and plot it on the ab plane.'

'Find all values of a that divide the plane into 6 parts.'

'Find the coordinates of the two intersection points of the two circles x^2 + y^2 = 10 and x^2 + y^2 - 2x + 6y + 2 = 0.'

'1. Find the equation of a circle with center (0,0) and radius √2.'

'Find the equation of the circle that has a center at (1,1) and is tangent to the line 4x+3y-12=0.'

'On the coordinate plane, let A be the point (-3,2) and B be the point (4,0). Find the coordinates of the points that are equidistant from the x-axis and y-axis, respectively.'

'Find the coordinates of the point that divides point A (x1, y1) and point B (x2, y2) externally. The external division ratio is m:n.'

'Find the equation of a circle that passes through the point (2,3), is tangent to the y-axis, and has its center on the line y=x+2.'

'In a triangle ABC, let D, E, F be the points dividing the sides BC, CA, AB in the ratio 3:1, respectively. Show that the centroid of triangle DEF coincides with the centroid of triangle ABC.'

'Find the area of the triangle formed by the lines x-y+1=0, 2x+y-2=0, and x+2y=0.'

'Determine the number of intersection points between a line and a circle in geometry and equations.'

'Find the equations of the following circles:'

'Description question about the polar lines of a circle with respect to point A. Given a point A(p, q) outside the circle x^2+y^2=r^2, find the equation of the line β passing through the points of tangency P, Q of the two tangents drawn from point A to the circle. Using the equation p x+q y=r^2, prove that the polar line of point A passing through another point B implies that the polar line of point B passes through point A.'

'Let S be the area of a regular pentagon inscribed in a circle of radius 1'

'In the regular tetrahedron OABC with side length a, points P, Q, R are taken on edges AB, BC, and OC respectively. Starting from vertex O, passing through points P, Q, R in sequence, what is the length of the shortest path to vertex A?'

'In quadrilateral ABCD, AB=4, BC=5, CD=t, DA=3-t (0<t<3). Also, assume that quadrilateral ABCD has a circumscribed circle.'

'There are two lines that intersect the line y=x-1 and form an angle of 15 degrees, passing through the point (0,1). Find the equations of these lines.'

'For a regular tetrahedron ABCD with edge length of 6, let E be the point on edge BC such that 2BE=EC, and let M be the midpoint of edge CD.'

'Consider a triangle with side lengths a, a+2, a+4.'

'In triangle ABC, AB = 8, AC = 5, and ∠A = 120°. Let D be the intersection of the angle bisector of ∠A and side BC. Find the length of segment AD.'

'A cone with a radius of 2 and a slant height of 6 is tangent to a sphere O both on its lateral surface and at its center of the base. Determine the radius, volume, and surface area of this sphere.'

'(2) BD=3, AD=3 \\sqrt{7}'

'Choose the most appropriate term from (A) to (E) to fill in the blank.'

'Let the area of hexagon ABCDEF be S'

'Practice 1: On the sides AB, BC, CA of an equilateral triangle ABC with side length 1, points D, E, F are taken such that AD=x, BE=2x, CF=3x. (1) Express the area of triangle DEF, S, in terms of x. (2) Find the value of x that minimizes S in (1) and the minimum value.'

'In △ABC, where a=2, b=√2, c=1. Find:\n(1) cos B, sin B\n(2) The area of △ABC, S\n(3) The radius of the incircle of △ABC, r\n(4) The radius of the circumcircle of △ABC, R\nRefer to p. 265, Basic Concept 3, Basic 162.'

'In the quadrilateral ABCD inscribed in a circle, with AB=4, BC=5, CD=7, and DA=10. Find: (1) the value of cos A (2) the area of quadrilateral ABCD.'

'188\nMathematics I\n(1) As shown in the figure, take the vertices A, B, C of triangle T to be AB=5, BC=6, CA=7.'

'Given b=4, c=4√3, and B=30°, find a, A, C, and R.'

'In a right triangle where the sum of the lengths of the two sides is 20, find the right triangle with the minimum length of the hypotenuse, and determine the length of the hypotenuse.'

'In triangle ABC, let the radius of the circumscribed circle be R. Find the following values: (1) When A=60°, C=45°, a=3, find c and R'

'Practice (1) In the diagram on the right, find the lengths of line segments DE and AE.\n(2) Using the diagram on the right, find the following values: sin 15°, cos 15°, tan 15°'

'Please explain the Circle Theorem.'

'Consider an equilateral triangle ABC with side length of 1. Points D, E, F are taken on sides AB, BC, CA respectively such that AD=x, BE=2x, CF=3x. (1) Express the area of triangle DEF, denoted by S, in terms of x. (2) Find the value of x that minimizes S and the minimum value.'

'Please explain what a chart is based on the following text: In the Concise Oxford Dictionary (C.O.D.), a chart is described as a sea map of CHARTNavigator, including coast outlines, rocks, shoals, etc.'

'In triangle ABC, let D be the intersection point of the angle bisector of angle A and edge BC. Find the lengths of segments BD and AD in the following cases:'

'Find the area S of a regular octagon inscribed in a circle of radius a.'

'Chapter 4 Geometry and Measurement\n15 Basics of Trigonometry\nProblem\nIf one angle of a triangle is 90 degrees and another angle is 30 degrees, find the measure of the remaining angle and trigonometric ratios.'

'In △ABC, let the radius of the circumcircle be R. Find the following: (2) When a=√2, B=50°, and R=1, find the values of A and C'

'How to represent the coordinates of point P(a, b) on the coordinate plane?'

"To check if it is possible to draw more than four sets using circles, let's try to represent four sets A, B, C, D with circles. First, draw the Venn diagrams for sets A, B, C, and then try to add the Venn diagram for set D to observe the result.\n\nNext, to verify if we can draw four sets using circles, draw four different circles on a plane and count the intersections. In this case, the four circles must follow the following rules:\n- Any two circles intersect at two points\n- Any three circles do not intersect at the same point\n\nCalculate the number of regions created by the intersections of the four circles and check if this number matches the number of common parts formed by the four sets and their complements."

'Please explain the Pythagorean theorem.'

'On the coordinate plane, there is one line and two parabolas, Line L: y=ax+b, Curve C_{1}: y=-2x^{2}, Curve C_{2}: y=x^{2}-12x+33. When Line L intersects Curve C_{1} and Curve C_{2} at two points each, the inequality a^{2}-a<b<a^{2} holds, where a>0.'

'In triangle ABC, find the following.'

'Find the area of the following figure. (2) \ \\mathrm{AB}=3, \\mathrm{AC}=3 \\sqrt{3}, \\angle \\mathrm{B}=60^{\\circ} \ parallelogram \ \\mathrm{ABCD} \'

'Triangle Solution (1)'

'Calculate the values of trigonometric functions from the given coordinates. (1) P(-1,1) (2) P(-√3, 1)'

'In triangle ABC, when b=2, c=√5+1, and A=60 degrees, determine whether C is an acute angle, right angle, or obtuse angle.'

"Define the conditions p, q, r regarding triangles as follows: p: all three internal angles are different q: not a right triangle r: no internal angle is 45 degrees Choose the correct options from each choice: (1) The contrapositive of the proposition 'r implies (p or q)' is 'a _______ implies not r'. [Choices] 0 (force and q) (1) (not p and not q) (2) (to be as q) (2) What triangle serves as a counterexample to the proposition '(p or q) implies r'? [Choices] Right isosceles triangle (1) A triangle with internal angles of 30 degrees, 45 degrees, 105 degrees (2) Equilateral triangle (3) Triangle with side lengths 3, 4, 5 (4) Isosceles triangle with a vertex angle of 45 degrees (3) r is the causal relationship for (p or q) as a _______. [Choices] (0) Necessary and sufficient condition (1) Only necessary but not sufficient condition (2) Only sufficient but not necessary condition (3) Neither necessary nor sufficient condition"

'In triangle ABC, when the radius of the circumscribed circle is R, find the following: (1) Side c when C=120° and R=4. (2) Side b and radius R when A=45°, B=60°, and a=2.'

'Parabolic Antenna'

'In an acute triangle ABC, let BD and CE be the perpendiculars dropped from vertices B and C to their respective opposite sides. If BC=a and the measure of angle A is represented by A, express the length of segment DE in terms of a and A. Also, you may use the property that if angle PRQ=90 degrees for segment PQ, then point R lies on the circumference of the circle with segment PQ as its diameter.'

'There is a quadrilateral ABCD inscribed in a circle. If AB=4, BC=5, CD=7, DA=10, find the area S of quadrilateral ABCD.'

'In the quadrilateral ABCD inscribed in a circle, with AB = BC = 1, BD = √7, DA = 2, find: (1) A (2) The length of side CD (3) The area of quadrilateral ABCD'

'Minimum area of a triangle'

'Maximum angle of a triangle'

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'Area of a rectangle inscribed in a circle (1)'

'Determine the range of values for x such that the triangle with side lengths 3, 5, and x forms an acute triangle.'

'Folded a piece of origami in the shape of an equilateral triangle with side length 10 cm, denoted as ABC.'

'In a quadrilateral ABCD inscribed in a circle, with AB=8, BC=10, and CD=DA=3. Find the area S of the quadrilateral ABCD.'

'In triangle ABC, if B=30°, b=√2, and c=2, find the values of A, C, and a.'

'In triangle ABC where AB=6, BC=4, and CA=5, find the length of segment BD where the angle bisector of angle B intersects side AC at point D.'

'In triangle ABC with AB=13, BC=15, and CA=8, let AD be the perpendicular from point A to side BC. Find the following values when AD is 90 degrees: (1) Length of BD, (2) sin angle B, (3) tan angle C'

'On the line x=1, let the point with y-coordinate sqrt{3} be T. The point P where the line OT intersects the semicircle with radius 1 is the point in the figure. The angle θ we seek is ∠AOP.'

'In triangle ABC, when a=5, b=7, and c=8, find angle B.'

'In triangle ABC, where b=3, c=√2, and A=45°, find the length of side a.'

'In the diagram on the right, find the lengths of the line segments AB, BC, and CA.'

'Please explain the theorems and formulas related to the following triangle.'

'Find the equation of the parabola after symmetrically moving the parabola y = -2x^2 + 3x - 5 with respect to the following line or point. (1) x-axis (2) y-axis (3) origin'

'(1)\\[ \egin{aligned} A & =180^{\\circ}-(B+C) \\\\ & =180^{\\circ}-(30^{\\circ}+105^{\\circ}) \\\\ & =45^{\\circ} \\end{aligned} \\] Therefore, the area of triangle ABC is \\[ \egin{aligned} \\frac{1}{2} b c \\sin 45^{\\circ} & =\\frac{1}{2}(\\sqrt{6}-\\sqrt{2}) \\cdot 2 \\cdot \\frac{1}{\\sqrt{2}} \\\\ & =\\frac{\\sqrt{6}-\\sqrt{2}}{\\sqrt{2}} \\\\ & =\\frac{\\sqrt{2}(\\sqrt{3}-1)}{\\sqrt{2}} \\\\ & =\\sqrt{3}-1 \\end{aligned} \\]'

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

'In a rectangle with a length of 40 cm, what is the minimum length of the diagonal? Also, what kind of rectangle is it at that time? If the vertical length of the rectangle is x cm, then the horizontal length is (20-x) cm. Additionally, x>0 and 20-x>0, hence 0<x<20. Let the length of the diagonal of the rectangle be l cm. l^2 = x^2+(20-x)^2 = 2x^2-40x+400 = 2(x-10)^2+200. The minimum value of l^2 occurs at x=10 with a value of 200. Since l>0, when l^2 is at its minimum, l is also at its minimum. Therefore, the minimum length of the diagonal is sqrt(200)=10 sqrt(2) cm. At this point, the horizontal length is 20-x=10 cm, making the diagonal length minimum when the rectangle is a square.'

'Explain about the quadrants in a coordinate plane and give an example of the second quadrant.'

'Radius of circumcircle and incircle of a triangle'

'203 Basic triangle obtuse angle triangle conditions'

'In quadrilateral ABCD, if AB=8, BC=5, CD=DA=3, and A=60 degrees, what is the length of the diagonal BD?'

'The diagram on the right is a box plot of the scores of 30 students in a science test. When the scores that formed this box plot are represented in a histogram, which of the following 0 to 2 corresponds to it?'

'In a rectangle with a perimeter of 40 cm, find the minimum length of the diagonal. Also, determine the characteristics of the rectangle at that time.'

'A regular tetrahedron OABC with edge length 6 is given. Let L be the midpoint of edge OA, M be the point that divides edge OB into 2:1, and N be the point that divides edge OC into 1:2. Find the area of triangle LMN.'

'There is a quadrilateral ABCD inscribed in a circle. If AB = 4, BC = 5, CD = 7, DA = 10, find the area S of quadrilateral ABCD.'

'When the lengths of the three sides of ∆ABC are as follows, determine whether angle A is acute, right, or obtuse.'

'On the semicircle with radius 1, the point where the x-coordinate is 1/2 is point P. The angle we seek is ∠AOP.'

'In triangle ABC, where AB=3, AC=2, and ∠BAC=60°, if the angle bisector of angle A intersects BC at D, find the length of segment AD.'

'Given the lengths of the diagonals AC and BD of the quadrilateral ABCD as p and q, and one of the angles they form as θ, express the area S of the quadrilateral ABCD in terms of p, q, and θ.'

'In the quadrilateral ABCD, which is not a parallelogram, AD=BC. Let the midpoints of AB and CD be P and Q respectively, and the midpoints of AC and BD be M and N respectively. (1) Express →PQ, →MN in terms of →AD, →BC. (2) Prove that PQ is perpendicular to MN.'

'Polar equation of a straight line (1)'

'A circle passing through point A(-3,0) and tangent to the line x=3 has center at P(x, y). Find the locus of point P.'

'Polar equation of a circle (2)'

'Find the coordinates of point R'

'(1) Ellipse $\\frac{x^{2}}{4}+\\frac{y^{2}}{9}=1$ (2) Circle $(x-1)^{2}+(y+2)^{2}=1$ (3) Hyperbola $\\frac{(x-2)^{2}}{16}-\\frac{(y+1)^{2}}{9}=1$'

'Consider the ellipse $x^{2}+4 y^{2}=4$ and the line $y=t(x+2)$ and their intersection point $P(x, y)$. Express the ellipse excluding the point $(-2, 0)$ using the parameter $t$.'

"Ellipse\nIn this section, we will learn about the locus of points where the sum of distances from two fixed points remains constant.\nEquation of an ellipse\nIn a plane, the locus of points where the sum of distances from two fixed points F and F' remains constant. The two fixed points, F(c, 0) and F'(-c, 0) [c>0], are called the foci of the ellipse. Let's find the equation of an ellipse C with a sum of distances 2a from these two points using the concept of locus."

'Polar equation of a trajectory'

'Let k be a constant. Determine the number of intersection points between the ellipse x^{2} + 4y^{2} = 20 and the line y = (1/2)x + k.'

"What is Archimedes' spiral?"

'Find the locus of point P, such that the ratio of distances from point TR (F(0,1)) and line l: y=-1 is given. 107 (1) 1: 1 (2) 1: 2 (3) 2: 1 Let P(x, y), and let PH be the perpendicular from P to line l, then PH=|y-(-1)|=|y+1|'

'At the incenter of a triangle\n(1) In triangle ABC, where AB = 6, BC = 3, CA = 4, and the incenter is I. Express AI in terms of AB and AC.'

"Let's consider the following curves and their trajectories. (1) Let C1 be a circle with radius 7 centered at the origin, and let C2 be a circle with radius 1 centered at point F(4,0). Let P be the center of the circle that is tangent to C1 and tangent externally to C2, then, what is the relation of P? Answer choices: (0) OP*FP is constant (1) |OP-FP| is constant (2) OP+FP is constant (3) OP^2+FP^2 is constant. Therefore, point P lies on an ellipse with foci O and F, and the length of the major axis determined."

'If a circle with radius 2, x^2+y^2=25, is scaled by 3/5 in the x-axis direction with respect to the y-axis, what kind of curve will it become?'

'Circle with center at (-2i) and radius 2 (1) Circle with center at (-1/2i) and radius 3/2'

'What kind of shape does the parametric representation represent?'

'Find the minimum distance between a point P on the ellipse x^2+4y^2=4 and a point Q on the line x+2y=3.'

'Find the part obtained by removing point 2 from a circle with center at point 1 and radius 1.'

'Chapter 4 Form and Curve - Simplifying to y^{2} = -12x, therefore, point P lies on the parabola y^{2} = -12x. Conversely, all points P(x, y) on this parabola satisfy the condition. Thus, the required trajectory is the parabola y^{2} = -12x.'

"What is Apollonius' circle?"

'(1) Perpendicular bisector of the line segment connecting points 0 and 1 (2) Circle with radius 2 centered at point 3'

'Find the minimum distance between the point P on the ellipse $x^{2} + 4y^{2} = 4$ and the point Q on the line $x + 2y = 3$.'

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'Let TR be a constant. Find the number of intersection points between the ellipse x^2 + 4y^2 = 20 and the line y = \\frac{1}{2}x + k.'

'Prove that when a line passing through the focus \ \\mathrm{F} \ of the parabola \\( y^{2}=4 p x(p \\neq 0) \\) intersects the parabola at points \\\mathrm{A}, \\mathrm{B}\, the product of the \ y \ coordinates of points \\\mathrm{A}\ and \\\mathrm{B}\ remains constant.'

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'Find the locus of point P such that the ratio of distances from point F(1,0) and line ℓ: x=-2 to point P is 1:2.'

'Find the focus and directrix of the parabola $y^{2}=3 x$. Also, sketch its general shape.'

'For triangle ABC with vertices A(-1+i), B(2√3-1), and C(6+(√3+1)i), find the measure of angle BAC.'

'Find the locus of the center P(x, y) of a circle passing through the point A(4,0) and tangent to the line x = -4.'

'Given three different points A(α), B(β), C(γ) and the following relation between them, find the measures of the 86 angles in triangle ABC with these three points as vertices.'

'TRAINING 41\nFrom point P(1,3), draw a perpendicular line to the line ℓ: 2x-3y+4=0, with the intersection point being H.\n(1) Find the coordinates of point H using vectors.\n(2) Find the distance between point P and line ℓ.'

'Find the equation of a parabola with a focus and directrix as follows, and sketch its graph. (A) Point (-1,0), line x=1 (B) Point (0,2), line y=-2'

'Find triangle OAB.'