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6.44*. On the arc $A_{1} A_{2 n+1}$ of the circumscribed circle $S$ of a regular $(2 n+1)$-gon $A_{1} \ldots A_{2 n+1}$, a point $A$ is taken. Prove that:
a) $d_{1}+d_{3}+\ldots+d_{2 n+1}=d_{2}+d_{4}+\ldots+d_{2 n}$, where $d_{i}=A A_{i}$
b) $l_{1}+\ldots+l_{2 n+1}=l_{2}+\ldots+l_{2 n}$, where $l_{i}$ is the length of... | 6.44. a) Let's write Ptolemy's theorem for all quadrilaterals with vertices at point $A$ and three consecutive vertices of the given polygon; then group the factors in the obtained equations that involve $d_{i}$ with even indices to the right side. Adding these equations, we get $(2 a+b)\left(d_{1}+\ldots+d_{2 n+1}\rig... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,118 |
6.45*. Circles of radius $x$ and $y$ touch a circle of radius $R$, and the distance between the points of tangency is $a$. Calculate the length of the following common tangent to the first two circles:
a) external, if both tangencies are external or internal simultaneously;
b) internal, if one tangency is internal an... | 6.45. Let both tangencies be external and $x \leqslant y$. A line passing through the center $O$ of the circle of radius $x$ parallel to the segment connecting the points of tangency intersects the circle of radius $y-x$ (with the center at the center of the circle of radius $y$) at points $A$ and $B$ (Fig. 6.9). Then ... | (R)^{2}(R-x)(R-y) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,119 |
6.46*. Circles $\alpha, \beta, \gamma$, and $\delta$ touch a given circle at the vertices $A, B, C$, and $D$ of a convex quadrilateral $ABCD$. Let $t_{\alpha \beta}$ be the length of the common tangent to circles $\alpha$ and $\beta$ (external if both contacts are internal or external simultaneously, and internal if on... | 6.46. Let $R$ be the radius of the circumscribed circle of quadrilateral $ABCD$; $r_a$, $r_b$, $r_c$, and $r_d$ be the radii of circles $\alpha, \beta, \gamma$, and $\delta$. Let further $a = \sqrt{R \pm r_a}$, where the plus sign is taken in the case of external tangency, and the minus sign in the case of internal tan... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,120 |
6.48*. a) Diagonals $A C$ and $B E$ of a regular pentagon $A B C D E$ intersect at point $K$. Prove that the circumcircle of triangle $C K E$ is tangent to the line $B C$.
b) Let $a$ be the length of a side of a regular pentagon, and $d$ be the length of its diagonal. Prove that $d^{2}=a^{2}+a d$. | 6.48. a) Let $O$ be the center of the circumcircle of triangle $C K E$. It is sufficient to check that $\angle C O K=2 \angle K C B$. Both these angles are easily calculated: $\angle C O K=180^{\circ}-2 \angle O K C=180^{\circ}-\angle E K C=180^{\circ}-\angle E D C=72^{\circ}$ and $\angle K C B=$ $=\left(180^{\circ}-\a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,121 |
6.49*. Prove that a square can be inscribed in a regular pentagon such that its vertices will lie on four sides of the pentagon. | 6.49. Let the perpendiculars erected to the line $A B$ at points $A$ and $B$ intersect the sides $D E$ and $C D$ at points $P$ and $Q$. Any point on the segment $C Q$ is a vertex of a rectangle inscribed in the pentagon $A B C D E$ (the sides of this rectangle are parallel to $A B$ and $A P$), and as this point moves f... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,122 |
6.51*. The opposite sides of a convex hexagon $A B C D E F$ are pairwise parallel. Prove that:
a) the area of triangle $A C E$ is at least half the area of the hexagon.
b) the areas of triangles $A C E$ and $B D F$ are equal. | 6.51. Draw lines \( l_{1}, l_{2} \) and \( l_{3} \) through points \( A, C \) and \( E \), parallel to lines \( BC, DE \) and \( FA \) respectively. Denote the points of intersection of lines \( l_{1} \) and \( l_{2}, l_{2} \) and \( l_{3}, l_{3} \) and \( l_{1} \) by \( P, Q, R \) respectively (Fig. 6.11). Then
\[
S_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,124 |
6.52*. All angles of the convex hexagon $A B C D E F$ are equal. Prove that $|B C-E F|=|D E-A B|=|A F-C D|$. | 6.52. Let's construct triangle $P Q R$, as in the previous problem. This triangle is equilateral, and $P Q=|A B-D E|, Q R=$ $=|C D-A F|, P R=|E F-B C|$. Therefore, $|A B-D E|=|C D-A F|=|E F-B C|$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,125 |
6.53*. The sums of the angles at vertices $A, C, E$ and $B, D, F$ of a convex hexagon $A B C D E F$ with equal sides are equal. Prove that the opposite sides of this hexagon are parallel. | 6.53. The sum of the angles at vertices $A, C$ and $E$ is $360^{\circ}$, so from the isosceles triangles $A B F, C B D$ and $E D F$, a triangle can be formed by attaching $A B$ to $C B$, and $E D$ and $E F$ to $C D$ and $A F$. The sides of the resulting triangle are equal to the sides of triangle $B D F$. Therefore, un... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,126 |
6.54*. Prove that if in a convex hexagon each of the three diagonals connecting opposite vertices divides the area in half, then these diagonals intersect at one point. | 6.54. Suppose the diagonals of a hexagon form a triangle $P Q R$. Let's denote the vertices of the hexagon as follows: vertex $A$ lies on the ray $Q P$, $B$ on $R P$, $C$ on $R Q$, and so on. Since the lines $A D$ and $B E$ divide the area of the hexagon in half, we have $S_{A P E F} + S_{P E D} = S_{P D C B} + S_{A B ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,127 |
6.55*. Prove that if in a convex hexagon each of the three segments connecting the midpoints of opposite sides divides the area in half, then these segments intersect at one point.
See also problems $1.45,2.11,2.20,2.47,3.67,4.6,4.28,4.31,5.88,6.95$, 9.45 a), 9.76-9.78, 13.3, 14.6, 18.16, 18.17, 18.24, 18.25, 29.29 a)... | 6.55. Let the midpoints of the sides of the convex hexagon $A B C D E F$ be denoted as shown in Fig. 6.12. Let $O$ be the point of intersection of segments $K M$ and $L N$. The areas of the triangles into which the segments connecting point $O$ with the vertices and midpoints of the sides divide the hexagon are denoted... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,128 |
6.56. The number of sides of the polygon $A_{1} \ldots A_{n}$ is odd. Prove that:
a) if this polygon is inscribed and all its angles are equal, then it is regular;
b) if this polygon is circumscribed and all its sides are equal, then it is regular. | 6.56. a) Let $O$ be the center of the circumscribed circle. Since $\angle A_{k} O A_{k+2} = 360^{\circ} - 2 \angle A_{k} A_{k+1} A_{k+2} = \varphi$ is a constant value, then under a rotation centered at $O$ by the angle $\varphi$, point $A_{k}$ transitions to $A_{k+2}$. For odd $n$, it follows from this that all sides ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,129 |
6.59*. On the sides $AB, BC, CD$, and $DA$ of the square $ABCD$, internal equilateral triangles $ABK, BCL, CDM$, and $DAN$ are constructed. Prove that the midpoints of the sides of these triangles (which are not sides of the square) and the midpoints of the segments $KL, LM, MN$, and $NK$ form a regular dodecagon. | 6.59. Triangle $B M C$ is isosceles with an angle of $30^{\circ}$ at the vertex and an angle of $\left(180^{\circ}-30^{\circ}\right) / 2=75^{\circ}$ at the base. Therefore, triangles $B A M$ and $B C N$ are isosceles with an angle of $15^{\circ}$ at the base. Hence, triangle $B M N$ is equilateral. Let $O$ be the cente... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,131 |
6.60*. Does there exist a regular polygon in which the length of one diagonal is equal to the sum of the lengths of two other diagonals? | 6.60. Consider a regular dodecagon \(A_{1} \ldots A_{12}\) inscribed in a circle of radius \(R\). It is clear that \(A_{1} A_{7}=2 R, A_{1} A_{3}=A_{1} A_{11} = R\). Therefore, \(A_{1} A_{7}=A_{1} A_{3}+A_{1} A_{11}\).
$-gon is inscribed in a circle of radius $R$ with center $O$. Prove that the sum of the lengths of the segments cut off by the angle $A_{k} O A_{k+1}$ on the lines $A_{1} A_{2 k}, A_{2} A_{2 k-1}, \ldots, A_{k} A_{k+1}$, is $R$. | 6.61. For $k=3$ the solution of the problem is clear from Fig. 6.15. Indeed, $A_{3} A_{4}=O Q, K L=Q P$ and $M N=P A_{14}$, therefore $A_{3} A_{4}+$ $+K L+M N=O Q+Q P+P A_{14}=O A_{14}=R$. The proof is conducted similarly for any $k$.
 $\{k, m, n\}=\{p, q, r\}$
b) $\{k, m... | 6.62. To prove it, it is sufficient to apply the results of problems 5.86 and 5.77, b) to the triangle $A_{a} A_{c} A_{e}$ and the lines $A_{a} A_{d}, A_{c} A_{f}$ and $A_{e} A_{b}$. In solving problem b), it is also necessary to note that $\sin 20^{\circ} \sin 70^{\circ}=\sin 20^{\circ} \cos 20^{\circ}=$ $=\left(\sin ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,134 |
6.63*. In a regular triacontagon, three diagonals are drawn. Let's define sets $\{k, m, n\}$ and $\{p, q, r\}$ for them in the same way as in the previous problem. Prove that if $\{k, m, n\}=\{1,3,14\}$ and $\{p, q, r\}=\{2,2,8\}$, then the diagonals intersect at one point. | 6.63. As in the previous problem, we need to verify the equality $\sin 2 \alpha \sin 2 \alpha \sin 8 \alpha = \sin \alpha \sin 3 \alpha \sin 14 \alpha$, where $\alpha = 180^{\circ} / 30 = 6^{\circ}$. It is clear that $\sin 14 \alpha = \cos \alpha$, so $2 \sin \alpha \sin 3 \alpha \sin 14 \alpha = \sin 2 \alpha \sin 3 \... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,135 |
6.64*. In a regular $n$-gon ( $n \geqslant 3$ ), the midpoints of all sides and diagonals are marked. What is the maximum number of marked points that can lie on one circle? | 6.64. Let first $n=2 m$. The diagonals and sides of a regular $2 m$-gon have $m$ different lengths. Therefore, the marked points lie on $m-1$ concentric circles (with $n$ points on each) or at the common center of these circles. Since different circles have no more than two common points, a circle not belonging to this... | n | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,136 |
6.65*. The vertices of a regular $n$-gon are colored in several colors such that the points of one color are the vertices of a regular polygon. Prove that among these polygons, there will be two equal ones.
6.66* ${ }^{*}$ Prove that for $n \geqslant 6$, a regular $(n-1)$-gon cannot be inscribed in a regular $n$-gon i... | 6.65. Let the center of the polygon be denoted by $O$, and the vertices by $A_{1}, \ldots, A_{n}$. Suppose that among the monochromatic polygons, there are no equal ones, i.e., they have $m=m_{1}<m_{2}<m_{3}<\ldots<m_{k}$ sides respectively. Consider the transformation $f$ defined on the set of vertices of the $n$-poly... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,137 |
6.67. Let $O$ be the center of a regular $n$-gon $A_{1} \ldots A_{n}, X$ be an arbitrary point. Prove that $\overrightarrow{O A}_{1}+\ldots+\overrightarrow{O A}_{n}=\overrightarrow{0}$ and $\overrightarrow{X A}_{1}+\ldots+\overrightarrow{X A}_{n}=$ $=n \overrightarrow{X O}$ | 6.67. Let $\boldsymbol{a}=\overrightarrow{O A}_{1}+\ldots+\overrightarrow{O A}_{n}$. When rotating around point $O$ by an angle of $360^{\circ} / n$, point $A_{i}$ transitions to $A_{i+1}$, and thus vector $\boldsymbol{a}$ transitions to itself, i.e., $\boldsymbol{a}=\mathbf{0}$.
Since $\overrightarrow{X A}_{i}=\overr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,138 |
6.68. Prove that in the vertices of a regular $n$-gon, one can place real numbers $x_{1}, \ldots, x_{n}$, all different from zero, such that for any regular $k$-gon, all vertices of which are vertices of the original $n$-gon, the sum of the numbers standing at its vertices is zero. | 6.68. Draw a line $l$ through the center of a regular polygon $A_{1} \ldots A_{n}$, not passing through its vertices. Let $x_{i}$ be the projection of the vector $\overrightarrow{O A}_{i}$ onto a line perpendicular to line $l$. Then all $x_{i}$ are non-zero and the sum of the numbers $x_{i}$ at the vertices of a regula... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 27,139 |
6.69. Point $A$ lies inside the regular decagon $X_{1} \ldots X_{10}$, and point $B$ lies outside it. Let $\boldsymbol{a}=\overrightarrow{A X}_{1}+\ldots+\overrightarrow{A X}_{10}$ and $\boldsymbol{b}=\overrightarrow{B X}_{1}+\ldots+\overrightarrow{B X}_{10}$. Can it happen that $|\boldsymbol{a}|>|\boldsymbol{b}|$? | 6.69. According to problem $6.67 \boldsymbol{a}=10 \overrightarrow{A O}$ and $\boldsymbol{b}=10 \overrightarrow{B O}$, where $O$ is the center of the polygon $X_{1} \ldots X_{10}$. It is clear that if point $A$ is located very close to a vertex of the polygon, and point $B$ is very close to the midpoint of a side, then... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,140 |
6.70. A regular polygon $A_{1} \ldots A_{n}$ is inscribed in a circle of radius $R$ with center $O$; $X$ is an arbitrary point. Prove that $A_{1} X^{2}+\ldots+$ $+A_{n} X^{2}=n\left(R^{2}+d^{2}\right)$, where $d=O X$. | 6.70. Since
$$
A_{i} X^{2}=\overrightarrow{A_{i} O}+\overrightarrow{O X}^{2}=A_{i} O^{2}+O X^{2}+2\left(\overrightarrow{A_{i} O}, \overrightarrow{O X}\right)=R^{2}+d^{2}+2\left(\overrightarrow{A_{i} O}, \overrightarrow{O X}\right)
$$
then $\sum A_{i} X^{2}=n\left(R^{2}+d^{2}\right)+2 \quad \sum \overrightarrow{A_{i} ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,141 |
6.71. Find the sum of the squares of the lengths of all sides and diagonals of a regular $n$-gon inscribed in a circle of radius $R$. | 6.71. Let $S_{k}$ denote the sum of the squares of the distances from vertex $A_{k}$ to all other vertices. Then
$$
\begin{aligned}
S_{k}= & A_{k} A_{1}^{2}+A_{k} A_{2}^{2}+\ldots+A_{k} A_{n}^{2}= \\
& =A_{k} O^{2}+2\left(\overrightarrow{A_{k} O}, \overrightarrow{O A}_{1}\right)+A_{1} O^{2}+\ldots+A_{k} O^{2}+2\left(\... | n^{2}R^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,142 |
6.72. Prove that the sum of the distances from an arbitrary point $X$ to the vertices of a regular $n$-gon will be the smallest if $X$ is the center of the $n$-gon. | 6.72. Let $X_{k}$ be the image of point $X$ under the rotation about the center $O$ of the given $n$-gon, which maps $A_{k}$ to $A_{1}$. Under this rotation, the segment $A_{k} X$ is transformed into $A_{1} X_{k}$. Therefore, $A_{1} X+\ldots+A_{n} X=A_{1} X_{1}+\ldots+A_{1} X_{n}$. Since the $n$-gon $X_{1} \ldots X_{n}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,143 |
6.73. A regular $n$-gon $A_{1} \ldots A_{n}$ is inscribed in a circle of radius $R$ with center $O ; \boldsymbol{e}_{i}=\overrightarrow{O A}_{i}, \boldsymbol{x}_{i}=\overrightarrow{O X}-$ an arbitrary vector. Prove that $\sum\left(\boldsymbol{e}_{i}, \boldsymbol{x}\right)^{2}=n R^{2} \cdot O X^{2} / 2$. | 6.73. Let $B_{i}$ be the projection of point $X$ onto the line $O A_{i}$. Then $\left(\boldsymbol{e}_{i}, \boldsymbol{x}\right)=$ $=\left(\overrightarrow{O A}_{i}, \overrightarrow{O B}_{i}+\overrightarrow{B_{i} X}\right)=\left(\overrightarrow{O A}_{i}, \overrightarrow{O B}_{i}\right)= \pm R \cdot O B_{i}$. The points $... | \sumOB_{i}^{2}=nOX^{2}/2 | Geometry | proof | Yes | Yes | olympiads | false | 27,144 |
6.74. Find the sum of the squares of the distances from the vertices of a regular $n$-gon inscribed in a circle of radius $R$ to an arbitrary line passing through the center of the polygon. | 6.74. Let $e_{1}, \ldots, e_{n}$ be vectors from the center of a given $n$-gon to its vertices; $\boldsymbol{x}$ be a unit vector perpendicular to the line $l$. The desired sum is $\sum\left(\boldsymbol{e}_{i}, \boldsymbol{x}\right)^{2}=n R^{2} / 2$ (see problem 6.73 ). | nR^{2}/2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,145 |
6.75. The distance from point $X$ to the center of a regular $n$-gon is $d$, and $r$ is the radius of the inscribed circle of the $n$-gon. Prove that the sum of the squares of the distances from point $X$ to the lines containing the sides of the $n$-gon is $n\left(r^{2}+d^{2} / 2\right)$. | 6.75. Let $e_{1}, \ldots, e_{n}$ be unit vectors directed from the center $O$ of a regular $n$-gon to the midpoints of its sides; $\boldsymbol{x}=\overrightarrow{O X}$. Then the distance from point $X$ to the $i$-th side is $\left|\left(\boldsymbol{x}, \boldsymbol{e}_{i}\right)-r\right|$. Therefore, the desired sum is ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,146 |
6.76. Prove that the sum of the squares of the lengths of the projections of the sides of a regular $n$-gon onto any line is equal to $n a^{2} / 2$, where $a$ is the side length of the $n$-gon.
6.77* . A regular $n$-gon $A_{1} \ldots A_{n}$ is inscribed in a circle of radius $R$; $X$ is a point on this circle. Prove t... | 6.76. Let $\boldsymbol{x}$ be a unit vector parallel to the line $l, \boldsymbol{e}_{i}={\overrightarrow{A_{i} A_{i+1}}}$. Then the square of the length of the projection of the side $A_{i} A_{i+1}$ onto the line $l$ is $\left(\boldsymbol{x}, \boldsymbol{e}_{i}\right)^{2}$. According to problem $6.73 \sum\left(\boldsym... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,147 |
6.78*. a) A regular $n$-gon $A_{1} \ldots A_{n}$ is inscribed in a circle of radius 1 with center $O ; \boldsymbol{e}_{i}=\overrightarrow{O A}_{i}, \boldsymbol{u}$ is an arbitrary vector. Prove that $\sum\left(\boldsymbol{u}, \boldsymbol{e}_{i}\right) \boldsymbol{e}_{i}=n \boldsymbol{u} / 2$.
b) Perpendiculars $X A_{1... | 6.78. a) First, let's prove the required relation for $\boldsymbol{u}=\boldsymbol{e}_{1}$. Let $\boldsymbol{e}_{i}=$ $=\left(\sin \varphi_{i}, \cos \varphi_{i}\right)$, and $\cos \varphi_{1}=1$. Then $\sum\left(\boldsymbol{e}_{1}, \boldsymbol{e}_{i}\right) \boldsymbol{e}_{i}=\sum \cos \varphi_{i} \boldsymbol{e}_{i}=$ $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,148 |
6.79*. Prove that if a number $n$ is not a power of a prime number, then there exists a convex $n$-gon with side lengths $1, 2, \ldots, n$, all of whose angles are equal.
See also problems $2.9, 4.59, 4.62, 6.39, 6.44, 6.48-6.50, 9.83, 9.84, 11.46$, $11.48, 17.31, 18.32, 19.48, 23.8, 24.2$.
## §7. Inscribed and Circu... | 6.79. Let $\boldsymbol{e}_{0}, \ldots, \boldsymbol{e}_{n-1}$ be the vectors of the sides of a regular $n$-gon. It is sufficient to prove that by reordering these vectors, we can obtain a set of vectors $\left\{\boldsymbol{a}_{1}, \ldots, \boldsymbol{a}_{n}\right\}$ such that $\sum_{k=1}^{n} k \boldsymbol{a}_{k}=\mathbf... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,149 |
6.80*. On the sides of a triangle, three squares are constructed externally. What should the angles of the triangle be so that the six vertices of these squares, distinct from the vertices of the triangle, lie on one circle? | 6.80. Suppose that on the sides of triangle $A B C$, squares $A B B_{1} A_{1}, B C C_{2} B_{2}, A C C_{3} A_{3}$ are constructed externally, and the vertices $A_{1}, B_{1}, B_{2}, C_{2}, C_{3}, A_{3}$ lie on a single circle $S$. The perpendicular bisectors of segments $A_{1} B_{1}$, $B_{2} C_{2}$, $A_{3} C_{3}$ pass th... | Thetrianglemustbeeitherequilateralorisoscelesright. | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,150 |
6.81*. A $2 n$-gon $A_{1} \ldots A_{2 n}$ is inscribed in a circle. Let $p_{1}, \ldots, p_{2 n}$ be the distances from an arbitrary point $M$ on the circle to the sides $A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{2 n} A_{1}$. Prove that $p_{1} p_{3} \ldots p_{2 n-1}=p_{2} p_{4} \ldots p_{2 n}$. | 6.81. In any triangle, the relation $h_{c}=a b / 2 R$ (Problem 12.33) holds, so $p_{k}=M A_{k} \cdot M A_{k+1} / 2 R$. Therefore,
$$
p_{1} p_{3} \ldots p_{2 n-1}=M A_{1} \cdot M A_{2} \ldots M A_{2 n} /(2 R)^{n}=p_{2} p_{4} \ldots p_{2 n}
$$ | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,151 |
6.82*. An inscribed polygon is divided into triangles by non-intersecting diagonals. Prove that the sum of the radii of all the circles inscribed in these triangles does not depend on the division. | 6.82. Let $A B C$ be a triangle inscribed in a circle $S$. Denote the distances from the center $O$ of the circle to the sides $B C, C A$, and $A B$ by $a, b$, and $c$ respectively. Then $R+r=a+b+c$ if the point $O$ lies inside the triangle $A B C$, and $R+r=-a+b+c$ if the points $O$ and $A$ lie on opposite sides of th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,152 |
6.83*. Two $n$-gons are inscribed in the same circle, and the sets of their side lengths are the same, but the corresponding sides are not necessarily equal. Prove that the areas of these polygons are equal. | 6.83. Let the polygon $A_{1} \ldots A_{n}$ be inscribed in a circle. Consider the point $A_{2}^{\prime}$, which is symmetric to the point $A_{2}$ with respect to the perpendicular bisector of the segment $A_{1} A_{3}$. Then the polygon $A_{1} A_{2}^{\prime} A_{3} \ldots A_{n}$ is also inscribed, and its area is equal t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,153 |
6.84*. Positive numbers $a_{1}, \ldots, a_{n}$ are such that $2 a_{i}<a_{1}+\ldots+a_{n}$ for all $i=1, \ldots, n$. Prove that there exists an inscribed $n$-gon, the lengths of whose sides are $a_{1}, \ldots, a_{n}$.
$$
* * *
$$ | 6.84. Without loss of generality, we can assume that $a_{n}$ is the largest of the numbers $a_{1}, \ldots, a_{n}$. Let the $n$-gon $A_{1} \ldots A_{n}$ be inscribed in a circle with center $O$. Then $A_{i} A_{i+1}: A_{1} A_{n}=\sin \left(\angle A_{i} O A_{i+1} / 2\right): \sin \left(\angle A_{1} O A_{n} / 2\right)$. Th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,154 |
6.85. A point lying inside a circumscribed $n$-gon is connected by segments to all vertices and points of tangency. The triangles formed in this way are alternately colored red and blue. Prove that the product of the areas of the red triangles is equal to the product of the areas of the blue triangles.
6.86* ${ }^{*}$... | 6.85. Let $h_{1}, \ldots, h_{n}$ be the distances from a given point to the corresponding sides, and $a_{1}, \ldots, a_{n}$ be the distances from the vertices of the polygon to the points of tangency. Then the product of the areas of both the red and blue triangles is $a_{1} \ldots a_{n} h_{1} \ldots h_{n} / 2^{n}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,155 |
6.87*. A circle of radius $r$ touches the sides of a polygon at points $A_{1}, \ldots, A_{n}$, and the length of the side containing point $A_{i}$ is $a_{i}$. Point $X$ is at a distance $d$ from the center of the circle. Prove that $a_{1} X A_{1}^{2}+\ldots+a_{n} X A_{n}^{2}=P\left(r^{2}+d^{2}\right)$, where $P-$ is th... | 6.87. Let $O$ be the center of the given circle. Then $\overrightarrow{X A}_{i}=\overrightarrow{X O}+\overrightarrow{O A}_{i}$, and therefore, $X A_{i}^{2}=X O^{2}+O A_{i}^{2}+2\left(\overrightarrow{X O}, \overrightarrow{O A}_{i}\right)=d^{2}+r^{2}+2\left(\overrightarrow{X O}, \overrightarrow{O A}_{i}\right)$. Since $a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,156 |
6.88*. A circle is circumscribed around an $n$-sided polygon $A_{1} \ldots A_{n} ; l$ is an arbitrary tangent to the circle, not passing through the vertices of the $n$-sided polygon. Let $a_{i}$ be the distance from vertex $A_{i}$ to the line $l$, and $b_{i}$ be the distance from the point of tangency of side $A_{i} A... | 6.88. According to problem $5.8, b_{i-1} b_{i} / a_{i}^{2}=\sin ^{2}\left(A_{i} / 2\right)$. To solve problem a), it is sufficient to multiply all such equations, and for solving problem b), the product of equations with even index $i$ needs to be divided by the product of equations with odd index $i$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,157 |
6.89*. Some sides of a convex polygon are red, the others are blue. The sum of the lengths of the red sides is less than half the perimeter, and there is no pair of adjacent blue sides. Prove that it is impossible to inscribe a circle in this polygon.
See also problems $2.12,4.39,19.6$.
## §8. Arbitrary Convex Polygo... | 6.89. Let $BC$ be the blue side, and $AB$ and $CD$ be the sides adjacent to $BC$. According to the problem, the sides $AB$ and $CD$ are red. Suppose the polygon is circumscribed; $P, Q, R$ are the points of tangency of the sides $AB, BC, CD$ with the inscribed circle. Clearly, $BP = BQ$, $CR = CQ$, and the segments $BP... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,158 |
6.90. What is the maximum number of acute angles a convex polygon can have
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 6.90. Let a convex $n$-gon have $k$ acute angles. Then the sum of its angles is less than $k \cdot 90^{\circ} + (n-k) \cdot 180^{\circ}$. On the other hand, the sum of the angles of an $n$-gon is $(n-2) \cdot 180^{\circ}$. Therefore, $(n-2) \cdot 180^{\circ} < k \cdot 90^{\circ} + (n-k) \cdot 180^{\circ}$, i.e., $k < 4... | 3 | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 27,159 |
6.92*. For which $n$ does there exist a convex $n$-gon, one of whose sides has length 1, and the lengths of all diagonals are integers? | 6.92. We will prove that $n \leqslant 5$. Let $AB=1$, and let $C$ be a vertex not adjacent to either $A$ or $B$. Then $|AC - BC| < AB = 1$. Therefore, $AC = BC$, i.e., point $C$ lies on the perpendicular bisector of side $AB$. Thus, besides vertices $A, B, C$, the polygon can have only two more vertices.
An example of... | n\leqslant5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,161 |
6.93*. Can a convex irregular pentagon have exactly four sides of the same length and exactly four diagonals of the same length?
Can the fifth side of such a pentagon share a point with the fifth diagonal? | 6.93. An example of a pentagon satisfying the condition of the problem is shown in Fig. 6.20. Let's explain how it is constructed. Take an isosceles right triangle \(EAB\), draw the perpendicular bisectors of the sides \(EA\) and \(AB\), and on these bisectors construct points \(C\) and \(D\) such that \(ED = BC = AB\)... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,162 |
6.95*. Prove that the points of intersection of opposite sides (if these sides are not parallel) of an inscribed hexagon lie on one straight line (Pascal).
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 6.95. Let lines $A B$ and $D E$ intersect at point $G$, $B C$ and $E F$ at point $H$, and $C D$ and $F A$ at point $K$. Let $X$ and $Y$ be the points of intersection of the circumcircle of triangle $E B H$ with lines $A B$ and $D E$. We will show that the corresponding sides of triangles $A D K$ and $X Y H$ are paralle... | proof | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 27,164 |
6.96*. Point $M$ lies on the circumcircle of triangle $ABC$; $R$ is an arbitrary point. Lines $AR, BR$, and $CR$ intersect the circumcircle at points $A_{1}, B_{1}$, and $C_{1}$. Prove that the points of intersection of lines $M A_{1}$ and $BC, M B_{1}$ and $CA, M C_{1}$ and $AB$ lie on a single line passing through po... | 6.96. Let $A_{2}, B_{2}$ and $C_{2}$ be the specified points of intersection of the lines. Applying Pascal's theorem to the points $M, A_{1}, A, C, B, B_{1}$, we obtain that the points $A_{2}, B_{2}$ and $R$ lie on the same line. Similarly, the points $A_{2}, C_{2}$ and $R$ lie on the same line. Therefore, the points $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,165 |
6.97*. Given a triangle $A B C$ and some point $T$. Let $P$ and $Q$ be the feet of the perpendiculars dropped from point $T$ to the lines $A B$ and $A C$ respectively, and let $R$ and $S$ be the feet of the perpendiculars dropped from point $A$ to the lines $T C$ and $T B$ respectively. Prove that the intersection poin... | 6.97. Since the angles $A P T, A R T, A S T$ and $A Q T$ are right angles, the points $A, P, R$, $T, S, Q$ lie on a circle constructed on the segment $A T$ as its diameter. Therefore, by Pascal's theorem (problem 30.50) the points $B, C$ and $X$ lie on the same line. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,166 |
6.98*. In triangle $A B C$, the altitudes $A A_{1}$ and $B B_{1}$ and the angle bisectors $A A_{2}$ and $B B_{2}$ are drawn; the incircle touches the sides $B C$ and $A C$ at points $A_{3}$ and $B_{3}$. Prove that the lines $A_{1} B_{1}, A_{2} B_{2}$, and $A_{3} B_{3}$ either intersect at one point or are parallel. | 6.98. Points $A_{1}$ and $B_{1}$ lie on the circle $S$ with diameter $A B$. Let $A_{4}$ and $B_{4}$ be the points of intersection of the lines $A A_{2}$ and $B B_{2}$ with the line $A_{3} B_{3}$. According to problem 2.42, a) these points lie on the circle $S$. The lines $A_{1} B$ and $A_{4} A$ intersect at point $A_{2... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,167 |
6.99*. Quadrilateral $ABCD$ is inscribed in circle $S$; $X$ is an arbitrary point, $M$ and $N$ are the second points of intersection of lines $XA$ and $XD$ with circle $S$. Lines $DC$ and $AX$, $AB$ and $DX$ intersect at points $E$ and $F$. Prove that the point of intersection of lines $MN$ and $EF$ lies on line $BC$. | 6.99. Let $K$ be the point of intersection of the lines $B C$ and $M N$. Applying Pascal's theorem to the points $A, M, N, D, C, B$, we get that the points $E, K$ and $F$ lie on the same line, which means that $K$ is the point of intersection of the lines $M N$ and $E F$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,168 |
6.100*. Quadrilateral $ABCD$ is inscribed in a circle with center $O$. Point $X$ is such that $\angle BAX = \angle CDX = 90^{\circ}$. Prove that the intersection point of the diagonals of quadrilateral $ABCD$ lies on the line $XO$. | 6.100. Let points $B_{1}$ and $C_{1}$ be symmetric to points $B$ and $C$ with respect to point $O$. Then point $X$ lies on the lines $A B_{1}$ and $C_{1} D$. Applying Pascal's theorem to the hexagon $A B_{1} B D C_{1} C$. The lines $A B_{1}$ and $D C_{1}$ intersect at point $X$, the lines $B B_{1}$ and $C C_{1}$ inters... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,169 |
6.101*. Points $A$ and $A_{1}$, lying inside a circle with center $O$, are symmetric with respect to point $O$. Rays $A P$ and $A_{1} P_{1}$ are collinear, rays $A Q$ and $A_{1} Q_{1}$ are also collinear. Prove that the point of intersection of lines $P_{1} Q$ and $P Q_{1}$ lies on the line $A A_{1}$. (Points $P, P_{1}... | 6.101. Let rays $P A$ and $Q A$ intersect the circle at points $P_{2}$ and $Q_{2}$, i.e., $P_{1} P_{2}$ and $Q_{1} Q_{2}$ are diameters of the given circle. Apply Pascal's theorem to the hexagon $P P_{2} P_{1} Q Q_{2} Q_{1}$. The lines $P P_{2}$ and $Q Q_{2}$ intersect at point $A$, and the lines $P_{1} P_{2}$ and $Q_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,170 |
6.102*. Two circles touch the circumcircle of triangle $A B C$ at point $K$; moreover, one of these circles touches side $A B$ at point $M$, and the other touches side $A C$ at point $N$. Prove that the center of the inscribed circle of triangle $A B C$ lies on the line $M N$. | 6.102. Let $B_{1}$ and $C_{1}$ be the midpoints of the arcs $A C$ and $A B$ (the arcs not containing points $B$ and $C$ respectively). According to problem 3.42, a) points $M$ and $N$ lie on segments $K C_{1}$ and $K B_{1}$.
Applying Pascal's theorem to the hexagon $C_{1} C A B B_{1} K$. The lines $C C_{1}$ and $B B_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,171 |
6.103*. Given five points of a certain circle. Using only a straightedge, construct a sixth point of this circle.
保留源文本的换行和格式，翻译结果如下：
6.103*. Given five points of a certain circle. Using only a straightedge, construct a sixth point of this circle. | 6.103. Let points $A, B, C, D, E$ lie on the same circle. Suppose we construct point $F$ on the same circle. Denote by $K, L, M$ the points of intersection of the lines $A B$ and $D E$, $B C$ and $E F$, $C D$ and $F A$, respectively. Then, by Pascal's theorem, points $K, L, M$ lie on the same line.
From this, the foll... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,172 |
6.104*. Points $A_{1}, \ldots, A_{6}$ lie on the same circle, and points $K, L, M$ and $N$ lie on the lines $A_{1} A_{2}, A_{3} A_{4}, A_{1} A_{6}$ and $A_{4} A_{5}$ respectively, such that $K L\left\|A_{2} A_{3}, L M\right\| A_{3} A_{6}$ and $M N \| A_{6} A_{5}$. Prove that $N K \| A_{5} A_{2}$.
## Problems for Indep... | 6.104. Let $P$ and $Q$ be the points of intersection of the line $A_{3} A_{4}$ with $A_{1} A_{2}$ and $A_{1} A_{6}$, and let $R$ and $S$ be the points of intersection of the line $A_{4} A_{5}$ with $A_{1} A_{6}$ and $A_{1} A_{2}$. Then $A_{2} K: A_{3} L = A_{2} P: A_{3} P$, $A_{3} L: A_{6} M = A_{3} Q: A_{6} Q$, and $A... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,173 |
7.1. Two wheels of radii $r_{1}$ and $r_{2}$ roll along a straight line $l$. Find the set of points of intersection $M$ of their common internal tangents. | 7.1. Let $O_{1}$ and $O_{2}$ be the centers of wheels with radii $r_{1}$ and $r_{2}$ respectively. If $M$ is the point of intersection of the internal tangents, then $O_{1} M: O_{2} M=r_{1}: r_{2}$. From this condition, it is easy to obtain that the distance from point $M$ to line $l$ is $2 r_{1} r_{2} /\left(r_{1}+r_{... | \frac{2r_{1}r_{2}}{r_{1}+r_{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,174 |
7.2. Sides $A B$ and $C D$ of quadrilateral $A B C D$ with area $S$ are not parallel. Find the locus of points $X$ lying inside the quadrilateral for which $S_{A B X} + S_{C D X} = S / 2$. | 7.2. Let $O$ be the point of intersection of the lines $A B$ and $C D$. On the rays $O A$ and $O D$, lay off segments $O K$ and $O L$, equal to $A B$ and $C D$ respectively. Then $S_{A B X}+S_{C D X}=$ $=S_{K O X}+S_{L O X} \pm S_{K X L}$. Therefore, the area of triangle $K X L$ is constant, i.e., point $X$ lies on a l... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,175 |
7.3. Given two lines intersecting at point $O$. Find the locus of points $X$ for which the sum of the lengths of the projections of the segment $O X$ onto these lines is constant. | 7.3. Let $\boldsymbol{a}$ and $\boldsymbol{b}$ be unit vectors parallel to the given lines; $\boldsymbol{x} = \overrightarrow{O X}$. The sum of the lengths of the projections of the vector $\boldsymbol{x}$ onto the given lines is $|(\boldsymbol{a}, \boldsymbol{x})| + |(\boldsymbol{b}, \boldsymbol{x})| = |(\boldsymbol{a... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,176 |
7.4. Given a rectangle $A B C D$. Find the locus of points $X$ for which $A X+$ $+B X=C X+D X$ | 7.4. Let $l$ be a line passing through the midpoints of sides $B C$ and $A D$. Suppose that point $X$ does not lie on line $l$, for example, that points $A$ and $X$ lie on the same side of line $l$. Then $A X < D X$ and $B X < C X$, so $A X + B X < C X + D X$. Therefore, line $l$ is the desired locus of points. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,177 |
7.5*. Find the geometric locus of points $M$, lying inside the rhombus $A B C D$ and having the property that $\angle A M D + \angle B M C = 180^{\circ}$.
$$
* * *
$$ | 7.5. Let $N$ be such a point that $\overrightarrow{M N}=\overrightarrow{D A}$. Then $\angle N A M=\angle D M A$ and $\angle N B M=\angle B M C$, so the quadrilateral $A M B N$ is cyclic. The diagonals of the cyclic quadrilateral $A M B N$ are equal, so $A M \| B N$ or $B M \| A N$. In the first case, $\angle A M D=\ang... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,178 |
7.6. On a plane, points $A$ and $B$ are given. Find the locus of points $M$ for which the difference of the squares of the lengths of segments $A M$ and $B M$ is constant. | 7.6. Let's introduce a coordinate system, choosing point $A$ as the origin and directing the $O x$ axis along the ray $A B$. Let point $M$ have coordinates $(x, y)$. Then $A M^{2}=x^{2}+y^{2}$ and $B M^{2}=(x-a)^{2}+y^{2}$, where $a=A B$. Therefore, $A M^{2}-B M^{2}=$ $=2 a x-a^{2}$. This value is equal to $k$ for poin... | \frac{^2+k}{2a} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,179 |
7.7. Given a circle $S$ and a point $M$ outside it. Through the point $M$, all possible circles $S_{1}$ intersecting the circle $S$ are drawn; $X$ is the point of intersection of the tangent at point $M$ to the circle $S_{1}$ with the extension of the common chord of circles $S$ and $S_{1}$. Find the locus of $X$. | 7.7. Let $A$ and $B$ be the points of intersection of circles $S$ and $S_{1}$. Then $X M^{2}=$ $=X A \cdot X B=X O^{2}-R^{2}$, where $O$ and $R$ are the center and radius of circle $S$. Therefore, $X O^{2}-X M^{2}=R^{2}$, which means that points $X$ lie on the perpendicular to the line $O M$ (see problem 7.6). | XO^{2}-XM^{2}=R^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,180 |
7.8. Given two non-intersecting circles. Find the geometric locus of the centers of circles that bisect the given circles (i.e., intersect them at diametrically opposite points). | 7.8. Let $O_{1}$ and $O_{2}$ be the centers of the given circles, and $R_{1}$ and $R_{2}$ their radii. A circle of radius $r$ with center $X$ intersects the first circle at diametrically opposite points if and only if $r^{2}=X O_{1}^{2}+R_{1}^{2}$, therefore the desired locus of points consists of points $X$ such that ... | XO_{1}^{2}+R_{1}^{2}=XO_{2}^{2}+R_{2}^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,181 |
7.9. Inside a circle, a point $A$ is taken. Find the geometric locus of the points of intersection of the tangents to the circle drawn through the ends of all possible chords containing point $A$. | 7.9. Let $O$ be the center of the circle, $R$ its radius, $M$ the point of intersection of the tangents drawn through the ends of a chord containing point $A$, and $P$ the midpoint of this chord. Then $O P \cdot O M=R^{2}$ and $O P=O A \cos \varphi$, where $\varphi=\angle A O P$. Therefore, $A M^{2}=O M^{2}+O A^{2}-2 O... | OM^{2}-AM^{2}=2R^{2}-OA^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,182 |
7.10*. a) Given a parallelogram $A B C D$. Prove that the value $A X^{2}+$ $+C X^{2}-B X^{2}-D X^{2}$ does not depend on the choice of point $X$.
b) Quadrilateral $A B C D$ is not a parallelogram. Prove that all points $X$ satisfying the relation $A X^{2}+C X^{2}=B X^{2}+$ $+D X^{2}$ lie on a single line perpendicular... | 7.10. Let $P$ and $Q$ be the midpoints of the diagonals $A C$ and $B D$. Then $A X^{2}+C X^{2}=$ $=2 P X^{2}+A C^{2} / 2$ and $B X^{2}+D X^{2}=2 Q X^{2}+B D^{2} / 2$ (see problem 12.11, a)), therefore in problem b) the desired locus of points $X$ consists of points such that $P X^{2}-Q X^{2}=$ $=\left(B D^{2}-A C^{2}\r... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,183 |
7.11. A segment of constant length moves in a plane such that its ends slide along the sides of the right angle $A B C$. What is the trajectory of the midpoint of this segment? | 7.11. Let $M$ and $N$ be the endpoints of a given segment, $O$ be its midpoint. Point $B$ lies on the circle with diameter $M N$, so $O B = M N / 2$. The trajectory of point $O$ is a part of a circle with radius $M N / 2$ centered at $B$, contained within the angle $A B C$. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,184 |
7.12. Find the geometric locus of the midpoints of the chords of a given circle that pass through a given point. | 7.12. Let $M$ be a given point, $O$ be the center of a given circle. If $X$ is the midpoint of the chord $A B$, then $X O \perp A B$. Therefore, the required locus of points is a circle with diameter $M O$. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,185 |
7.13. Given two points $A$ and $B$. Two circles touch the line $A B$ (one at point $A$, the other at point $B$) and touch each other at point $M$. Find the locus of $M$. | 7.13. Let's draw a common tangent through point $M$ to the circles. Let $O$ be the point of intersection of this tangent with the line $A B$. Then $A O = M O = B O$, i.e., $O$ is the midpoint of segment $A B$. Point $M$ lies on a circle with center $O$ and radius $A B / 2$. The set of points $M$ is a circle with diamet... | ThesetofpointsMiscirclewithdiameterAB(pointsABshouldbeexcluded) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,186 |
7.14. On a plane, two points $A$ and $B$ are given. Find the locus of points $M$ such that $A M: B M=k$ (Apollonian circle). | 7.14. For $k=1$, we obtain the perpendicular bisector of the segment $A B$. In the following, we will assume that $k \neq 1$.
We introduce a coordinate system on the plane such that points $A$ and $B$ have coordinates $(-a, 0)$ and $(a, 0)$, respectively. If point $M$ has coordinates $(x, y)$, then $\frac{A M^{2}}{B M... | 圆心为(-\frac{1+k^{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,187 |
7.15. Let $S$ be the Apollonian circle for points $A$ and $B$, and point $A$ lies outside the circle $S$. Tangents $A P$ and $A Q$ are drawn from point $A$ to the circle $S$. Prove that $B$ is the midpoint of segment $P Q$. | 7.15. Let the line $A B$ intersect the circle $S$ at points $E$ and $F$, with point $E$ lying on the segment $A B$. Then $P E$ is the bisector of triangle $A P B$, so $\angle E P B = \angle E P A = \angle E F P$. Since $\angle E P F = 90^{\circ}$, then $P B \perp E F$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,188 |
7.16*. Let $A D$ and $A E$ be the bisectors of the internal and external angles of triangle $A B C$, and let $S_{a}$ be the circle with diameter $D E$. Circles $S_{b}$ and $S_{c}$ are defined similarly. Prove that:
a) the circles $S_{a}, S_{b}$ and $S_{c}$ have two common points $M$ and $N$, and the line $M N$ passes ... | 7.16. a) The considered circles are Apollonian circles for the pairs of vertices of triangle $ABC$, so if $X$ is a common point of circles $S_{a}$ and $S_{b}$, then $X B: X C = A B: A C$ and $X C: X A = B C: B A$, i.e., $X B: X A = C B: C A$, which means that point $X$ lies on circle $S_{c}$. It is also clear that if $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,189 |
7.17*. Triangle $A B C$ is equilateral, and $M$ is some point. Prove that if the numbers $A M, B M$, and $C M$ form a geometric progression, then the common ratio of this progression is less than 2.
See also problems $14.21 \mathrm{a}), 18.15$.
## §3. Inscribed Angle | 7.17. Let $O_{1}$ and $O_{2}$ be such points that $\overrightarrow{B O}_{1}=4 \overrightarrow{B A} / 3$ and $\overrightarrow{C O_{2}}=4 \overrightarrow{C B} / 3$. It is easy to verify that if $B M>2 A M$, then the point $M$ lies inside the circle $S_{1}$ with radius $2 A B / 3$ centered at $O_{1}$ (see problem 7.14), a... | q<2 | Geometry | proof | Yes | Yes | olympiads | false | 27,190 |
7.18. Points $A$ and $B$ are fixed on a circle, while point $C$ moves along this circle. Find the set of intersection points of: a) the altitudes; b) the angle bisectors of triangles $A B C$. | 7.18. a) Let $O$ be the intersection point of the altitudes $A A_{1}$ and $B B_{1}$. Points $A_{1}$ and $B_{1}$ lie on the circle with diameter $C O$. Therefore, $\angle A O B = 180^{\circ} - \angle C$. Hence, the desired locus of points is a circle symmetric to the given one with respect to the line $A B$ (points proj... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,191 |
7.19. Point $P$ moves along the circumcircle of square $A B C D$. Lines $A P$ and $B D$ intersect at point $Q$, and the line through point $Q$ parallel to $A C$ intersects line $B P$ at point $X$. Find the locus of point $X$. | 7.19. Points $P$ and $Q$ lie on the circle with diameter $D X$, so $\angle(Q D, D X)=\angle(Q P, P X)=\angle(A P, P B)=45^{\circ}$, i.e., point $X$ lies on the line $C D$. | XliesonthelineCD | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,192 |
7.20. a) On a circle, points $A$ and $B$ are fixed, while points $A_{1}$ and $B_{1}$ move along the same circle such that the length of the arc $A_{1} B_{1}$ remains constant; $M$ is the point of intersection of the lines $A A_{1}$ and $B B_{1}$. Find the locus of $M$.
b) Triangles $A B C$ and $A_{1} B_{1} C_{1}$ are i... | 7.20. a) If point $A_{1}$ travels along the circumference of an arc of magnitude $2 \varphi$, then point $B_{1}$ will also travel an arc of magnitude $2 \varphi$, which means that lines $A A_{1}$ and $B B_{1}$ will rotate by an angle $\varphi$ and the angle between them will not change. Therefore, point $M$ moves along... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,193 |
7.21*. On a plane, four points are given. Find the set of centers of rectangles formed by four lines passing through the given points, respectively. | 7.21. Suppose points $A$ and $C$ lie on opposite sides of a rectangle. Let $M$ and $N$ be the midpoints of segments $A C$ and $B D$ respectively. Draw a line $l_{1}$ through point $M$, parallel to the sides of the rectangle on which points $A$ and $C$ lie, and a line $l_{2}$ through point $N$, parallel to the sides of ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,194 |
7.22*. Find the locus of points $X$ inside a regular triangle $ABC$ that have the property $\angle XAB + \angle XBC + \angle XCA = 90^{\circ}$.
See also problems $2.5, 2.38$.
## §4. Auxiliary Congruent Triangles | 7.22. It is easy to verify that the feet of the altitudes of triangle $ABC$ have the required property. Suppose that a point $X$, not lying on any of the altitudes of triangle $ABC$, also has the required property. Then the line $BX$ intersects the altitudes $AA_1$ and $CC_1$ at points $X_1$ and $X_2$. Since $\angle XA... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,195 |
7.23*. Given a semicircle with center $O$. From each point $X$, lying on the extension of the diameter of the semicircle, a tangent ray to the semicircle is drawn and on it, a segment $X M$ is laid off, equal to the segment $X O$. Find the locus of points $M$, obtained in this way. | 7.23. Let $K$ be the point of tangency of the line $M X$ and the given semicircle, and $P$ the projection of point $M$ onto the diameter. In the right triangles $M P X$ and $O K X$, the hypotenuses are equal and $\angle P X M = \angle O X K$, so these triangles are congruent, and in particular, $M P = K O = R$, where $... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,196 |
7.24*. Let $A$ and $B$ be fixed points in the plane. Find $\Gamma \mathrm{MT} C$, possessing the following property: the height $h_{b}$ of triangle $A B C$ is equal to $b$.
Translate the above text into English, please retain the line breaks and format of the source text, and output the translation result directly. | 7.24. Let $H$ be the foot of the altitude $h_{b}$ of triangle $ABC$ and $h_{b}=b$. Denote by $B^{\prime}$ the intersection point of the perpendicular to line $AB$ through point $A$ and the perpendicular to line $AH$ through point $C$.
Right triangles $AB^{\prime}C$ and $BAH$ are equal, since $\angle AB^{\prime}C = \an... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,197 |
7.25*. Given a circle and a point $P$ inside it. Through each point $Q$ of the circle, we draw a tangent. The perpendicular dropped from the center of the circle to the line $P Q$ and the tangent intersect at point $M$. Find the locus of points $M$
## §5. Homothety
Translate the above text into English, please retain... | 7.25. Let $O-$ be the center of the circle, $N-$ the intersection point of the lines $O M$ and $Q P$. Drop a perpendicular $M S$ from point $M$ to the line $O P$. From the similarity of triangles $O N Q$ and $O Q M$, $O P N$ and $O M S$, we get $O N: O Q = O Q: O M$ and $O P: O N = O M: O S$. Multiplying these equaliti... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,198 |
7.26. Points $A$ and $B$ are fixed on a circle. Point $C$ moves along this circle. Find the set of points of intersection of the medians of triangles $A B C$.
untranslated text:
7.26. На окружности фиксированы точки $A$ and $B$. Точка $C$ перемещается по этой окружности. Найдите множество точек пересечения медиан тре... | 7.26. Let $O$ be the midpoint of segment $AB$, and $M$ be the point of intersection of the medians of triangle $ABC$. Under a homothety with center $O$ and coefficient $1/3$, point $C$ is mapped to point $M$. Therefore, the points of intersection of the medians of triangle $ABC$ lie on the circle $S$, which is the imag... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,199 |
7.27. Given a triangle $A B C$. Find the set of centers of rectangles $P Q R S$, vertices $Q$ and $P$ of which lie on side $A C$, vertices $R$ and $S$ - on sides $A B$ and $B C$ respectively. | 7.27. Let $O$ be the midpoint of the altitude $B H$, $M$ be the midpoint of the segment $A C$, and $D$ and $E$ be the midpoints of sides $R Q$ and $P S$ respectively (Fig. 7.2).
Points $D$ and $E$ lie on the lines $A O$ and $C O$ respectively.
, $O$ be the midpoint of segment $O_{1} O_{2}$; $P^{\prime}, Q^{\prime}$, and $O^{\prime}$ be the projections of points $O_{1}, O_{2}$, and $O$ onto line $P Q$. As line $P Q$ rotates, point $O^{\prime}$ t... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,201 |
7.29. Points $A, B$ and $C$ lie on the same line, with $B$ between $A$ and $C$. Find the locus of points $M$ such that the radii of the circumscribed circles of triangles $A M B$ and $C M B$ are equal.
See also problems $19.10,19.22,19.39$.
## §6. Method of Loci | 7.29. Let $P$ and $Q$ be the centers of the circumcircles of triangles $A M B$ and $C M B$. Point $M$ belongs to the desired locus if $B P M Q$ is a rhombus, i.e., point $M$ is the image of the midpoint of segment $P Q$ under a homothety with center $B$ and coefficient 2. Since the projections of points $P$ and $Q$ ont... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,202 |
7.30. Points $P$ and $Q$ move with the same constant speed $v$ along two lines intersecting at point $O$. Prove that there exists a fixed point $A$ on the plane such that the distances from $A$ to points $P$ and $Q$ are equal at any moment in time. | 7.30. Point $P$ passes through point $O$ at moment $t_{1}$, point $Q$ - at moment $t_{2}$. At the moment $\left(t_{1}+t_{2}\right) / 2$, points $P$ and $Q$ are at the same distance from point $O$, equal to $\left|t_{1}-t_{2}\right| v / 2$. At this moment, draw perpendiculars to the lines at points $P$ and $Q$. It is ea... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,203 |
7.31. Through the midpoint of each diagonal of a convex quadrilateral, a line is drawn parallel to the other diagonal. These lines intersect at point $O$. Prove that the segments connecting point $O$ to the midpoints of the sides of the quadrilateral divide its area into equal parts. | 7.31. Let the midpoints of the diagonals $AC$ and $BD$ of quadrilateral $ABCD$ be denoted by $M$ and $N$ respectively. It is clear that $S_{AMB}=S_{BMC}$ and $S_{AMD}=S_{DMC}$, i.e., $S_{DABM}=S_{BCDM}$. Since the areas of quadrilaterals $DABM$ and $BCDM$ do not change when point $M$ is moved parallel to $BD$, then $S_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,204 |
7.33. Inside a convex polygon, points $P$ and $Q$ are taken. Prove that there exists a vertex of the polygon that is closer to $Q$ than to $P$. | 7.33. Suppose that all vertices of the polygon are no closer to point $Q$ than to point $P$. Then all vertices of the polygon lie in the same half-plane defined by the perpendicular bisector of segment $P Q$, as does point $P$, and point $Q$ lies in the other half-plane. Therefore, point $Q$ lies outside the polygon, w... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,206 |
7.34. Points $A, B$ and $C$ are such that for any fourth point $M$, either $M A \leqslant M B$ or $M A \leqslant M C$. Prove that point $A$ lies on the segment $B C$. | 7.34. Let's find the locus of points \( M \) for which \( M A > M B \) and \( M A > M C \). Draw the perpendicular bisectors \( l_{1} \) and \( l_{2} \) of segments \( A B \) and \( A C \). \( M A > M B \) for points lying inside the half-plane defined by the line \( l_{1} \) and not containing point \( A \). Therefore... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,207 |
7.35. Given a quadrilateral $A B C D$, where $A B<B C$ and $A D<D C$. Point $M$ lies on the diagonal $B D$. Prove that $A M<M C$.
## §7. GMT with non-zero area | 7.35. Let $O$ be the midpoint of diagonal $A C$. The projections of points $B$ and $D$ onto line $A C$ lie on segment $A O$, so the projection of point $M$ also lies on segment $A O$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,208 |
7.36. Let $O$ be the center of rectangle $A B C D$. Find the locus of points $M$ for which $A M \geqslant O M, B M \geqslant O M, C M \geqslant O M$ and $D M \geqslant O M$. | 7.36. Let's draw the perpendicular bisector $l$ of the segment $A O$. It is clear that $A M \geqslant O M$ if and only if the point $M$ lies on the same side of the line $l$ as the point $O$ (or lies on the line $l$). Therefore, the desired locus of points is a rhombus formed by the perpendicular bisectors of the segme... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,209 |
7.39. On a plane, there are two non-intersecting circles. Is it necessarily true that there exists a point \( M \), lying outside these circles, satisfying the following condition: every line passing through point \( M \) intersects at least one of these circles?
Find the locus of points \( M \) that satisfy this cond... | 7.39. Let's draw the common tangents to the given circles (Fig. 7.4). It is easy to verify that the points belonging to the shaded areas (but not their boundaries) satisfy the required condition, while points not lying in these areas,
. | 7.40. Let the perpendiculars dropped from points $A_{1}, B_{1}, C_{1}$ to the lines $B C, C A, A B$ intersect at point $M$. Since points $B_{1}$ and $M$ lie on the same perpendicular to the line $A C$, we have $B_{1} A^{2}-B_{1} C^{2}=M A^{2}-M C^{2}$. Similarly, $C_{1} B^{2}-C_{1} A^{2}=M B^{2}-M A^{2}$ and $A_{1} C^{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,213 |
7.43*. a) The perpendiculars dropped from the vertices of triangle $A B C$ to the corresponding sides of triangle $A_{1} B_{1} C_{1}$ intersect at one point. Prove that the perpendiculars dropped from the vertices of triangle $A_{1} B_{1} C_{1}$ to the corresponding sides of triangle $A B C$ also intersect at one point... | 7.43. a) This problem is an obvious consequence of problem 7.40.
b) Let a $90^{\circ}$ rotation about some point transform triangle $A_{1} B_{1} C_{1}$ into $A_{2} B_{2} C_{2}$. Perpendiculars to the sides of triangle $A_{2} B_{2} C_{2}$ are parallel to the corresponding sides of triangle $A_{1} B_{1} C_{1}$, so the p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,215 |
7.44*. On the line $l$, points $A_{1}, B_{1}$, and $C_{1}$ are taken, and from the vertices of triangle $A B C$ perpendiculars $A A_{2}, B B_{2}$, and $C C_{2}$ are dropped to this line. Prove that the perpendiculars dropped from points $A_{1}, B_{1}$, and $C_{1}$ to the lines $B C, C A$, and $A B$ intersect at one poi... | 7.44. We need to determine in which case the equality $A B_{1}^{2}+B C_{1}^{2}+C A_{1}^{2}=B A_{1}^{2}+C B_{1}^{2}+A C_{1}^{2}$ holds. By subtracting the quantity $A A_{2}^{2}+B B_{2}^{2}+C C_{2}^{2}$ from both sides of this equality, we transition to the relation $A_{2} B_{1}^{2}+B_{2} C_{1}^{2}+C_{2} A_{1}^{2}=B_{2} ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,216 |
7.45*. Triangle $A B C$ is equilateral, $P$ is an arbitrary point. Prove that the perpendiculars dropped from the centers of the inscribed circles of triangles $P A B, P B C$, and $P C A$ to the lines $A B, B C$, and $C A$ intersect at one point. | 7.45. It can be assumed that the side length of the given equilateral triangle is 2. Let $P A=2 a, P B=2 b$ and $P C=2 c ; A_{1}, B_{1}$ and $C_{1}$ - the projections of the centers of the inscribed circles of triangles $P B C, P C A$ and $P A B$ on the lines $B C, C A$ and $A B$. According to problem $3.2 A B_{1}^{2}+... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,217 |
7.46*. Prove that if the perpendiculars erected from the bases of the bisectors of a triangle intersect at one point, then the triangle is isosceles.
## §9. Fermat-Apollonius Circle
## §9. Fermat-Apollonius Circle | 7.46. The segments into which the bisectors divide the sides of a triangle are easily calculated. As a result, we get that if the perpendiculars erected
from the bases of the bisectors intersect, then
$$
\frac{a c}{b+c}^{2}+\frac{a b}{a+c}^{2}+\frac{b c}{a+b}^{2}=\frac{a b}{b+c}^{2}+\frac{b c}{a+c}^{2}+\frac{a c}{a+b}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,218 |
7.47*. Prove that the set of points $X$, possessing the property that $k_{1} A_{1} X^{2}+\ldots+k_{n} A_{n} X^{2}=c$:
a) when $k_{1}+\ldots+k_{n} \neq 0$ is a circle or an empty set;
b) when $k_{1}+\ldots+k_{n}=0$ is a line, a plane, or an empty set. | 7.47. Let $\left(a_{i}, b_{i}\right)$ be the coordinates of point $A_{i}$, and $(x, y)$ be the coordinates of point $X$. Then the equation that point $X$ satisfies can be rewritten as $c=$ $=\sum k_{i}\left(\left(x-a_{i}\right)^{2}+\left(x-b_{i}\right)^{2}\right)=\left(\sum k_{i}\right)\left(x^{2}+y^{2}\right)-\left(2 ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,219 |
7.48*. A line $l$ intersects two circles at four points. Prove that the quadrilateral formed by the tangents at these points is cyclic, and the center of its circumscribed circle lies on the line connecting the centers of the given circles. | 7.48. Let the line $l$ cut arcs $A_{1} B_{1}$ and $A_{2} B_{2}$ of size $2 \alpha_{1}$ and $2 \alpha_{2}$ on the given circles; $O_{1}$ and $O_{2}$ are the centers of the circles, $R_{1}$ and $R_{2}$ are their radii. Let $K$ be the point of intersection of the tangents at points $A_{1}$ and $A_{2}$. By the Law of Sines... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,220 |
7.49*. Points $M$ and $N$ are such that $A M: B M: C M = A N: B N: C N$. Prove that the line $M N$ passes through the center $O$ of the circumcircle of triangle $A B C$.
See also problems $7.6, 7.14, 8.59-8.63$.
## Problems for independent solving | 7.49. Let $A M: B M: C M=p: q: r$. All points $X$ satisfying the relation $\left(q^{2}-r^{2}\right) A X^{2}+\left(r^{2}-p^{2}\right) B X^{2}+\left(p^{2}-q^{2}\right) C X^{2}=0$ lie on one straight line (see problem 7.47), and points $M, N$ and $O$ satisfy this relation. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,221 |
8.1. Construct triangle $A B C$ given $a, h_{a}$ and $R$.
Construct triangle $A B C$ given $a, h_{a}$ and $R$. | 8.1. Construct segment $B C$ of length $a$. The center $O$ of the circumcircle of triangle $A B C$ is the intersection point of two circles of radius $R$ with centers at points $B$ and $C$. Choose one of these intersection points and construct the circumcircle $S$ of triangle $A B C$. Point $A$ is the intersection poin... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,222 |
8.2. Construct a point $M$ inside the given triangle such that $S_{A B M}: S_{B C M}: S_{A C M}=1: 2: 3$. | 8.2. Let's construct points $A_{1}$ and $B_{1}$ on sides $B C$ and $A C$ respectively such that $B A_{1}: A_{1} C=1: 3$ and $A B_{1}: B_{1} C=1: 2$. Let point $X$ lie inside triangle $A B C$. It is clear that $S_{A B X}: S_{B C X}=1: 2$ if and only if point $X$ lies on segment $B B_{1}$, and $S_{A B X}: S_{A C X}=1: 3$... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,223 |
8.3. Through a given point $P$ lying inside a given circle, draw a chord such that the difference in the lengths of the segments into which $P$ divides the chord has a given value $a$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly. | 8.3. Let $O$ be the center of the given circle, $AB$ be a chord passing through point $P$, and $M$ be the midpoint of $AB$. Then $|AP - BP| = 2PM$. Since $\angle PMO = 90^{\circ}$, point $M$ lies on the circle $S$ with diameter $OP$. Construct the chord $PM$ of circle $S$ such that $PM = a / 2$ (there are two such chor... | Other | math-word-problem | Yes | Yes | olympiads | false | 27,224 | |
8.4. Given a line and a circle. Construct a circle of a given radius $r$ that is tangent to them. | 8.4. Let $R$ be the radius of the given circle, and $O$ its center. The center of the desired circle lies on a circle $S$ of radius $|R \pm r|$ centered at $O$. On the other hand, its center lies on a line $l$ parallel to the given line and at a distance $r$ from it (there are two such lines). Any point of intersection... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,225 |
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