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18.16. The hexagon $A B C D E F$ is regular, $K$ and $M$ are the midpoints of segments $B D$ and $E F$. Prove that triangle $A M K$ is equilateral. | 18.16. Let $O$ be the center of the hexagon. Consider a rotation centered at $A$ by $60^{\circ}$, which maps point $B$ to $O$. Under this rotation, segment $O C$ is mapped to segment $F E$. Point $K$ is the midpoint of diagonal $B D$ of parallelogram $B C D O$, so it is also the midpoint of diagonal $C O$. Therefore, p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,811 |
18.17. Let $M$ and $N$ be the midpoints of sides $CD$ and $DE$ of a regular hexagon $ABCDEF$, and $P$ be the point of intersection of segments $AM$ and $BN$.
a) Find the measure of the angle between the lines $AM$ and $BN$.
b) Prove that $S_{ABP} = S_{MDNP}$. | 18.17. When rotating by $60^{\circ}$ around the center of this hexagon, which maps vertex $A$ to $B$, segment $C D$ transitions to $D E$, so point $M$ transitions to $N$. Thus, under this rotation, segment $A M$ transitions to $B N$, meaning the angle between these segments is $60^{\circ}$. Additionally, under this rot... | S_{ABP}=S_{MDNP} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,812 |
18.18. On the sides $AB$ and $BC$ of an equilateral triangle $ABC$, points $M$ and $N$ are taken such that $MN \| AC$, $E$ is the midpoint of segment $AN$, and $D$ is the center of triangle $BMN$. Find the measures of the angles of triangle $CDE$.
保留源文本的换行和格式，直接输出翻译结果。 | 18.18. Consider a rotation by $60^{\circ}$ centered at $C$, which maps point $B$ to $A$. In this process, points $M, N$, and $D$ are transformed to $M^{\prime}, N^{\prime}$, and $D^{\prime}$. Since $A M N N^{\prime}$ is a parallelogram, the midpoint $E$ of diagonal $A N$ is its center of symmetry. Therefore, under symm... | 30,60,90 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,813 |
18.19. On the sides of triangle $A B C$, equilateral triangles $A B C_{1}, A B_{1} C$ and $A_{1} B C$ are constructed externally. Let $P$ and $Q$ be the midpoints of segments $A_{1} B_{1}$ and $A_{1} C_{1}$. Prove that triangle $A P Q$ is equilateral. | 18.19. Consider a rotation centered at $A$, which maps point $C_{1}$ to point $B$. Under this rotation, the equilateral triangle $A_{1} B C$ is transformed into triangle $A_{2} F B_{1}$, and the segment $A_{1} C_{1}$ is transformed into segment $A_{2} B$. It remains to note that $B A_{1} A_{2} B_{1}$ is a parallelogram... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,814 |
18.20. On the sides $AB$ and $AC$ of triangle $ABC$, equilateral triangles $ABC'$ and $AB'C$ are constructed externally. Point $M$ divides side $BC$ in the ratio $BM: MC = 3: 1$; $K$ and $L$ are the midpoints of sides $AC'$ and $B'C$. Prove that the angles of triangle $KLM$ are $30^{\circ}$, $60^{\circ}$, and $90^{\cir... | 18.20. Let $\overrightarrow{A B}=4 \boldsymbol{a}, \overrightarrow{C A}=4 b$. Further, let $R$ be the rotation that maps the vector $\overrightarrow{A B}$ to $\overrightarrow{A C^{\prime}}$ (and thus the vector $\overrightarrow{C A}$ to $\overrightarrow{C B^{\prime}}$). Then $\overrightarrow{L M}=(\boldsymbol{a}+\bolds... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,815 |
18.21. Equilateral triangles $A B C, C D E, E H K$ (vertices are traversed counterclockwise) are placed on a plane such that $\overrightarrow{A D}=\overrightarrow{D K}$. Prove that triangle $B H D$ is also equilateral. | 18.21. When rotating around point $C$ by $60^{\circ}$ counterclockwise, point $A$ moves to point $B$, point $D$ moves to point $E$, and thus the vector $\overrightarrow{D K}=\overrightarrow{A D}$ moves to the vector $\overrightarrow{B E}$. Since when rotating around point $H$ by $60^{\circ}$ counterclockwise, point $K$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,816 |
18.22*. a) Inside an acute-angled triangle, find the point for which the sum of the distances to the vertices is minimal.
b) Inside triangle $ABC$, all angles of which are less than $120^{\circ}$, a point $O$ is taken from which its sides are seen at an angle of $120^{\circ}$. Prove that the sum of the distances from ... | 18.22. a) Let $O$ be a point inside triangle $ABC$. When rotated $60^{\circ}$ around point $A$, points $B$, $C$, and $O$ move to some points $B^{\prime}$, $C^{\prime}$, and $O^{\prime}$ (Fig. 18.2). Since $A O = O O^{\prime}$ and $O C = O^{\prime} C^{\prime}$, we have $B O + A O + C O = B O + O O^{\prime} + O^{\prime} ... | (^{2}+b^{2}+^{2})/2+2\sqrt{3}S | Geometry | proof | Yes | Yes | olympiads | false | 27,817 |
18.23*. Given a point $X$ and an equilateral triangle $A B C$. Prove that from the segments $X A, X B$, and $X C$ a triangle can be formed, and this triangle is degenerate if and only if the point $X$ lies on the circumcircle of triangle $A B C$ (Pompeiu). | 18.23. Let $O$ be the center of the equilateral triangle $ABC$, $\boldsymbol{x}=\overrightarrow{XO}$, and $\boldsymbol{a}=\overrightarrow{OA}$. Then $\overrightarrow{OB}=R^{\alpha} a$ and $\overrightarrow{OC}=R^{2 \alpha} a$, where $\alpha=120^{\circ}$. Therefore,
$$
\begin{aligned}
\overrightarrow{XA}+R^{\alpha}(\ove... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,818 |
18.24*. Hexagon $A B C D E F$ is inscribed in a circle of radius $R$, and $A B=C D=E F=R$. Prove that the midpoints of the sides $B C, D E$ and $F A$ form an equilateral triangle. | 18.24. Let $P, Q$ and $R$ be the midpoints of sides $B C, D E$ and $F A$, and $O$ be the center of the circumscribed circle. Suppose that triangle $P Q R$ is equilateral. We will prove that then the midpoints of sides $B C, D E^{\prime}$ and $F^{\prime} A$ of the hexagon $A B C D E^{\prime} F^{\prime}$, where vertices ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,819 |
18.25*. On the sides of a convex centrally symmetric hexagon $A B C D E F$, regular triangles are constructed externally. Prove that the midpoints of the segments connecting the vertices of adjacent triangles form a regular hexagon.
## §3. Rotations by Arbitrary Angles | 18.25. Let $K, L, M$ and $N$ be the vertices of equilateral triangles constructed on the sides $B C, A B, A F$ and $F E$; $B_{1}, A_{1}$ and $F_{1}$ be the midpoints of segments $K L, L M$ and $M N$ (Fig. 18.4). Let $\boldsymbol{a}=\overrightarrow{B C}=\overrightarrow{F E}, \boldsymbol{b}=\overrightarrow{A B}$ and $\bo... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,820 |
18.26. Given points $A$ and $B$ and a circle $S$. Construct points $C$ and $D$ on the circle $S$ such that $A C \| B D$ and the arc $C D$ has a given measure $\alpha$. | 18.26. Let a rotation by an angle $\alpha$ with the center at the center of the circle $S$, which maps point $C$ to point $D$, map point $A$ to some point $A^{\prime}$. Then $\angle\left(B D, D A^{\prime}\right)=\alpha$, i.e., point $D$ lies on the circle from whose points the segment $A^{\prime} B$ is seen at an orien... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,821 |
18.27. A rotation with center $O$ translates the line $l_{1}$ into the line $l_{2}$, and the point $A_{1}$, lying on the line $l_{1}$, into the point $A_{2}$. Prove that the point of intersection of the lines $l_{1}$ and $l_{2}$ lies on the circumcircle of triangle $A_{1} O A_{2}$. | 18.27. Let $P$ be the point of intersection of lines $l_{1}$ and $l_{2}$. Then $\angle\left(O A_{1}, A_{1} P\right)=$ $=\angle\left(O A_{1}, l_{1}\right)=\angle\left(O A_{2}, l_{2}\right)=\angle\left(O A_{2}, A_{2} P\right)$. Therefore, points $O, A_{1}, A_{2}$ and $P$ lie
on the same circle. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,822 |
18.28. Two identical letters Г lie on a plane. The ends of the short bars of these letters are denoted by $A$ and $A^{\prime}$. The long bars are divided into $n$ equal parts by points $A_{1}, \ldots, A_{n-1} ; A_{1}^{\prime}, \ldots, A_{n-1}^{\prime}$ (the division points are numbered from the ends of the long bars). ... | 18.28. Identical letters $\Gamma$ can be superimposed by a rotation with some center $O$ (if they are superimposed by a parallel translation, then $A A_{i} \| A^{\prime} A_{i}^{\prime}$). According to problem 18.27, the point $X_{i}$ lies on the circumcircle of triangle $A^{\prime} O A$. It is clear that points lying o... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,823 |
18.29. Two points move uniformly with the same speed along two lines intersecting at point $P$: point $A$ along one line, and point $B$ along the other. They pass through point $P$ at different times. Prove that at any moment, the circumcircle of triangle $A B P$ passes through some fixed point, different from $P$. | 18.29. Let $O$ be the center of rotation $R$ that maps the segment $A\left(t_{1}\right) A\left(t_{2}\right)$ to the segment $B\left(t_{1}\right) B\left(t_{2}\right)$, where $t_{1}$ and $t_{2}$ are some moments in time. Then this rotation maps $A(t)$ to $B(t)$ at any moment in time $t$. Therefore, according to problem 1... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,824 |
18.30. Triangle $A_{1} B_{1} C_{1}$ is obtained from triangle $A B C$ by rotating it by an angle $\alpha\left(\alpha<180^{\circ}\right)$ around the center of its circumscribed circle. Prove that the points of intersection of the sides $A B$ and $A_{1} B_{1}, B C$ and $B_{1} C_{1}, C A$ and $C_{1} A_{1}$ (or their exten... | 18.30. Let $A$ and $B$ be points on a circle with center $O$, and $A_{1}$ and $B_{1}$ be the images of these points after a rotation by angle $\alpha$ about the center $O$; $P$ and $P_{1}$ be the midpoints of segments $A B$ and $A_{1} B_{1}$; $M$ be the point of intersection of the lines $A B$ and $A_{1} B_{1}$. The ri... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,825 |
18.31*. Given a triangle $A B C$. Construct a line that bisects both its area and perimeter.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 18.31. According to problem 5.52, a line that bisects both the area and the perimeter of a triangle passes through the center of its inscribed circle. It is also clear that if a line passes through the center of the inscribed circle of a triangle and bisects its perimeter, then it also bisects its area. Therefore, we n... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,826 |
18.32*. On the vectors $\overrightarrow{A_{i} B_{i}}$, where $i=1, \ldots, k$, regular $n$-gons $A_{i} B_{i} C_{i} D_{i} \ldots(n \geqslant 4)$ are constructed, all with the same orientation. Prove that the $k$-gons $C_{1} \ldots C_{k}$ and $D_{1} \ldots D_{k}$ are regular and have the same orientation if and only if t... | 18.32. Suppose that $k$-gons $C_{1} \ldots C_{k}$ and $D_{1} \ldots D_{k}$ are regular and identically oriented. Let $C$ and $D$ be the centers of these $k$-gons, $\boldsymbol{c}_{i}=\overrightarrow{C C_{i}}$ and $\boldsymbol{d}_{i}=\overrightarrow{D D_{i}}$. Then $\overrightarrow{C_{i} D_{i}}=\overrightarrow{C_{i} C}+... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,827 |
18.33*. Prove that three lines, symmetric to an arbitrary line passing through the orthocenter of a triangle with respect to the sides of the triangle, intersect at one point.
Translate the above text into English, please retain the line breaks and format of the source text, and output the translation result directly. | 18.33. Let $H$ be the orthocenter of triangle $ABC$, and $H_{1}, H_{2}$, and $H_{3}$ be the points symmetric to $H$ with respect to the sides $BC$, $CA$, and $AB$. The points $H_{1}, H_{2}$, and $H_{3}$ lie on the circumcircle of triangle $ABC$ (Problem 5.10). Let $l$ be a line passing through point $H$. The line symme... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,828 |
18.34*. In the circus arena, which is a circle with a radius of 10 m, a lion is running. Moving along a broken line, he ran $30 \mathrm{km}$. Prove that the sum of all his turning angles is not less than 2998 radians.
## §4. Compositions of Rotations | 18.34. Suppose the lion ran along the broken line $A_{1} A_{2} \ldots A_{n}$. Let's straighten the lion's trajectory as follows. Rotate the circus arena and the subsequent trajectory around point $A_{2}$ so that point $A_{3}$ falls on the ray $A_{1} A_{2}$. Then rotate the circus arena and the subsequent trajectory aro... | 2998 | Geometry | proof | Yes | Yes | olympiads | false | 27,829 |
18.35. Prove that the composition of two rotations by angles, the sum of which is not a multiple of $360^{\circ}$, is a rotation. At what point is its center located and what is the angle of rotation? Also, investigate the case when the sum of the angles of rotation is a multiple of $360^{\circ}$.
$$
＊ \quad * \quad *... | 18.35. Consider the composition of rotations $R_{B}^{\beta} \circ R_{A}^{\alpha}$. If $A=B$, the statement of the problem is obvious, so we will assume that $A \neq B$. Let $l=AB$, and let lines $a$ and $b$ pass through points $A$ and $B$ respectively, such that $\angle(a, l)=\alpha / 2$ and $\angle(l, b)=\beta / 2$. T... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,830 |
18.36. On the sides of an arbitrary convex quadrilateral, squares are constructed externally. Prove that the segments connecting the centers of opposite squares are equal in length and perpendicular. | 18.36. Let $P, Q, R$ and $S$ be the centers of squares constructed externally on the sides $AB, BC, CD$ and $DA$ respectively. Construct isosceles right triangles internally on segments $QR$ and $SP$ with vertices $O_{1}$ and $O_{2}$. Then $D=R_{R}^{90^{\circ}} \circ R_{Q}^{90^{\circ}}(B)=R_{O_{1}}^{180^{\circ}}(B)$ an... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,831 |
18.37. On the sides of a parallelogram, squares are constructed externally. Prove that their centers form a square. | 18.37. Let $P, Q, R$ and $S$ be the centers of squares constructed externally on the sides $AB, BC, CD$ and $DA$ of parallelogram $ABCD$. According to the previous problem, $PR = QS$ and $PR \perp QS$. Moreover, the center of symmetry of parallelogram $ABCD$ is the center of symmetry of quadrilateral $PQRS$, i.e., $PQR... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,832 |
18.38. On the sides of triangle $ABC$, squares are constructed externally with centers $P$, $Q$, and $R$. On the sides of triangle $PQR$, squares are constructed internally. Prove that their centers are the midpoints of the sides of triangle $ABC$. | 18.38. Let $P, Q$ and $R$ be the centers of squares constructed externally on the sides $AB, BC$, and $CA$. Consider a $90^{\circ}$ rotation centered at $R$ that maps $C$ to $A$. A $90^{\circ}$ rotation in the same direction centered at $P$ maps point $A$ to $B$. The composition of these two rotations is a $180^{\circ}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,833 |
18.39. Inside a convex quadrilateral $A B C D$, isosceles right triangles $A B O_{1}, \mathrm{BCO}_{2}, \mathrm{CDO}_{3}$, and $D A O_{4}$ are constructed. Prove that if $O_{1}=O_{3}$, then $O_{2}=O_{4}$.
$$
* * *
$$ | 18.39. If $O_{1}=O_{3}$, then $R_{D}^{90^{\circ}} \circ R_{C}^{90^{\circ}} \circ R_{B}^{90^{\circ}} \circ R_{A}^{90^{\circ}}=R_{O_{3}}^{180^{\circ}} \circ R_{O_{1}}^{180^{\circ}}=E$. Therefore, $E=R_{A}^{90^{\circ}} \circ E \circ R_{A}^{-90^{\circ}}=R_{A}^{90^{\circ}} \circ R_{D}^{90^{\circ}} \circ R_{C}^{90^{\circ}} \... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,834 |
18.40*. a) On the sides of an arbitrary triangle, regular triangles are constructed externally. Prove that their centers form a regular triangle.
b) Prove the analogous statement for triangles constructed internally.
c) Prove that the difference in the areas of the regular triangles obtained in problems a) and b) is ... | 18.40. a) See the solution of the more general problem 18.44 (it is sufficient to set $\alpha=$ $=\beta=\gamma=120^{\circ}$ ). For case b), the proof is analogous.
c) Let $Q$ and $R$ (respectively $Q_{1}$ and $R_{1}$) be the centers of the equilateral triangles constructed externally (respectively internally) on the s... | S_{ABC} | Geometry | proof | Yes | Yes | olympiads | false | 27,835 |
18.41*. On the sides of triangle $A B C$, equilateral triangles $A^{\prime} B C$ and $B^{\prime} A C$ are constructed externally, $C^{\prime} A B-$ internally, $M-$ is the center of triangle $C^{\prime} A B$. Prove that $A^{\prime} B^{\prime} M$ is an isosceles triangle, and $\angle A^{\prime} M B^{\prime}=120^{\circ}$... | 18.41. The composition of a $60^{\circ}$ rotation about point $A^{\prime}$, which maps $B$ to $C$, a $60^{\circ}$ rotation about point $B^{\prime}$, which maps $C$ to $A$, and a $120^{\circ}$ rotation about point $M$, which maps $A$ to $B$, has a fixed point $B$. Since the first two rotations are performed in the direc... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,836 |
18.42*. Let the angles $\alpha, \beta, \gamma$ be such that $0<\alpha, \beta, \gamma<\pi$ and $\alpha+\beta+\gamma=$ $=\pi$. Prove that if the composition of rotations $R_{C}^{2 \gamma} \circ R_{B}^{2 \beta} \circ R_{A}^{2 \alpha}$ is the identity transformation, then the angles of triangle $A B C$ are $\alpha, \beta$,... | 18.42. From the condition of the problem, it follows that $R_{C}^{-2 \gamma}=R_{B}^{2 \beta} \circ R_{A}^{2 \alpha}$, i.e., point $C$ is the center of the composition of rotations $R_{B}^{2 \beta} \circ R_{A}^{2 \alpha}$. This means that $\angle B A C=\alpha$ and $\angle A B C=\beta$ (see problem 18.35). Therefore, $\a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,837 |
18.44*. On the sides of an arbitrary triangle $A B C$, outside it, isosceles triangles $A^{\prime} B C, A B^{\prime} C$ and $A B C^{\prime}$ are constructed with vertices $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ and angles $\alpha, \beta$ and $\gamma$ at these vertices, respectively, such that $\alpha+\beta+\gamma=2... | 18.44. Since $R_{C^{\prime}}^{\gamma} \circ R_{B^{\prime}}^{\beta} \circ R_{A^{\prime}}^{\alpha}(B)=R_{C^{\prime}}^{\gamma} \circ R_{B^{\prime}}^{\beta}(C)=R_{C^{\prime}}^{\gamma}(A)=B$, then $B$ is a fixed point of the composition of rotations $R_{C^{\prime}}^{\gamma} \circ R_{B^{\prime}}^{\beta} \circ R_{A^{\prime}}^... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,839 |
18.45*. Let $A K L$ and $A M N$ be similar isosceles triangles with vertex $A$ and angle $\alpha$ at the vertex; $G N K$ and $G' L M$ be similar isosceles triangles with angle $\pi-\alpha$ at the vertex. Prove that $G=G'$. (The triangles are oriented.) | 18.45. Since $R_{G^{\prime}}^{\pi-\alpha} \circ R_{A}^{\alpha}(N)=L$ and $R_{G}^{\pi-\alpha} \circ R_{A}^{\alpha}(L)=N$, the transformations $R_{G^{\prime}}^{\pi-\alpha} \circ R_{A}^{\alpha}$ and $R_{G}^{\pi-\alpha} \circ R_{A}^{\alpha}$ are central symmetries with respect to the midpoint of segment $L N$, i.e., $R_{G^... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,840 |
18.46*. On the sides $AB$, $BC$, and $CA$ of triangle $ABC$, points $P$, $Q$, and $R$ are taken respectively. Prove that the centers of the circumcircles of triangles $APR$, $BPQ$, and $CQR$ form a triangle similar to triangle $ABC$.
## Problems for Independent Solving | 18.46. Let $A_{1}, B_{1}$ and $C_{1}$ be the centers of the circumcircles of triangles $A P R, B P Q$ and $C Q R$. With successive rotations centered at $A_{1}, B_{1}$ and $C_{1}$ by angles $2 \alpha, 2 \beta$ and $2 \gamma$, point $R$ first moves to $P$, then to $Q$, and finally returns to its original position. Since... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,841 |
19.1. A quadrilateral is cut by its diagonals into four triangles. Prove that the points of intersection of the medians of these triangles form a parallelogram. | 19.1. Under a homothety with the center at the intersection point of the diagonals of a quadrilateral and a coefficient of $3 / 2$, the points of intersection of the medians of the specified triangles are transformed into the midpoints of the sides of the quadrilateral. It remains to use the result of problem 1.2. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,842 |
19.2. The extensions of the lateral sides $AB$ and $CD$ of trapezoid $ABCD$ intersect at point $K$, and its diagonals intersect at point $L$. Prove that points $K$, $L$, $M$, and $N$, where $M$ and $N$ are the midpoints of the bases $BC$ and $AD$, lie on one straight line. | 19.2. Under a homothety with center $K$, which maps $\triangle K B C$ to $\triangle K A D$, point $M$ is mapped to $N$, so point $K$ lies on the line $M N$. Under a homothety with center $L$, which maps $\triangle L B C$ to $\triangle L D A$, point $M$ is mapped to $N$. Therefore, point $L$ lies on the line $M N$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,843 |
19.3. In a trapezoid, the point of intersection of the diagonals is equidistant from the lines on which the lateral sides lie. Prove that the trapezoid is isosceles. | 19.3. Let the extensions of the lateral sides $A B$ and $C D$ intersect at point $K$, and the diagonals of the trapezoid intersect at point $L$. According to the previous problem, the line $K L$ passes through the midpoint of segment $A D$, and by the condition of the problem, this same line bisects angle $A K D$. Ther... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,844 |
19.4. The medians $A A_{1}, B B_{1}$, and $C C_{1}$ of triangle $A B C$ intersect at point $M$; $P$ is an arbitrary point. Line $l_{a}$ passes through point $A$ parallel to line $P A_{1}$; lines $l_{b}$ and $l_{c}$ are defined similarly. Prove that:
a) lines $l_{a}, l_{b}$, and $l_{c}$ intersect at a single point $Q$;... | 19.4. Under a homothety with center $M$ and coefficient -2, the lines $P A_{1}, P B_{1}$, and $P C_{1}$ transform into the lines $l_{a}, l_{b}$, and $l_{c}$, and thus the desired point $Q$ is the image of point $P$ under this homothety. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,845 |
19.5. Circle $S$ touches the equal sides $A B$ and $B C$ of isosceles triangle $A B C$ at points $P$ and $K$, and also touches the circumcircle of triangle $A B C$ internally. Prove that the midpoint of segment $P K$ is the center of the inscribed circle of triangle $A B C$. | 19.5. Consider the homothety $H_{B}^{k}$ with center $B$, mapping the segment $A C$ to the segment $A^{\prime} C^{\prime}$, which is tangent to the circumcircle of triangle $A B C$. Denote the midpoints of segments $P K$ and $A^{\prime} C^{\prime}$ by $O_{1}$ and $D$, and the center of circle $S$ by $O$.
The circle $S... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,846 |
19.6*. A convex polygon has the following property: if all its sides are moved outward by a distance of 1, the resulting lines form a polygon similar to the original. Prove that this polygon is circumscribed. | 19.6. Let $k$ be the similarity coefficient of polygons, with $k<1$. By shifting the sides of the original polygon inward sequentially by $k, k^{2}, k^{3}, \ldots$, we obtain a contracting system of nested convex polygons, similar to the original with coefficients $k, k^{2}, k^{3}, \ldots$ The unique common point of th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,847 |
19.7*. Let $R$ and $r$ be the radii of the circumscribed and inscribed circles of a triangle. Prove that $R \geqslant 2 r$, and equality is achieved only for an equilateral triangle. | 19.7. Let $A_{1}, B_{1}$, and $C_{1}$ be the midpoints of sides $BC, AC$, and $AB$ respectively. Under a homothety with the center at the intersection of the medians of triangle $ABC$ and a homothety coefficient of $-1/2$, the circumcircle $S$ of triangle $ABC$ transforms into the circumcircle $S_{1}$ of triangle $A_{1... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,848 |
19.8*. Let $M$ be the center of mass of an $n$-gon $A_{1} \ldots A_{n} ; M_{1}, \ldots, M_{n}$ be the centers of mass of the $(n-1)$-gons obtained from this $n$-gon by removing the vertices $A_{1}, \ldots, A_{n}$ respectively. Prove that the polygons $A_{1} \ldots A_{n}$ and $M_{1} \ldots M_{n}$ are homothetic. | 19.8. Since $\overrightarrow{M M_{i}}=\left(\overrightarrow{M A_{1}}+\ldots+\overrightarrow{M A_{n}}-\overrightarrow{M A_{i}}\right) /(n-1)=-\overrightarrow{M A_{i}} /(n-1)$, the point $A_{i}$ is mapped to the point $M_{i}$ under a homothety with center $M$ and coefficient $-1 /(n-1)$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,849 |
19.9*. Prove that any convex polygon $\Phi$ contains two non-intersecting polygons $\Phi_{1}$ and $\Phi_{2}$, similar to $\Phi$ with a coefficient of $1 / 2$.
See also problem 5.97.
## §2. Homothetic Circles | 19.9. Let $A$ and $B$ be the pair of points in the polygon $\Phi$ that are farthest apart from each other. Then $\Phi_{1}=H_{A}^{1 / 2}(\Phi)$ and $\Phi_{2}=H_{B}^{1 / 2}(\Phi)$ are the desired figures. Indeed, $\Phi_{1}$ and $\Phi_{2}$ do not intersect, as they lie on opposite sides of the perpendicular bisector of th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,850 |
19.10. On a circle, points $A$ and $B$ are fixed, while point $C$ moves along this circle. Find the geometric locus of the points of intersection of the medians of triangles $A B C$.
untranslated text:
19.10. На окружности фиксированы точки $A$ и $B$, а точка $C$ движется по этой окружности. Найдите геометрическое ме... | 19.10. Let $M$ be the point of intersection of the medians of triangle $ABC$, and $O$ be the midpoint of segment $AB$. It is clear that $3 \overrightarrow{O M} = \overrightarrow{O C}$, so the points $M$ fill a circle obtained from the original circle by a homothety with coefficient $1 / 3$ and center $O$. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,851 |
19.11*. a) The inscribed circle of triangle $ABC$ touches side $AC$ at point $D$, and $DM$ is its diameter. Line $BM$ intersects side $AC$ at point $K$. Prove that $AK = DC$.
b) In a circle, perpendicular diameters $AB$ and $CD$ are drawn. From point $M$, lying outside the circle, tangents to the circle are drawn, int... | 19.11. a) Under a homothety with center $B$, which maps the inscribed circle to the excircle tangent to side $AC$, point $M$ is transformed into some point $M'$. Point $M'$ is the endpoint of a diameter perpendicular to line $AC$, so $M'$ is the point of tangency of the inscribed circle with side $AC$, and thus the int... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,852 |
19.12*. Let $O$ be the center of the inscribed circle of triangle $ABC$, $D$ be the point of tangency of the circle with side $AC$, and $B_{1}$ be the midpoint of side $AC$. Prove that the line $B_{1} O$ bisects the segment $BD$. | 19.12. Using the solution and notations from problem 19.11, a). Since $A K = D C$, then $B_{1} K = B_{1} D$, which means $B_{1} O$ is the midline of triangle $M K D$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,853 |
19.13*. Circles $\alpha, \beta$, and $\gamma$ have the same radii and are tangent to the sides of angles $A, B$, and $C$ of triangle $ABC$ respectively. Circle $\delta$ is externally tangent to all three circles $\alpha, \beta$, and $\gamma$. Prove that the center of circle $\delta$ lies on the line passing through the... | 19.13. Let $O_{\alpha}, O_{\beta}, O_{\gamma}$ and $O_{\delta}$ be the centers of circles $\alpha, \beta, \gamma$ and $\delta; O_{1}$ and $O_{2}$ be the centers of the inscribed and circumscribed circles of triangle $ABC$. The triangle $O_{\alpha} O_{\beta} O_{\gamma}$ transforms into triangle $ABC$ under a homothety w... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,854 |
19.14*. Given a triangle $A B C$. Four circles of equal radius $\rho$ are constructed such that one of them touches the other three, and each of these three touches two sides of the triangle. Find $\rho$, if the radii of the inscribed and circumscribed circles of the triangle are $r$ and $R$ respectively. | 19.14. Let $A_{1}, B_{1}$ and $C_{1}$ be the centers of the circles touching the sides of the triangle, $O$ be the center of the circle touching these circles, $O_{1}$ and $O_{2}$ be the centers of the inscribed and circumscribed circles of triangle $A B C$. The lines $A A_{1}, B B_{1}$ and $C C_{1}$ are the angle bise... | \rho=\frac{rR}{2r+R} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,855 |
19.15*. A circle is inscribed in each angle of triangle $ABC$, touching the circumcircle. Let $A_{1}, B_{1}$ and $C_{1}$ be the points of tangency of these circles with the circumcircle. Prove that the lines $A A_{1}, B B_{1}$ and $C C_{1}$ intersect at one point.
## §3. Constructions and Geometric Loci | 19.15. Let $X$ be the center of homothety (with a positive coefficient) that maps the incircle of triangle $ABC$ to the circumcircle. The line $AX$ intersects the incircle at points $A'$ and $A''$, one of which (for definiteness, $A''$) is mapped to point $A$ by the given homothety, and the other to some point $A_2$ ly... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,856 |
19.16. Given an angle $A B C$ and a point $M$ inside it. Construct a circle that is tangent to the sides of the angle and passes through the point $M$. | 19.16. Let's take an arbitrary point \( O \) on the bisector of angle \( ABC \) and construct a circle \( S \) with center \( O \), tangent to the sides of the angle. The line \( BM \) intersects the circle \( S \) at points \( M_1 \) and \( M_2 \). The problem has two solutions: under a homothety with center \( B \) t... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,857 |
19.17. Inscribe two equal circles in a triangle, each of which touches two sides of the triangle and the other circle. | 19.17. It is clear that both circles touch one of the sides of the triangle. Let's show how to construct circles that touch the side $AB$. Take a line $c'$ parallel to the line $AB$. Construct circles $S_{1}'$ and $S_{2}'$ of the same radius, touching each other and the line $c'$. Construct tangents $a'$ and $b'$ to th... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,858 |
19.19. Construct triangle $A B C$ given sides $A B$ and $A C$ and the bisector $A D$. | 19.19. Let's take the segment $A D$ and draw circles $S_{1}$ and $S_{2}$ with center $A$ and radii $A B$ and $A C$ respectively. The vertex $B$ is the intersection point of circle $S_{1}$ with the image of circle $S_{2}$ under the homothety with center $D$ and coefficient $-D B / D C = -A B / A C$. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,860 |
19.20. Solve problem 16.18 using homothety.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
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19.20. Solve problem 16.18 using homothety. | 19.20. Let's take an arbitrary point $X$ on the larger circle $S_{2}$. Let $S_{2}^{\prime}$ be the image of the circle $S_{2}$ under the homothety with center $X$ and coefficient $1 / 3$, and let $Y$ be the point of intersection of the circles $S_{2}^{\prime}$ and $S_{1}$. Then $X Y$ is the desired line. | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,861 | |
19.21. Construct a point on side $B C$ of the given triangle $A B C$ such that the line connecting the feet of the perpendiculars dropped from this point to sides $A B$ and $A C$ is parallel to $B C$. | 19.21. Draw perpendiculars from points $B$ and $C$ to the lines $A B$ and $A C$; let $P$ be their point of intersection. Then the point of intersection of the lines $A P$ and $B C$ is the desired one. | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,862 | |
19.22*. In a right triangle $ABC$, the vertex $A$ of the right angle remains fixed, while the vertices $B$ and $C$ slide along fixed circles $S_{1}$ and $S_{2}$, which are externally tangent at point $A$. Find the geometric locus of the feet $D$ of the altitudes $AD$ of the triangles $ABC$.
See also problems $7.26-7.2... | 19.22. Let's draw the common external tangents $l_{1}$ and $l_{2}$ to the circles $S_{1}$ and $S_{2}$. The lines $l_{1}$ and $l_{2}$ intersect at point $K$, which is the center of homothety $H$ that maps circle $S_{1}$ to circle $S_{2}$. Let $A_{1}=H(A)$. Points $A$ and $K$ lie on the line connecting the centers of the... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,863 |
19.23. Transformation $f$ has the following property: if $A'$ and $B'$ are the images of points $A$ and $B$, then $\overrightarrow{A' B'} = k \overrightarrow{A B}$, where $k$ is a constant number. Prove that:
a) if $k=1$, then transformation $f$ is a parallel translation;
b) if $k \neq 1$, then transformation $f$ is ... | 19.23. It follows from the condition of the problem that the mapping $f$ is bijective.
a) Let point $A$ be mapped to point $A^{\prime}$ by $f$, and point $B$ to point $B^{\prime}$. Then $\overrightarrow{B B^{\prime}}=\overrightarrow{B A}+\overrightarrow{A A^{\prime}}+\overrightarrow{A^{\prime} B^{\prime}}=-\overrighta... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,864 |
19.24. Prove that the composition of two homotheties with coefficients $k_{1}$ and $k_{2}$, where $k_{1} k_{2} \neq 1$, is a homothety with coefficient $k_{1} k_{2}$, and that its center lies on the line connecting the centers of these homotheties. Investigate the case $k_{1} k_{2}=1$. | 19.24. Let $H=H_{2} \circ H_{1}$, where $H_{1}$ and $H_{2}$ are homotheties with centers $O_{1}$ and $O_{2}$ and coefficients $k_{1}$ and $k_{2}$. Introduce the notation $A^{\prime}=H_{1}(A), B^{\prime}=H_{1}(B), A^{\prime \prime}=$ $=H_{2}\left(A^{\prime}\right), B^{\prime \prime}=H_{2}\left(B^{\prime}\right)$. Then $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,865 |
19.25. The common external tangents to the pairs of circles $S_{1}$ and $S_{2}, S_{2}$ and $S_{3}, S_{3}$ and $S_{1}$ intersect at points $A, B$ and $C$ respectively. Prove that points $A, B$ and $C$ lie on a single straight line. | 19.25. Point $A$ is the center of homothety that maps $S_{1}$ to $S_{2}$, and point $B$ is the center of homothety that maps $S_{2}$ to $S_{3}$. The composition of these homotheties maps $S_{1}$ to $S_{3}$, and its center lies on the line $A B$. On the other hand, the center of the homothety that maps $S_{1}$ to $S_{3}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,866 |
19.26. Trapezoids $A B C D$ and $A P Q D$ have a common base $A D$, and the lengths of all their bases are pairwise distinct. Prove that the following pairs of lines intersect at points lying on the same straight line:
a) $A B$ and $C D, A P$ and $D Q, B P$ and $C Q$;
b) $A B$ and $C D, A Q$ and $D P, B Q$ and $C P$.
... | 19.26. a) Let $K, L, M$ be the points of intersection of the lines $A B$ and $C D, A P$ and $D Q$, $B P$ and $C Q$. These points are the centers of homotheties $H_{K}, H_{L}$, and $H_{M}$ with positive coefficients, mapping the segment $B C$ to $A D$, $A D$ to $P Q$, and $B C$ to $P Q$, respectively. It is clear that $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,867 |
19.27. Circles $S_{1}$ and $S_{2}$ intersect at points $A$ and $B$. Lines $p$ and $q$, passing through point $A$, intersect circle $S_{1}$ at points $P_{1}$ and $Q_{1}$,
and circle $S_{2}$ at points $P_{2}$ and $Q_{2}$. Prove that the angle between lines $P_{1} Q_{1}$ and $P_{2} Q_{2}$ is equal to the angle between cir... | 19.27. Since $\angle\left(P_{1} A, A B\right)=\angle\left(P_{2} A, A B\right)$, the oriented angular measures of arcs $B P_{1}$ and $B P_{2}$ are equal. Therefore, under the rotational homothety centered at $B$, which maps $S_{1}$ to $S_{2}$, point $P_{1}$ is mapped to $P_{2}$, and line $P_{1} Q_{1}$ is mapped to line ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,868 |
19.28. Circles $S_{1}$ and $S_{2}$ intersect at points $A$ and $B$. Under a rotational homothety $P$ centered at $A$, which maps $S_{1}$ to $S_{2}$, point $M_{1}$ on circle $S_{1}$ is mapped to $M_{2}$. Prove that the line $M_{1} M_{2}$ passes through point $B$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 19.28. Since the oriented angular measures of arcs $A M_{1}$ and $A M_{2}$ are equal, then $\angle\left(M_{1} B, B A\right)=\angle\left(M_{2} B, B A\right)$, which means points $M_{1}, M_{2}$, and $B$ lie on the same line. | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,869 | |
19.29. Circles $S_{1}, \ldots, S_{n}$ pass through the point $O$. A grasshopper jumps from a point $X_{i}$ on circle $S_{i}$ to a point $X_{i+1}$ on circle $S_{i+1}$ such that the line $X_{i} X_{i+1}$ passes through the point of intersection of circles $S_{i}$ and $S_{i+1}$, different from point $O$. Prove that after $... | 19.29. Let $P_{i}$ be a spiral similarity with center $O$, mapping circle $S_{i}$ to $S_{i+1}$. Then $X_{i+1}=P_{i}\left(X_{i}\right)$ (see problem 19.28). It remains to note that the composition $P_{n} \circ \ldots \circ P_{2} \circ P_{1}$ is a spiral similarity with center $O$, mapping $S_{1}$ to $S_{1}$, i.e., it is... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,870 |
19.30. Two circles intersect at points $A$ and $B$, and chords $A M$ and $A N$ are tangent to these circles. Triangle $M A N$ is completed to parallelogram $M A N C$ and segments $B N$ and $M C$ are divided in equal ratios by points $P$ and $Q$. Prove that $\angle A P Q=\angle A N C$. | 19.30. Since $\angle A M B = \angle N A B$ and $\angle B A M = \angle B N A$, then $\triangle A M B \sim \triangle N A B$, which means $A N: A B = M A: M B = C N: M B$. Moreover, $\angle A B M = 180^{\circ} - \angle M A N = \angle A N C$. Therefore, $\triangle A M B \sim \triangle A C N$, i.e., a rotational homothety c... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,871 |
19.31. Given two non-concentric circles $S_{1}$ and $S_{2}$. Prove that there exist exactly two rotational homotheties with a rotation angle of $90^{\circ}$, mapping $S_{1}$ to $S_{2}$.
＊＊＊ | 19.31. Let $O_{1}$ and $O_{2}$ be the centers of the given circles, and $r_{1}$ and $r_{2}$ their radii. The coefficient $k$ of the rotational homothety that transforms $S_{1}$ into $S_{2}$ is $r_{1} / r_{2}$, and its center $O$ lies on the circle with diameter $O_{1} O_{2}$, and, moreover, $O O_{1}: O O_{2}=k=$ $=r_{1... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,872 |
19.32. Given a square $A B C D$. Points $P$ and $Q$ lie on sides $A B$ and $B C$ respectively, such that $B P = B Q$. Let $H$ be the foot of the perpendicular dropped from point $B$ to segment $P C$. Prove that $\angle D H Q = 90^{\circ}$. | 19.32. Consider the transformation that maps triangle $B H C$ to triangle $P H B$, i.e., the composition of a $90^{\circ}$ rotation about point $H$ and a homothety with coefficient $B P: C B$ and center $H$. Since this transformation maps the vertices of the square to the vertices of the square, and points $C$ and $B$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,873 |
19.33. On the sides of triangle $A B C$, similar triangles are constructed externally: $\triangle A_{1} B C \sim \triangle B_{1} C A \sim \triangle C_{1} A B$. Prove that the points of intersection of the medians of triangles $A B C$ and $A_{1} B_{1} C_{1}$ coincide. | 19.33. Let $P$ be a rotational homothety that maps vector $\overrightarrow{C B}$ to vector $\overrightarrow{C A_{1}}$. Then $\overrightarrow{A A_{1}}+\overrightarrow{B B_{1}}+\overrightarrow{C C_{1}}=\overrightarrow{A C}+P(\overrightarrow{C B})+\overrightarrow{C B}+P(\overrightarrow{B A})+\overrightarrow{B A}+P(\overri... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,874 |
19.34. The midpoints of sides $B C$ and $B_{1} C_{1}$ of the regular triangles $A B C$ and $A_{1} B_{1} C_{1}$ coincide (the vertices of both triangles are listed in a clockwise direction). Find the measure of the angle between the lines $A A_{1}$ and $B B_{1}$, as well as the ratio of the lengths of segments $A A_{1}$... | 19.34. Let $M$ be the common midpoint of sides $B C$ and $B_{1} C_{1}, \boldsymbol{x}=\overrightarrow{M B}$ and $\boldsymbol{y}=\overrightarrow{M B_{1}}$. Let, further, $P$ be a rotational homothety with center $M$, rotation angle $90^{\circ}$, and coefficient $\sqrt{3}$, mapping point $B$ to $A$, and $B_{1}$ to $A_{1}... | 90,\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,875 |
19.35. Triangle $A B C$ under a rotational homothety transforms into triangle $A_{1} B_{1} C_{1} ; O$ - an arbitrary point. Let $A_{2}$ be the vertex of the parallelogram $O A A_{1} A_{2}$; points $B_{2}$ and $C_{2}$ are defined similarly. Prove that $\triangle A_{2} B_{2} C_{2} \sim \triangle A B C$. | 19.35. Let $P$ be a rotational homothety that maps triangle $A B C$ to triangle $A_{1} B_{1} C_{1}$. Then $\overrightarrow{A_{2} B_{2}}=\overrightarrow{A_{2} O}+\overrightarrow{O B_{2}}=\overrightarrow{A_{1} A}+\overrightarrow{B B_{1}}=\overrightarrow{B A}+\overrightarrow{A_{1} B_{1}}=$ $=-\overrightarrow{A B}+P(\overr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,876 |
19.36*. A map of a rectangular area was placed on top of another map of the same area but at a smaller scale. Prove that it is possible to pierce both maps with a needle so that the puncture point represents the same point on both maps.
19.37*. Rotational homotheties $P_{1}$ and $P_{2}$ with centers $A_{1}$ and $A_{2}... | 19.36. The original map is a rectangle $K_{0}$ on the plane, the smaller map is a rectangle $K_{1}$ contained in $K_{0}$. Consider the rotational homothety $f$ that maps the rectangle $K_{0}$ onto $K_{1}$. Let $K_{i+1}=f\left(K_{i}\right)$. Since the sequence $K_{i}$ is a contracting sequence of nested polygons, there ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,877 |
19.38*. Triangles $M A B$ and $M C D$ are similar but have opposite orientations. Let $O_{1}$ be the center of rotation by the angle $2 \angle(\overrightarrow{A B}, \overrightarrow{B M})$, which maps $A$ to $C$, and $O_{2}$ be the center of rotation by the angle $2 \angle(\overrightarrow{A B}, \overrightarrow{A M})$, w... | 19.38. Let $P_{1}$ be a rotational homothety centered at $B$, mapping $A$ to $M$, and $P_{2}$ be a rotational homothety centered at $D$, mapping $M$ to $C$. Since the product of the coefficients of these rotational homotheties is $(B M: B A) \cdot(D C: D M)=1$, their composition $P_{2} \circ P_{1}$ is a rotation (mappi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,878 |
19.39*. Given a semicircle with diameter $A B$. For each point $X$ on this semicircle, a point $Y$ is marked on the ray $X A$ such that $X Y = k X B$. Find the locus of points $Y$. | 19.39. It is easy to check that $\operatorname{tg} X B Y=k$ and $B Y: B X=\sqrt{k^{2}+1}$, i.e., point $Y$ is obtained from $X$ by a rotational homothety with center $B$, rotation angle $\operatorname{arctg} k$, and coefficient $\sqrt{k^{2}+1}$. The desired locus of points is the image of the given semicircle under thi... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,879 |
19.40*. On the side $A B$ of triangle $A B C$, a point $P$ is given. Inscribe in triangle $A B C$ a triangle $P X Y$, similar to the given triangle $L M N$.
保留源文本的换行和格式，直接输出翻译结果如下：
19.40*. On the side $A B$ of triangle $A B C$, a point $P$ is given. Inscribe in triangle $A B C$ a triangle $P X Y$, similar to the give... | 19.40. Suppose that triangle $P X Y$ is constructed, with points $X$ and $Y$ lying on sides $A C$ and $C B$ respectively. We know the transformation that maps $X$ to $Y$, specifically a rotational homothety centered at $P$, with an angle of rotation $\varphi=\angle X P Y=\angle M L N$ and a homothety coefficient $k=P Y... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,880 |
19.41*. Construct a quadrilateral $A B C D$ given $\angle B+\angle D, a=A B, b=$ $=B C, c=C D$ and $d=D A$.
See also problem 5.133.
## §6. Center of Rotational Homothety | 19.41. Suppose that the quadrilateral $A B C D$ is constructed. Consider the rotational homothety centered at $A$, mapping $B$ to $D$. Let $C^{\prime}$ be the image of point $C$ under this homothety. Then $\angle C D C^{\prime}=\angle B+\angle D$ and $D C^{\prime}=(B C \cdot A D) / A B=$ $=b d / a$.
Triangle $C D C^{\... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,881 |
19.42. a) Let $P$ be the intersection point of the lines $A B$ and $A_{1} B_{1}$. Prove that if among the points $A, B, A_{1}, B_{1}$, and $P$ there are no coincident points, then the common point of the circumcircles of triangles $P A A_{1}$ and $P B B_{1}$ is the center of a spiral similarity that maps point $A$ to $... | 19.42. a) If $O$ is the center of a rotational homothety that transforms segment $A B$ into segment $A_{1} B_{1}$, then
$$
\angle(P A, A O)=\angle\left(P A_{1}, A_{1} O\right) \text { and } \angle(P B, B O)=\angle\left(P B_{1}, B_{1} O\right)
$$
which means that point $O$ is the intersection point of the circumcircle... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,882 |
19.43. Points $A$ and $B$ move along two intersecting lines with constant but unequal speeds. Prove that there exists a point $P$ such that at any moment in time $A P: B P=k$, where $k$ is the ratio of the speeds. | 19.43. Let $A_{1}$ and $B_{1}$ be the positions of the points at one moment, and $A_{2}$ and $B_{2}$ be the positions of the points at another moment. Then, as point $P$, one can take the center of the rotational homothety that maps segment $A_{1} A_{2}$ to segment $B_{1} B_{2}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,883 |
19.44. Construct the center $O$ of a rotational homothety with a given coefficient $k \neq 1$, which transforms the line $l_{1}$ into the line $l_{2}$, and the point $A_{1}$ lying on $l_{1}$, into the point $A_{2}$. | 19.44. Let $P$ be the intersection point of lines $l_{1}$ and $l_{2}$. According to problem 19.42, point $O$ lies on the circumcircle $S_{1}$ of triangle $A_{1} A_{2} P$. On the other hand, $O A_{2}: O A_{1}=k$. The geometric locus of points $X$ for which $X A_{2}: X A_{1}=k$ is the circle $S_{2}$ (problem 7.14). Point... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,884 |
19.45. Prove that the center of the spiral similarity that maps segment $A B$ to segment $A_{1} B_{1}$ coincides with the center of the spiral similarity that maps segment $A A_{1}$ to segment $B B_{1}$. | 19.45. Let $O$ be the center of a rotational homothety that maps segment $A B$ to segment $A_{1} B_{1}$. Then $\triangle A B O \sim \triangle A_{1} B_{1} O$, i.e., $\angle A O B = \angle A_{1} O B_{1}$ and $A O: B O = A_{1} O: B_{1} O$. Therefore, $\angle A O A_{1} = \angle B O B_{1}$ and $A O: A_{1} O = B O: B_{1} O$,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,885 |
19.46*. Four intersecting lines form four triangles. Prove that the four circles circumscribed about these triangles have one common point. | 19.46. Let lines $A B$ and $D E$ intersect at point $C$, and lines $B D$ and $A E$ at point $F$. The center of the rotational homothety that maps segment $A B$ to segment $E D$ is the point of intersection of the circumcircles of triangles $A E C$ and $B D C$, distinct from point $C$ (see problem 19.42), and the center... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,886 |
19.47*. In parallelogram $ABCD$ which is not a rhombus, the lines symmetric to lines $AB$ and $CD$ with respect to diagonals $AC$ and $DB$ respectively, intersect at point $Q$. Prove that $Q$ is the center of a rotational homothety that maps segment $AO$ to segment $OD$, where $O$ is the center of the parallelogram. | 19.47. The center $O$ of parallelogram $A B C D$ is equidistant from the following pairs of lines: $A Q$ and $A B, A B$ and $C D, C D$ and $D Q$, so $Q O$ is the bisector of angle $A Q D$. Let $\alpha=\angle B A O, \beta=\angle C D O$ and $\varphi=\angle A Q O=\angle D Q O$. Then $\alpha+\beta=\angle A O D=$ $=360^{\ci... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,887 |
19.48*. Two regular pentagons share a common vertex. The vertices of each pentagon are numbered from 1 to 5 clockwise, with the number 1 placed at the common vertex. The vertices with the same numbers are connected by straight lines. Prove that the four resulting lines intersect at a single point. | 19.48. Let's solve the problem in a slightly more general form. Let point $O$ be taken on the circle $S$, and $H$ be a rotational homothety with center $O$. We will prove that then all lines $X X^{\prime}$, where $X$ is a point on the circle $S$ and $X^{\prime}=H(X)$, intersect at one point.
Let $P$ be the intersectio... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,888 |
19.49*. On the sides $B C, C A$ and $A B$ of triangle $A B C$, points $A_{1}$, $B_{1}$ and $C_{1}$ are taken such that $\triangle A B C \sim \triangle A_{1} B_{1} C_{1}$. Pairs of segments $B B_{1}$ and $C C_{1}$, $C C_{1}$ and $A A_{1}$, $A A_{1}$ and $B B_{1}$ intersect at points $A_{2}$, $B_{2}$ and $C_{2}$, respect... | 19.49. Let $O$ be the center of the rotational homothety that maps triangle $A_{1} B_{1} C_{1}$ to triangle $A B C$. We will prove, for example, that the circumcircles of triangles $A B C_{2}$ and $A_{1} B_{1} C_{2}$ pass through the point $O$. Under the considered homothety, segment $A B$ is mapped to segment $A_{1} B... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,889 |
19.50*. Lines $A_{2} B_{2}$ and $A_{3} B_{3}$, $A_{3} B_{3}$ and $A_{1} B_{1}$, $A_{1} B_{1}$ and $A_{2} B_{2}$ intersect at points $P_{1}$, $P_{2}$, $P_{3}$ respectively.
a) Prove that the circumcircles of triangles $A_{1} A_{2} P_{3}$, $A_{1} A_{3} P_{2}$, and $A_{2} A_{3} P_{1}$ intersect at one point lying on the ... | 19.50. Points $A_{1}, A_{2}$, and $A_{3}$ lie on the lines $P_{2} P_{3}, P_{3} P_{1}$, and $P_{1} P_{2}$ (Fig. 19.2), so the circumcircles of triangles $A_{1} A_{2} P_{3}, A_{1} A_{3} P_{2}$, and $A_{2} A_{3} P_{1}$ have a common point $V$ (see problem 2.81, a)), and points $O_{3}, O_{2}$, and $O_{1}$ lie on these circ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,890 |
19.51*. Let $A_{1} B_{1}, A_{2} B_{2}$ and $A_{3} B_{3}$, as well as $A_{1} C_{1}, A_{2} C_{2}$ and $A_{3} C_{3}$, be corresponding segments of similar figures $F_{1}, F_{2}$ and $F_{3}$. Prove that the triangle formed by the lines $A_{1} B_{1}, A_{2} B_{2}$ and $A_{3} B_{3}$ is similar to the triangle formed by the li... | 19.51. Let $P_{1}$ be the point of intersection of the lines $A_{2} B_{2}$ and $A_{3} B_{3}$, and $P_{1}^{\prime}$ be the point of intersection of the lines $A_{2} C_{2}$ and $A_{3} C_{3}$; points $P_{2}, P_{3}, P_{2}^{\prime}$, and $P_{3}^{\prime}$ are defined similarly. Under the rotational homothety that maps $F_{1}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,891 |
19.52*. Let $l_{1}, l_{2}$ and $l_{3}$ be the corresponding lines of similar figures $F_{1}$, $F_{2}$ and $F_{3}$, intersecting at point $W$.
a) Prove that point $W$ lies on the circle of similarity of figures $F_{1}$, $F_{2}$ and $F_{3}$.
b) Let $J_{1}, J_{2}$ and $J_{3}$ be the points of intersection of lines $l_{1... | 19.52. a) Let $l_{1}^{\prime}, l_{2}^{\prime}$, and $l_{3}^{\prime}$ be the corresponding lines of figures $F_{1}, F_{2}$, and $F_{3}$, such that $l_{i}^{\prime} \| l_{i}$; these lines form triangle $P_{1} P_{2} P_{3}$. During the rotational homothety with center $O_{3}$, which transforms $F_{1}$ into $F_{2}$, the line... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,892 |
19.55*. Prove that the circle of similitude of triangle $ABC$ is the circle with diameter $KO$, where $K$ is the Lemoine point, and $O$ is the center of the circumscribed circle. | 19.55. Let $O_{a}$ be the intersection point of the circle passing through point $B$ and tangent to line $A C$ at point $A$, and the circle passing through point $C$ and tangent to line $A B$ at point $A$. According to problem 19.42, b) point $O_{a}$ is the center of the rotational homothety that maps segment $B A$ to ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,895 |
19.56*. Let $O$ be the center of the circumcircle of triangle $ABC$, $K$ the Lemoine point, $P$ and $Q$ the Brocard points, $\varphi$ the Brocard angle. Prove that points $P$ and $Q$ lie on the circle with diameter $KO$, and that $OP = OQ$ and $\angle POQ = 2\varphi$.
A triangle with vertices at the constant points of... | 19.56. If $P$ is the first Brocard point of triangle $ABC$, then $CP, AP$, and $BP$ are corresponding lines for similar figures constructed on segments $BC, CA$, and $AB$. Therefore, point $P$ lies on the circle of similitude $S$ (see problem 19.52, a)). Similarly, point $Q$ lies on the circle $S$. Moreover, lines $CP,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,896 |
19.58*. a) Prove that the lines passing through the vertices of triangle $A B C$ parallel to the sides of the Brocard triangle $A_{1} B_{1} C_{1}$ (a line through $A$ is parallel to $B_{1} C_{1}$, and so on) intersect at a single point $S$ (the Steiner point), and that this point lies on the circumcircle of triangle $A... | 19.58. a) Both statements directly follow from problem 5.105, a). Indeed, the Lemoine point $K$ lies on the circumcircle of triangle $A_{1} B_{1} C_{1}$ and $K A_{1} \| B C$ and so on.
b) This follows from problem 5.105, b). | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,898 |
20.1. Prove that if the lengths of all sides of a triangle are less than 1, then its area is less than $\sqrt{3} / 4$. | 20.1. Let $\alpha$ be the smallest angle of the triangle. Then $\alpha \leqslant 60^{\circ}$. Therefore $S=(b c \sin \alpha) / 2 \leqslant\left(\sin 60^{\circ}\right) / 2=\sqrt{3} / 4$. | \frac{\sqrt{3}}{4} | Geometry | proof | Yes | Yes | olympiads | false | 27,899 |
20.2. Prove that the circles constructed on the sides of a convex quadrilateral as diameters completely cover this quadrilateral. | 20.2. Let $X$ be an arbitrary point lying inside a convex quadrilateral. Since $\angle A X B+\angle B X C+\angle C X D+\angle A X D=360^{\circ}$, the largest of these angles is not less than $90^{\circ}$. Let for definiteness $\angle A X B \geqslant 90^{\circ}$. Then the point $X$ lies inside the circle with diameter $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,900 |
20.3. In a certain country, there are 100 airfields, and all pairwise distances between them are different. From each airfield, an airplane takes off and flies to the nearest airfield. Prove that no airfield can receive more than five airplanes. | 20.3. If planes from points $A$ and $B$ flew to point $O$, then $A B$ is the largest side of triangle $A O B$, i.e., $\angle A O B > 60^{\circ}$. Suppose that planes from points $A_{1}, \ldots, A_{n}$ flew to point $O$. Then one of the angles $A_{i} O A_{j}$ does not exceed $360^{\circ} / n$. Therefore, $360^{\circ} / ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,901 |
20.4. Inside a circle of radius 1, there are eight points. Prove that the distance between some two of them is less than 1. | 20.4. At least seven points are distinct from the center $O$ of the circle. Therefore, the smallest of the angles $A_{i} O A_{j}$, where $A_{i}$ and $A_{j}$ are the given points, does not exceed $360^{\circ} / 7 < 60^{\circ}$. If $A$ and $B$ are the points corresponding to the smallest angle, then $A B < 1$, since $A O... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,902 |
20.5. Six circles are arranged on a plane such that some point $O$ lies inside each of them. Prove that one of these circles contains the center of some other. | 20.5. One of the angles between the six segments connecting point $O$ with the centers of the circles does not exceed $360^{\circ} / 6=60^{\circ}$. Let $\angle O_{1} O O_{2} \leqslant 60^{\circ}$, where $O_{1}$ and $O_{2}$ are the centers of circles with radii $r_{1}$ and $r_{2}$, respectively. Since $\angle O_{1} O O_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,903 |
20.6*. Inside an acute-angled triangle, a point $P$ is taken. Prove that the greatest of the distances from point $P$ to the vertices of this triangle is less than twice the smallest of the distances from $P$ to its sides. | 20.6. Drop perpendiculars $P A_{1}, P B_{1}$ and $P C_{1}$ from point $P$ to the sides $B C$, $C A$ and $A B$ and choose the largest of the angles formed by these perpendiculars and the rays $P A, P B$ and $P C$. Let this be $\angle A P C_{1}$ for definiteness. Then $\angle A P C_{1} \geqslant 60^{\circ}$, so $P C_{1}:... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,904 |
20.7*. The lengths of the bisectors of a triangle do not exceed 1. Prove that its area does not exceed $1 / \sqrt{3}$.
## §2. Least or Greatest Distance | 20.7. Let $\alpha$ be the smallest angle of triangle $ABC$; $AD$ - the bisector. One of the sides $AB$ and $AC$ does not exceed $AD / \cos (\alpha / 2)$, since otherwise the segment $BC$ would not pass through point $D$. Let for definiteness $AB \leqslant AD / \cos (\alpha / 2) \leqslant AD / \cos 30^{\circ} \leqslant ... | \frac{1}{\sqrt{3}} | Geometry | proof | Yes | Yes | olympiads | false | 27,905 |
20.8. On the plane, there are $n \geqslant 3$ points, and not all of them lie on the same line. Prove that there exists a circle passing through three of the given points and not containing any of the remaining points inside it. | 20.8. Let $A$ and $B$ be the points among the given points for which the distance between them is minimal. Then, inside the circle with diameter $A B$, there are no given points. Let $C$ be the one of the remaining points from which the segment $A B$ is seen at the largest angle. Then, inside the circle passing through... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,906 |
20.9. On a plane, there are several points, all pairwise distances between which are distinct. Each of these points is connected to the nearest one. Can a closed broken line be obtained in this way? | 20.9. Suppose we have a closed broken line. Let $A B$ be the largest segment of this broken line, and $A C$ and $B D$ be the segments adjacent to it. Then $A C < A B$, i.e., $B$ is not the nearest point to $A$, and $B D < A B$, i.e., $A$ is not the nearest point to $B$. Therefore, points $A$ and $B$ cannot be connected... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,907 |
20.10. Prove that at least one of the bases of the perpendiculars dropped from an internal point of a convex polygon to its sides lies on the side itself, and not on its extension. | 20.10. Let $O$ be a given point. Draw the lines containing the sides of the polygon and choose among them the one that is the least distant from point $O$. Let side $A B$ lie on this line. We will prove that the foot of the perpendicular dropped from point $O$ to side $A B$ lies on the side itself. Suppose the foot of ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,908 |
20.11*. Prove that in any convex pentagon, there are three diagonals that can form a triangle. | 20.11. Let $B E$ be the largest diagonal of the pentagon $A B C D E$. We will prove that then a triangle can be formed from the segments $B E, E C$ and $B D$. For this, it is sufficient to check that $B E < E C + B D$. Let $O$ be the point of intersection of the diagonals $B D$ and $E C$. Then $B E < B O + O E < B D + ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,909 |
20.12*. Prove that a polygon cannot be covered by two polygons homothetic to it with a coefficient $k$, where $0<k<1$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 20.12. Let $O_{1}$ and $O_{2}$ be the centers of homotheties with coefficient $k$, transforming polygon $M$ into polygons $M_{1}$ and $M_{2}$. Then the point of polygon $M$ that is farthest from the line $O_{1} O_{2}$ is not covered by polygons $M_{1}$ and $M_{2}$. | Number Theory | proof | Yes | Yes | olympiads | false | 27,910 | |
20.13*. On a plane, there is a finite number of points, and any line passing through two of these points contains at least one more given point. Prove that all the given points lie on one line (Sylvester). | 20.13. Suppose that not all
Fig. 20.3 data points lie on the same line. Draw a line through each pair of these points (there are a finite number of such lines) and select the smallest non-z... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,911 |
20.14*. On a plane, there is a finite number of pairwise non-parallel lines, and through the intersection point of any two of them, there passes another one of the given lines. Prove that all these lines pass through one point. | 20.14. Suppose that not all
Fig. 20.4 lines pass through one point. Consider the intersection points of the lines and choose the smallest non-zero distance from these points to the given l... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,912 |
20.15*. On a plane, there are $n$ points, and the midpoints of all segments with endpoints at these points are marked. Prove that the number of distinct marked points is at least $2 n-3$.
See also problems $9.17,9.19$.
## §3. Minimum or Maximum Area | 20.15. Let $A$ and $B$ be the most distant points from each other. The midpoints of the segments connecting point $A$ (respectively point $B$) with the other points are all distinct and lie inside the circle of radius $AB / 2$ centered at $A$ (respectively $B$). The two obtained circles have only one common point, so t... | 2n-3 | Combinatorics | proof | Yes | Yes | olympiads | false | 27,913 |
20.16*. On a plane, there are $n$ points, and the area of any triangle with vertices at these points does not exceed 1. Prove that all these points can be placed inside a triangle of area 4.
20.17*. Polygon $M^{\prime}$ is homothetic to polygon $M$ with a homothety coefficient of $-1 / 2$. Prove that there exists a pa... | 20.16. Among all triangles with vertices at the given points, let's choose the triangle of the largest area. Let this be triangle $A B C$. Draw a line $l_{c} \| A B$ through vertex $C$. If points $X$ and $A$ lie on opposite sides of the line $l_{c}$, then $S_{A B X} > S_{A B C}$. Therefore, all given points lie on one ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,914 |
20.18*. Let $O$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$. Prove that if the perimeters of triangles $ABO$, $BCO$, $CDO$, and $DAO$ are equal, then $ABCD$ is a rhombus. | 20.18. For definiteness, we can assume that $A O \geqslant C O$ and $D O \geqslant B O$. Let points $B_{1}$ and $C_{1}$ be symmetric to points $B$ and $C$ with respect to point $O$ (see Fig. 20.5). Since triangle $B_{1} O C_{1}$ lies inside triangle $A O D$, then $P_{A O D} \geqslant P_{B_{1} O C_{1}}=P_{B O C}$, and e... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,915 |
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