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14.30*. Solve problem 13.44 using the properties of the center of mass.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 14.30. Place unit masses at the vertices of the polygon \(A_{1} \ldots A_{n}\). Then \(O\) is the center of mass of this system of points. Therefore, \(\overrightarrow{A_{i} O} = \left(\overrightarrow{A_{i} A_{1}} + \ldots + \overrightarrow{A_{i} A_{n}}\right) / n\) and \(A_{i} O \leqslant \left(A_{i} A_{1} + \ldots + ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,701 |
14.31*. On the sides $B C$ and $C D$ of the parallelogram $A B C D$, points $K$ and $L$ are taken such that $B K: K C = C L: L D$. Prove that the center of mass of triangle $A K L$ lies on the diagonal $B D$.
## §5. Barycentric Coordinates
Let a triangle $A_{1} A_{2} A_{3}$ be given on a plane. If the point $X$ is th... | 14.31. Let $k=B K / B C=1-(D L / D C)$. When projecting onto a line perpendicular to the diagonal $B D$, points $A, B, K$ and $L$ are transformed into points $A^{\prime}, B^{\prime}$, $K^{\prime}$ and $L^{\prime}$, such that $B^{\prime} K^{\prime}+B^{\prime} L^{\prime}=k A^{\prime} B^{\prime}+(1-k) A^{\prime} B^{\prime... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,702 |
14.32. Let a triangle $A_{1} A_{2} A_{3}$ be given. Prove that:
a) any point $X$ has some barycentric coordinates relative to it
b) under the condition $m_{1}+m_{2}+m_{3}=1$, the barycentric coordinates of point $X$ are uniquely determined. | 14.32. Let us introduce the following notations: $\boldsymbol{e}_{1}=\overrightarrow{A_{3} A_{1}}, \boldsymbol{e}_{2}=\overrightarrow{A_{3} A_{2}}$ and $\boldsymbol{x}=\overrightarrow{X A_{3}}$. The point $X$ is the center of mass of the vertices of the triangle $A_{1} A_{2} A_{3}$ with masses $m_{1}$, $m_{2}, m_{3}$ i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,703 |
14.34. Point $X$ lies inside triangle $A B C$. Lines passing through point $X$ parallel to $A C$ and $B C$ intersect side $A B$ at points $K$ and $L$ respectively. Prove that the barycentric coordinates of point $X$ are $(B L: A K: L K)$. | 14.34. When projecting onto line $A B$ parallel to line $B C$, the vector $\boldsymbol{u}=$ $=\overrightarrow{X A} \cdot B L+\overrightarrow{X B} \cdot A K+\overrightarrow{X C} \cdot L K$ transforms into the vector $\overrightarrow{L A} \cdot B L+\overrightarrow{L B} \cdot A K+\overrightarrow{L B} \cdot L K$. The latte... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,705 |
14.35. Find the barycentric coordinates of a) the circumcenter; b) the incenter; c) the orthocenter of the triangle. | 14.35. Using the result of problem 14.33, it is easy to verify that the answer is: a) $(\sin 2 \alpha: \sin 2 \beta: \sin 2 \gamma) ;$ b) $(a: b: c)$, c) $(\operatorname{tg} \alpha: \operatorname{tg} \beta: \operatorname{tg} \gamma)$.
If we require that the sum of the barycentric coordinates be equal to 1, then the an... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,706 |
14.36. Relative to triangle $ABC$, point $X$ has barycentric coordinates ( $\alpha: \beta: \gamma$ ), where $\alpha+\beta+\gamma=1$. Prove that $\overrightarrow{X A}=\beta \overrightarrow{B A}+\gamma \overrightarrow{C A}$ | 14.36. Adding the vector $(\beta+\gamma) \overrightarrow{X A}$ to both sides of the equation $\alpha \overrightarrow{X A}+\beta \overrightarrow{X B}+\gamma \overrightarrow{X C}=\overrightarrow{0}$, we get $\overrightarrow{X A}=(\beta+\gamma) \overrightarrow{X A}+\beta \overrightarrow{B X}+\gamma \overrightarrow{C X}=\b... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,707 |
14.37. Let ( $\alpha: \beta: \gamma$ ) be the barycentric coordinates of point $X$, with $\alpha+\beta+\gamma=1$; $M$ is the centroid of triangle $ABC$. Prove that $3 \overrightarrow{X M}=(\alpha-\beta) \overrightarrow{A B}+(\beta-\gamma) \overrightarrow{B C}+(\gamma-\alpha) \overrightarrow{C A}$ | 14.37. According to problem 14.1, b) $3 \overrightarrow{X M}=\overrightarrow{X A}+\overrightarrow{X B}+\overrightarrow{X C}$. In addition, $\overrightarrow{X A}=$ $=\beta \overrightarrow{B A}+\gamma \overrightarrow{C A}, \overrightarrow{X B}=\alpha \overrightarrow{A B}+\gamma \overrightarrow{C B}$ and $\overrightarrow{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,708 |
14.38*. Let $M$ be the centroid of triangle $ABC$, and $X$ be an arbitrary point. On the lines $BC$, $CA$, and $AB$, points $A_{1}$, $B_{1}$, and $C_{1}$ are taken such that $A_{1} X \parallel AM$, $B_{1} X \parallel BM$, and $C_{1} X \parallel CM$. Prove that the centroid $M_{1}$ of triangle $A_{1} B_{1} C_{1}$ coinci... | 14.38. Let the lines passing through point $X$ parallel to $A C$ and $B C$ intersect line $A B$ at points $K$ and $L$ respectively. If ( $\alpha: \beta: \gamma$ ) are the barycentric coordinates of point $X$, with $\alpha+\beta+\gamma=1$, then $2 \overrightarrow{X C_{1}}=\overrightarrow{X K}+\overrightarrow{X L}=$ $=\g... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,709 |
14.39*. Find the equation of the circumcircle of triangle $A_{1} A_{2} A_{3}$ in barycentric coordinates. | 14.39. Let $X$ be an arbitrary point, $O$ the center of the circumcircle of a given triangle, $\boldsymbol{e}_{i}=\overrightarrow{O A_{i}}$, and $\boldsymbol{a}=\overrightarrow{X O}$. If the point $X$ has barycentric coordinates $\left(x_{1}: x_{2}: x_{3}\right)$, then $\sum x_{i}\left(\boldsymbol{a}+\boldsymbol{e}_{i}... | \sum_{i<j}x_{i}x_{j}a_{ij}^{2}=0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,710 |
14.40*. a) Prove that the points with barycentric coordinates $(\alpha: \beta: \gamma)$ and ( $\alpha^{-1}: \beta^{-1}: \gamma^{-1}$ ) are isotomically conjugate with respect to triangle $A B C$.
b) The lengths of the sides of triangle $A B C$ are $a, b$, and $c$. Prove that the points with barycentric coordinates ( $... | 14.40. a) Let $X$ and $Y$ be points with barycentric coordinates $(\alpha: \beta: \gamma)$ and $(\alpha^{-1}: \beta^{-1}: \gamma^{-1})$; the lines $C X$ and $C Y$ intersect the line $A B$ at points $X_{1}$ and $Y_{1}$. Then $\overline{A X_{1}}: \overline{B X_{1}}=\beta: \alpha=\alpha^{-1}: \beta^{-1}=\overline{B Y_{1}}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,711 |
14.41*. On the lines $A B, B C, C A$, points $C_{1}$ and $C_{2}, A_{1}$ and $A_{2}, B_{1}$ and $B_{2}$ are given. Points $C_{1}$ and $C_{2}$ determine numbers $\gamma_{1}$ and $\gamma_{2}$, for which $\left(1+\gamma_{1}\right) \overrightarrow{A C_{1}}=\overrightarrow{A B}$ and $\left(1+\gamma_{2}\right) \overrightarrow... | 14.41. Points $A_{2}$ and $B_{1}$ have barycentric coordinates $\left(0: 1: \alpha_{2}\right)$ and $\left(1: 0: \beta_{1}\right)$, so in barycentric coordinates ( $\alpha: \beta: \gamma$ ) the line $A_{2} B_{1}$ is given by the equation $\alpha \beta_{1}+\beta \alpha_{2}=\gamma$. The lines $B_{2} C_{1}$ and $C_{2} A_{1... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,712 |
14.42*. The extensions of the sides of a convex quadrilateral $ABCD$ intersect at points $P$ and $Q$. Prove that the points of intersection of the bisectors of the external angles at vertices $A$ and $C$, $B$ and $D$, $P$ and $Q$ lie on one straight line. | 14.42. Consider the lines \( l_{1} = AB, l_{2} = BC, l_{3} = CD \) and \( l_{4} = AD \). Let \( x_{i} \) be the signed distance from point \( X \) to the line \( l_{i} \) (if point \( X \) and quadrilateral \( ABCD \) lie on the same side of the line \( l_{i} \), the sign is positive). Thus, \( \left(x_{1}: x_{2}: x_{3... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,713 |
14.43*. On the sides $A D$ and $D C$ of a convex quadrilateral $A B C D$, points $P$ and $Q$ are taken such that $\angle A B P=\angle C B Q$. Segments $A Q$ and $C P$ intersect at point $E$. Prove that $\angle A B E=\angle C B D$. | 14.43. Let $(x: y: z)$ be the trilinear coordinates relative to triangle $A B C$. From the equality $\angle A B P=\angle C B Q$, it follows that points $P$ and $Q$ have trilinear coordinates of the form $(p: u: q)$ and $(q: v: p)$. The lines $A P$ and $C Q$ are given by the equations $y: z=u: q$ and $x: y=q: v$, so the... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,714 |
14.44*. Find the trilinear coordinates of the Brocard points.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 14.44. Let $(x: y: z)$ be the trilinear coordinates of the first Brocard point $P$. Then $x: y: z = CP: AP: BP$. Moreover, $AP / \sin \varphi = AB / \sin \alpha = 2Rc / a$ (here $\varphi$ is the Brocard angle). Similarly, $BP = 2R \sin \varphi a / b$ and $CP = 2R \sin \varphi b / c$. Therefore, the first Brocard point ... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,715 | |
14.45*. Find the equations in trilinear coordinates for: a) the circumcircle; b) the incircle; c) the excircle.
保留了源文本的换行和格式。 | 14.45. a) The circumcircle is given by the equation $a y z + b x z + c x y = 0$, i.e., $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 0$ (here $a, b, c$ are the lengths of the sides of the triangle). One proof of this statement is contained in the solution to problem 5.11; another is in the solution to problem 14.39. Anoth... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,716 |
14.46*. Find the equation of the nine-point circle in trilinear coordinates. | 14.46. The nine-point circle is given in trilinear coordinates by the equation
$$
x^{2} \sin \alpha \cos \alpha+y^{2} \sin \beta \cos \beta+z^{2} \sin \gamma \cos \gamma=y z \sin \alpha+x z \sin \beta+x y \sin \gamma
$$
To prove this, it is sufficient to verify that the curve defined by this equation intersects each ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,717 |
14.47*. a) Prove that in trilinear coordinates any circle is given by the equation of the form
$$
(p x+q y+r z)(x \sin \alpha+y \sin \beta+z \sin \gamma)=y z \sin \alpha+x z \sin \beta+x y \sin \gamma
$$
b) Prove that the radical axis of two circles, given by equations of this form, is given by the equation
$$
p_{1}... | 14.47. The equation $y z \sin \alpha + x z \sin \beta + x y \sin \gamma = 0$ defines the circumcircle of a triangle. In Cartesian coordinates, the equation of any circle can be obtained by subtracting a linear function from the equation of a fixed circle. In trilinear coordinates, to maintain homogeneity, the "linear f... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,718 |
14.48*. Prove that the tangent to the inscribed circle at the point $\left(x_{0}: y_{0}: z_{0}\right)$ is given by the equation
$$
\frac{x}{\sqrt{x_{0}}} \cos \frac{\alpha}{2}+\frac{y}{\sqrt{y_{0}}} \cos \frac{\beta}{2}+\frac{z}{\sqrt{z_{0}}} \cos \frac{\gamma}{2}=0
$$ | 14.48. Let the points ( $x_{0}: y_{0}: z_{0}$ ) and ( $x_{1}: y_{1}: z_{1}$ ) lie on the inscribed circle. Then the line passing through these points is given by the equation
$$
x\left(\sqrt{y_{0} z_{1}}+\sqrt{y_{1} z_{0}}\right) \cos \frac{\alpha}{2}+y\left(\sqrt{x_{0} z_{1}}+\sqrt{x_{1} z_{0}}\right) \cos \frac{\bet... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,719 |
14.49*. Prove that the incircle touches the nine-point circle (Feuerbach). Find the trilinear coordinates of the point of tangency. | 14.49. The equation of the inscribed circle can be written in the form
$$
\left(x \frac{\cos ^{4} \frac{\alpha}{2}}{\sin \alpha}+\ldots\right)(x \sin \alpha+\ldots)=\frac{4 \cos ^{2} \frac{\alpha}{2} \cos ^{2} \frac{\beta}{2} \cos ^{2} \frac{\gamma}{2}}{\sin \alpha \sin \beta \sin \gamma}(y z \sin \alpha+\ldots)
$$
a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,720 |
15.1. Where should the bridge $M N$ be built across the river separating villages $A$ and $B$, so that the path $A M N B$ from $A$ to $B$ is the shortest? (The riverbanks are considered parallel lines, and the bridge is perpendicular to the banks.) | 15.1. Let $A^{\prime}$ be the image of point $A$ under a parallel translation by vector $\overrightarrow{M N}$. Then $A^{\prime} N = A M$, so the length of the path $A M N B$ is $A^{\prime} N + N B + M N$. Since the length of segment $M N$ is constant, we need to find the point $N$ for which the sum $A^{\prime} N + N B... | 0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,722 |
15.2. Given a triangle $A B C$. A point $M$, located inside the triangle, moves parallel to side $B C$ until it intersects side $C A$, then parallel to $A B$ until it intersects $B C$, then parallel to $A C$ until it intersects $A B$, and so on. Prove that after a certain number of steps, the trajectory of the point wi... | 15.2. Let's denote the consecutive points of the trajectory on the sides of the triangle as \(A_{1}, B_{1}\), \(B_{2}, C_{2}, C_{3}, A_{3}, A_{4} B_{4}, \ldots\) (Fig. 15.1). Since \(A_{1} B_{1} \| A B_{2}, B_{1} B_{2} \| C A_{1}\) and \(B_{1} C \| B_{2} C_{2}\), the triangle \(A B_{2} C_{2}\) is obtained from the tria... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,723 |
15.3. Let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$ of a convex quadrilateral $ABCD$.
a) Prove that $KM \leqslant (BC + AD) / 2$, and equality is achieved if and only if $BC \| AD$.
b) For fixed side lengths of the quadrilateral $ABCD$, find the maximum values of the lengths of the segment... | 15.3. a) Let's complete the triangle $C B D$ to a parallelogram $C B D E$. Then $2 K M = A E \leqslant A D + D E = A D + B C$, and equality is achieved only if $A D \| B C$.
b) Let $a = A B$, $b = B C$, $c = C D$, and $d = D A$. If $|a - c| = |b - d| \neq 0$, then according to part a), the maximum is achieved in the d... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,724 |
15.4. In trapezoid $ABCD$, sides $BC$ and $AD$ are parallel, $M$ is the point of intersection of the angle bisectors of angles $A$ and $B$, and $N$ is the point of intersection of the angle bisectors of angles $C$ and $D$. Prove that $2MN = |AB + CD - BC - AD|$. | 15.4. For the described trapezoid $A B C^{\prime} D^{\prime}$, the equality $2 M N^{\prime}=\left|A B+C^{\prime} D^{\prime}-B C^{\prime}-A D^{\prime}\right|$ is obvious, since $N^{\prime}=M$. When transitioning from trapezoid $A B C^{\prime} D^{\prime}$ to trapezoid $A B C D$, $2 N^{\prime} N$ is added to the left side... | 2MN=|AB+CD-BC-AD| | Geometry | proof | Yes | Yes | olympiads | false | 27,725 |
15.5*. From the vertex $B$ of the parallelogram $A B C D$, its heights $B K$ and $B H$ are drawn. It is known that $K H=a$ and $B D=b$. Find the distance from point $B$ to the intersection point of the heights of triangle $B K H$.
untranslated text:
15.5*. Из вершины $B$ параллелограмма $A B C D$ проведены его высоты... | 15.5. Let the intersection point of the altitudes of triangle $B K H$ be denoted as $H_{1}$. Since $H H_{1} \perp B K$ and $K H_{1} \perp B H$, it follows that $H H_{1} \| A D$ and $K H_{1} \| D C$, i.e., $H_{1} H D K$ is a parallelo-
 are parallel, and the distance between
Fig. 15.4
lines \( l_{11} ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,727 |
15.7*. In a square with side 1, there is a figure such that the distance between any two points of it is not equal to 0.001. Prove that the area of this figure does not exceed: a) 0.34; b) 0.287.
## §2. Constructions and Geometric Loci | 15.7. a) Let the figure lying inside the square \(ABCD\) with side 1 be denoted by \(F\), and its area by \(S\). Consider two vectors \(\overrightarrow{AA_1}\) and \(\overrightarrow{AA_2}\), where point \(A_1\) lies on side \(AD\) and \(AA_1 = 0.001\), and point \(A_2\) lies inside angle \(BAD\), \(\angle A_2AA_1 = 60^... | 0.287 | Geometry | proof | Yes | Yes | olympiads | false | 27,728 |
15.8. Given an angle $A B C$ and a line $l$. Construct a line parallel to line $l$, on which the sides of angle $A B C$ intercept a segment of a given length $a$. | 15.8. There are two vectors $\pm \boldsymbol{a}$, parallel to the line $l$ and having a given length $a$. Consider the images of the ray $B C$ under parallel translations by these vectors. The point of their intersection with the ray $B A$ lies on the desired line (if they do not intersect, then the problem has no solu... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,729 |
15.9. Given two circles $S_{1}, S_{2}$ and a line $l$. Draw a line $l_{1}$, parallel to the line $l$, such that:
a) the distance between the points of intersection of $l_{1}$ with the circles $S_{1}$ and $S_{2}$ has a given magnitude $a$;
b) $S_{1}$ and $S_{2}$ cut equal chords on $l_{1}$;
c) $S_{1}$ and $S_{2}$ cut... | 15.9. a) Let $S_{1}^{\prime}$ be the image of the circle $S_{1}$ under a parallel translation by a vector of length $a$, parallel to the line $l$ (there are two such vectors). The required line passes through the point of intersection of the circles $S_{1}^{\prime}$ and $S_{2}$.
b) Let $O_{1}$ and $O_{2}$ be the proje... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,730 |
15.10. Given non-intersecting chords \(AB\) and \(CD\) of a circle. Construct a point \(X\) on the circle such that the chords \(AX\) and \(BX\) intercept a segment \(EF\) on the chord \(CD\) with a given length \(a\). | 15.10. Suppose point $X$ is constructed. Translate point $A$ by vector $\overrightarrow{E F}$, i.e., construct point $A^{\prime}$ such that $\overrightarrow{E F}=\overrightarrow{A A^{\prime}}$. This construction can be done since vector $\overrightarrow{E F}$ is known: its length is $a$ and it is parallel to $C D$.
Si... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,731 |
15.11. Construct a quadrilateral $A B C D$ given four angles and the lengths of sides $A B=a$ and $C D=b$. | 15.11. Suppose that quadrilateral $A B C D$ is constructed. Denote the image of point $D$ under the translation by vector $\overrightarrow{C B}$ as
Fig. 15.6 as $D_{1}$. In triangle $A B D... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,732 |
15.12. Given circles $S_{1}, S_{2}$ and a point $A$. Draw a line $l$ through point $A$ such that $S_{1}$ and $S_{2}$ intercept equal chords on it. | 15.12. Suppose that points $M$ and $N$, where line $l$ intersects circle $S_{2}$, are constructed. Let $O_{1}$ and $O_{2}$ be the centers of circles $S_{1}$ and $S_{2}$; $O_{1}^{\prime}$ be the image of point $O_{1}$ under a parallel translation along line $l$ such that $O_{1}^{\prime} O_{2} \perp M N$, and $S_{1}^{\pr... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,733 |
15.13. a) Given circles $S_{1}$ and $S_{2}$, intersecting at points $A$ and $B$. Draw a line $l$ through point $A$ such that the segment of this line, enclosed within circles $S_{1}$ and $S_{2}$, has a given length.
b) Inscribe in a given triangle $ABC$ a triangle equal to a given triangle $PQR$. | 15.13. a) Draw a line $P Q$ through point $A$ (where $P$ lies on circle $S_{1}$ and $Q$ lies on circle $S_{2}$). Drop perpendiculars $O_{1} M$ and $O_{2} N$ from the centers $O_{1}$ and $O_{2}$ of circles $S_{1}$ and $S_{2}$ to the line $P Q$. Translate segment $M N$ parallel to vector $\overrightarrow{M O_{1}}$. Let $... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,734 |
15.15*. Find the geometric locus of points: a) the sum; b) the difference of the distances from which to two given lines has a given magnitude.
保留了源文本的换行和格式，这里直接输出了翻译结果。 | 15.15. Drop perpendiculars \( XA_1 \) and \( XA_2 \) from point \( X \) to the given lines \( l_1 \) and \( l_2 \). Take a point \( B \) on the ray \( A_1X \) such that \( A_1B = a \). Then, in the cases \( XA_1 \pm XA_2 = a \), we get \( XB = XA_2 \). Let \( l_1' \) be the image of the line \( l_1 \) under a parallel ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,736 |
15.16*. A corner made of a transparent material is moved so that two non-intersecting circles touch its sides internally. Prove that there is a point on it that describes an arc of a circle.
## Problems for independent solving | 15.16. Let side $A B$ of angle $B A C$ touch a circle of radius $r_{1}$ with center $O_{1}$, and side $A C$ touch a circle of radius $r_{2}$ with center $O_{2}$. Translate line $A B$ parallel to itself inside angle $B A C$ by a distance $r_{1}$, and line $A C$ by a distance $r_{2}$. Let $A_{1}$ be the point of intersec... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,737 |
16.1. Prove that if in a triangle the median and the bisector coincide, then the triangle is isosceles. | 16.1. Let in triangle $A B C$ the median $B D$ be a bisector. Consider the point $B_{1}$, symmetric to $B$ with respect to point $D$. Since $D$ is the midpoint of segment $A C$, the quadrilateral $A B C B_{1}$ is a parallelogram. And since $\angle A B B_{1}=\angle B_{1} B C=\angle A B_{1} B$, triangle $B_{1} A B$ is is... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,738 |
16.2. Two players take turns placing coins on a rectangular table. A coin can only be placed on a free spot. The player who cannot make a move loses. Prove that the first player can always win. | 16.2. The first player places a coin in the center of the table, and then places coins symmetrically to the coins of the second player relative to the center of the table. With this strategy, the first player always has the opportunity to make the next move. It is also clear that the game will end in a finite number of... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 27,739 |
16.3. A circle intersects the sides $B C, C A, A B$ of triangle $A B C$ at points $A_{1}$ and $A_{2}, B_{1}$ and $B_{2}, C_{1}$ and $C_{2}$ respectively. Prove that if the perpendiculars to the sides of the triangle, drawn through points $A_{1}$, $B_{1}$ and $C_{1}$, intersect at one point, then the perpendiculars to t... | 16.3. Let the perpendiculars to the sides, drawn through points $A_{1}, B_{1}$ and $C_{1}$, intersect at point $M$. Denote the center of the circle by $O$. The perpendicular to side $B C$, drawn through point $A_{1}$, is symmetric with respect to point $O$ to the perpendicular to side $B C$, drawn through point $A_{2}$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,740 |
16.4. Prove that the lines drawn through the midpoints of the sides of a cyclic quadrilateral perpendicular to the opposite sides intersect at one point. | 16.4. Let $P, Q, R$ and $S$ be the midpoints of sides $A B, B C, C D$ and $D A$; $M$ be the point of intersection of segments $P R$ and $Q S$ (i.e., the midpoint of both these segments; see problem 14.5); $O$ be the center of the circumscribed circle, and point $O^{\prime}$ be symmetric to $O$ with respect to point $M$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,741 |
16.5. Let $P$ be the midpoint of side $AB$ of a convex quadrilateral $ABCD$. Prove that if the area of triangle $PCD$ is half the area of quadrilateral $ABCD$, then $BC \parallel AD$. | 16.5. Let point $D^{\prime}$ be symmetric to $D$ with respect to point $P$. If the area of triangle $P C D$ is half the area of quadrilateral $A B C D$, then it is equal to the sum of the areas of triangles $P B C$ and $P A D$, i.e., the sum of the areas of triangles $P B C$ and $P B D^{\prime}$. Since $P$ is the midpo... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,742 |
16.6. Circles $S_{1}$ and $S_{2}$ with radius 1 touch at point $A$; the center $O$ of circle $S$ with radius 2 lies on $S_{1}$. Circle $S_{1}$ touches $S$ at point $B$. Prove that the line $A B$ passes through the point of intersection of circles $S_{2}$ and $S$. | 16.6. Circles $S_{1}$ and $S_{2}$ are symmetric with respect to point $A$. Since $O B$ is the diameter of circle $S_{1}$, $\angle B A O=90^{\circ}$, so when reflected over $A$, point $B$ again falls on circle $S$. Therefore, when reflected over $A$, point $B$ transitions to the point of intersection of circles $S_{2}$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,743 |
16.7*. In triangle $A B C$, medians $A F$ and $C E$ are drawn. Prove that if $\angle B A F=\angle B C E=30^{\circ}$, then triangle $A B C$ is equilateral. | 16.7. Since $\angle E A F=\angle E C F=30^{\circ}$, points $A, E, F$ and $C$ lie on the same circle $S$, and if $O$ is its center, then $\angle E O F=60^{\circ}$. Point $B$ is symmetric to $A$ with respect to point $E$, so it lies on the circle $S_{1}$, which is symmetric to circle $S$ with respect to point $E$. Simila... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,744 |
16.8*. Given a convex $n$-gon with pairwise non-parallel sides and a point $O$ inside it. Prove that no more than $n$ lines can be drawn through point $O$, each of which divides the area of the $n$-gon in half.
## §. Properties of symmetry | 16.8. Consider a polygon that is symmetric to the original one with respect to point $O$. Since the sides of the polygon are pairwise non-parallel, the contours of these polygons cannot have common segments, but can only have common points. And since the polygons are convex, there are no more than two intersection poin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,745 |
16.9. a) Prove that the composition of two central symmetries is a translation.
b) Prove that the composition of a translation and a central symmetry (in both orders) is a central symmetry. | 16.9. a) Let point $A$ under central symmetry with respect to point $O_{1}$ go to point $A_{1}$, and point $A_{1}$ under symmetry with respect to $O_{2}$ go to point $A_{2}$. Then $O_{1} O_{2}$ is the midline of triangle $A A_{1} A_{2}$, therefore $\overrightarrow{A A_{2}}=2 \overrightarrow{O_{1} O_{2}}$.
b) Let $O_{2... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,746 |
16.10. Prove that if a point is reflected symmetrically with respect to points $O_{1}, O_{2}$ and $O_{3}$, and then reflected symmetrically again with respect to these same points, it will return to its original position. | 16.10. According to the previous problem, $S_{B} \circ S_{A}=T_{2 \overrightarrow{A B}}$. Therefore, $S_{O_{3}} \circ S_{O_{2}} \circ S_{O_{1}} \circ S_{O_{3}} \circ S_{O_{2}} \circ S_{O_{1}}=T_{2\left(\overrightarrow{O_{2} O_{3}}+\overrightarrow{O_{3} O_{1}}+\overrightarrow{O_{1} O_{2}}\right)}$ - an identity transfor... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,747 |
16.11. a) Prove that a bounded figure cannot have more than one center of symmetry.
b) Prove that no figure can have exactly two centers of symmetry.
c) Let \( M \) be a finite set of points on a plane. A point \( O \) is called an "almost center of symmetry" of the set \( M \) if one point can be removed from \( M \... | 16.11. a) Suppose a bounded figure has two centers of symmetry \(O_{1}\) and \(O_{2}\). Let's introduce a coordinate system with the x-axis directed along the ray \(O_{1} O_{2}\). Since \(S_{O_{2}} \circ S_{O_{1}} = T_{2 \overrightarrow{O_{1} O_{2}}}\), the figure maps onto itself under the translation by the vector \(... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,748 |
16.12. On the segment $A B$, there are $n$ pairs of points symmetric with respect to its midpoint; $n$ points are painted blue, the others are painted red. Prove that the sum of the distances from $A$ to the blue points is equal to the sum of the distances from $B$ to the red points.
## §3. Symmetry helps solve the pr... | 16.12. If a pair of symmetric points is colored in different colors, then it can simply be discarded from consideration; discard all such pairs. In the remaining set of points, the number of blue pairs is equal to the number of red pairs. In addition, the sum of the distances from point $A$, as well as from point $B$, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,749 |
16.13. Through the common point $A$ of circles $S_{1}$ and $S_{2}$, draw a line such that these circles intercept equal chords on it. | 16.13. Consider the circle $S_{1}^{\prime}$, symmetric to the circle $S_{1}$ with respect to the point $A$. The desired line passes through the points of intersection of $S_{1}^{\prime}$ and $S_{2}$. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,750 |
16.14. Through the given point $A$, draw a line such that the segment enclosed between the points of intersection of this line with the given line and the given circle is bisected by point $A$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation r... | 16.14. Let $l^{\prime}$ be the image of the line $l$ under the symmetry with respect to the point $A$. The desired line passes through the point $A$ and the point of intersection of the line $l^{\prime}$ with the circle $S$. | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,751 | |
16.15. Given an angle $ABC$ and a point $D$ inside it. Construct a segment with endpoints on the sides of the given angle, such that its midpoint is located at point $D$. | 16.15. Let's construct points \( A' \) and \( C' \) as the intersections of the lines symmetric to lines \( BC \) and \( AB \) with respect to point \( D \), with lines \( AB \) and \( BC \) (Fig. 16.3). It is clear that point \( D \) is the midpoint of the constructed segment \( A' C' \), since points \( A' \) and \( ... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,752 | |
16.16. Given an angle and points $A$ and $B$ inside it. Construct a parallelogram for which points $A$ and $B$ are opposite vertices, and the other two vertices lie on the sides of the angle. | 16.16. Let $O$ be the midpoint of segment $AB$. It is necessary to construct points $C$ and $D$, lying on the sides of the angle, such that point $O$ is the midpoint of segment $CD$. This construction is described in the solution to the previous problem. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,753 |
16.17. Given four pairwise non-parallel lines and a point $O$, not lying on these lines. Construct a parallelogram with center $O$ and vertices lying on the given lines, - one on each.
翻译结果如下：
16.17. Given four pairwise non-parallel lines and a point $O$, not lying on these lines. Construct a parallelogram with center... | 16.17. Let's preliminarily divide the lines into pairs. This can be done in three ways. Let the opposite vertices $A$ and $C$ of the parallelogram $ABCD$ lie on one pair of lines, and $B$ and $D$ on the other. Considering the angle formed by the first pair of lines, we construct points $A$ and $C$ as described in the s... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,754 |
16.19*. Given non-intersecting chords $AB$ and $CD$ of a circle and a point $J$ on chord $CD$. Construct a point $X$ on the circle such that the chords $AX$ and $BX$ cut off a segment $EF$ on chord $CD$, which is bisected by point $J$. | 16.19. Suppose that point $X$ is constructed. Denote the images of points $A$, $B$, and $X$ under the symmetry with respect to point $J$ as $A^{\prime}$, $B^{\prime}$, and $X^{\prime}$, respectively (Fig. 16.4). The angle $\angle A^{\prime} F B=180^{\circ}-\angle A X B$ is known, so point $F$ is the intersection of seg... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,756 |
16.20*. Through the common point $A$ of circles $S_{1}$ and $S_{2}$, draw a line $l$ such that the difference in the lengths of the chords cut by $l$ on circles $S_{1}$ and $S_{2}$ has a given value $a$. | 16.20. Suppose that the line \( l \) is constructed. Consider the circle \( S_{1}^{\prime} \), which is symmetric to the circle \( S_{1} \) with respect to point \( A \). Let \( O_{1}, O_{1}^{\prime} \), and \( O_{2} \) be the centers of the circles \( S_{1}, S_{1}^{\prime} \), and \( S_{2} \) (Fig. 16.5). Draw lines \... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,757 |
16.21*. Given $m=2 n+1$ points - the midpoints of the sides of an $m$-gon. Construct its vertices.
## Problems for Independent Solution | 16.21. Let $B_{1}, B_{2}, \ldots, B_{m}$ be the midpoints of the sides $A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{m} A_{1}$ of the polygon $A_{1} A_{2} \ldots A_{m}$. Then $S_{B_{1}}\left(A_{1}\right)=A_{2}, S_{B_{2}}\left(A_{2}\right)=A_{3}, \ldots, S_{B_{m}}\left(A_{m}\right)=A_{1}$. Therefore, $S_{B_{m}} \circ \ldots \ci... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,758 |
17.1. Point $M$ lies on the diameter $A B$ of a circle. Chord $C D$ passes through $M$ and intersects $A B$ at an angle of $45^{\circ}$. Prove that the sum $C M^{2}+D M^{2}$ does not depend on the choice of point $M$. | 17.1. Let the points symmetric to points $C$ and $D$ with respect to the line $A B$ be denoted as $C^{\prime}$ and $D^{\prime}$, respectively. $\angle C^{\prime} M D=90^{\circ}$, therefore $C M^{2}+M D^{2}=C^{\prime} M^{2}+$ $+M D^{2}=C^{\prime} D^{2}$. Since $\angle C^{\prime} C D=45^{\circ}$, the chord $C^{\prime} D$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,759 |
17.2. Equal circles $S_{1}$ and $S_{2}$ touch the circle $S$ internally at points $A_{1}$ and $A_{2}$. An arbitrary point $C$ on the circle $S$ is connected by segments to points $A_{1}$ and $A_{2}$. These segments intersect $S_{1}$ and $S_{2}$ at points $B_{1}$ and $B_{2}$. Prove that $A_{1} A_{2} \| B_{1} B_{2}$. | 17.2. Let's draw a diameter of circle $S$ which is the axis of symmetry of circles $S_{1}$ and $S_{2}$. Let points $C^{\prime}$ and $B_{2}^{\prime}$ be symmetric to points $C$ and $B_{2}$ with respect to this diameter (Fig. 17.1).
Circles $S_{1}$ and $S$ are homothetic with the center of homothety at point $A_{1}$, an... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,760 |
17.3. Through a point $M$ on the base $AB$ of an isosceles triangle $ABC$, a line is drawn intersecting its lateral sides $CA$ and $CB$ (or their extensions) at points $A_1$ and $B_1$. Prove that $A_1A : A_1M = B_1B : B_1M$.
## §2. Constructions | 17.3. Let the line symmetric to the line $A_{1} B_{1}$ with respect to the line $A B$ intersect the sides $C A$ and $C B$ (or their extensions) at points $A_{2}$ and $B_{2}$. Since $\angle A_{1} A M=\angle B_{2} B M$ and $\angle A_{1} M A=\angle B_{2} M B$, we have $\triangle A_{1} A M \sim \triangle B_{2} B M$, i.e., ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,761 |
17.4. Construct a quadrilateral $A B C D$ in which the diagonal $A C$ is the bisector of angle $A$, given the lengths of its sides. | 17.4. Suppose quadrilateral $A B C D$ is constructed. Let, for definiteness, $A D>A B$. Denote by $B^{\prime}$ the point symmetric to point $B$ with respect to the diagonal $A C$. The point $B^{\prime}$ lies on side $A D$, and $B^{\prime} D=A D-A B$.
 $c, a-b (a>b)$ and angle $C$; b) $c$, $a+b$ and angle $C$. | 17.8. a) Suppose that triangle $A B C$ is constructed. Let $C^{\prime}$ be the point symmetric to point $A$ with respect to the bisector of angle $C$. Then $\angle B C^{\prime} A=$ $=180^{\circ}-\angle A C^{\prime} C=180^{\circ}-\left(180^{\circ}-\angle C\right) / 2=90^{\circ}+\angle C / 2$ and $B C^{\prime}=a-b$.
In ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,766 |
17.9. Given a line $l$ and points $A$ and $B$, lying on the same side of it. Construct a point $X$ on the line $l$ such that $A X + X B = a$, where $a$ is a given value. | 17.9. Let $S$ be a circle of radius $a$ with center $B$, $S^{\prime}$ be a circle of radius $AX$ with center $X$, and $A^{\prime}$ be the point symmetric to point $A$ with respect to line $l$. Then the circle $S^{\prime}$ is tangent to the circle $S$, and the point $A^{\prime}$ lies on the circle $S^{\prime}$. It remai... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,767 |
17.10. Given an acute angle $M O N$ and points $A$ and $B$ inside it. Find a point $X$ on the side $O M$ such that the triangle $X Y Z$, where $Y$ and $Z$ are the points of intersection of the lines $X A$ and $X B$ with $O N$, is isosceles: $X Y = X Z$. | 17.10. Let the projection of point $A$ onto line $O N$ be closer to point $O$ than the projection of point $B$. Suppose that the isosceles triangle $X Y Z$ is constructed.
Fig. 17.3
Consid... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,768 |
17.11. Given a line $M N$ and two points $A$ and $B$ on the same side of it. Construct a point $X$ on the line $M N$ such that $\angle A X M=2 \angle B X N$. | 17.11. Suppose that point $X$ is constructed. Let $B^{\prime}$ be the point symmetric to point $B$ with respect to the line $M N$; the circle of radius $A B^{\prime}$ centered at $B^{\prime}$ intersects the line $M N$ at point $A^{\prime}$. Then the ray $B^{\prime} X$ is the bisector of angle $A B^{\prime} A^{\prime}$.... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,769 |
17.12. Given three lines $l_{1}, l_{2}$ and $l_{3}$, intersecting at one point, and a point $A_{1}$ on the line $l_{1}$. Construct a triangle $A B C$ such that the point $A_{1}$ is the midpoint of its side $B C$, and the lines $l_{1}, l_{2}$ and $l_{3}$ are the perpendicular bisectors of the sides. | 17.12. Draw a line $BC$ through point $A_{1}$, perpendicular to line $l_{1}$. The vertex $A$ of the desired triangle $ABC$ is the intersection point of the lines symmetric to line $BC$ with respect to lines $l_{2}$ and $l_{3}$. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,770 |
17.13. Construct triangle $ABC$, if points $A, B$ and the line on which the bisector of angle $C$ lies are given. | 17.13. Let point $A^{\prime}$ be symmetric to point $A$ with respect to the bisector of angle $C$. Then $C$ is the point of intersection of the line $A^{\prime} B$ and the line on which the bisector of angle $C$ lies. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,771 |
17.14. Given three lines $l_{1}, l_{2}$ and $l_{3}$, intersecting at one point, and a point $A$ on line $l_{1}$. Construct a triangle $A B C$ such that point $A$ is its vertex, and the angle bisectors of the triangle lie on the lines $l_{1}, l_{2}$ and $l_{3}$. | 17.14. Let $A_{2}$ and $A_{3}$ be the points symmetric to point $A$ with respect to the lines $l_{2}$ and $l_{3}$. Then the points $A_{2}$ and $A_{3}$ lie on the line $B C$. Therefore, points $B$ and $C$ are the points of intersection of the line $A_{2} A_{3}$ with the lines $l_{2}$ and $l_{3}$. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,772 |
17.15. Construct a triangle given the midpoints of two sides and the line on which the bisector, drawn to one of these sides, lies.
## §3. Inequalities and Extrema | 17.15. Suppose that triangle $A B C$ is constructed, where $N$ is the midpoint of $A C$, $M$ is the midpoint of $B C$, and the bisector of angle $A$ lies on the given line $l$. Construct point $N^{\prime}$, which is symmetric to $N$ with respect to line $l$. The line $B A$ passes through point $N^{\prime}$ and is paral... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,773 |
17.16. On the bisector of the exterior angle $C$ of triangle $ABC$, a point $M$ is taken, different from $C$. Prove that $MA + MB > CA + CB$. | 17.16. Let points $A^{\prime}$ and $B^{\prime}$ be symmetric to $A$ and $B$ with respect to the line $C M$. Then $A M+M B=A^{\prime} M+M B>A^{\prime} B=A^{\prime} C+C B=A C+C B$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,774 |
17.17. In triangle $ABC$, the median $AM$ is drawn. Prove that $2 AM \geqslant (b+c) \cos (\alpha / 2)$. | 17.17. Let points $B^{\prime}, C^{\prime}$, and $M^{\prime}$ be symmetric to points $B, C$, and $M$ with respect to the external angle bisector at vertex $A$. Then $A M + A M^{\prime} \geqslant M M^{\prime} =$
$= \left(B B^{\prime} + C C^{\prime}\right) / 2 = (b + c) \sin \left(90^{\circ} - (\alpha / 2)\right) = (b + c... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,775 |
17.19. Prove that the area of any convex quadrilateral does not exceed half the sum of the products of opposite sides. | 17.19. Let $D^{\prime}$ be the point symmetric to point $D$ with respect to the perpendicular bisector of segment $A C$. Then $S_{A B C D}=S_{A B C D^{\prime}}=S_{B A D^{\prime}}+$ $+S_{B C D^{\prime}} \leqslant A B \cdot A D^{\prime} / 2+B C \cdot C D^{\prime} / 2=(A B \cdot C D+B C \cdot A D) / 2$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,777 |
17.20. Given a line $l$ and two points $A$ and $B$ on the same side of it. Find a point $X$ on the line $l$ such that the length of the broken line $A X B$ is minimized. | 17.20. Let point $A^{\prime}$ be symmetric to point $A$ with respect to line $l$. Let $X$ be a point on line $l$. Then $A X+X B=A^{\prime} X+X B \geqslant A^{\prime} B$, and equality is achieved if and only if point $X$ lies on the segment $A^{\prime} B$. Therefore, the desired point is the intersection of line $l$ and... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,778 |
17.22. a) Lines $l_{1}$ and $l_{2}$ are parallel. Prove that $S_{l_{1}} \circ S_{l_{2}}=T_{2 \boldsymbol{a}}$, where $T_{\boldsymbol{a}}$ is the parallel translation that maps $l_{1}$ to $l_{2}$, and $\boldsymbol{a} \perp l_{1}$.
b) Lines $l_{1}$ and $l_{2}$ intersect at point $O$. Prove that $S_{l_{2}} \circ S_{l_{1}... | 17.22. Let $X$ be an arbitrary point, $X_{1}=S_{l_{1}}(X)$ and $X_{2}=S_{l_{2}}\left(X_{1}\right)$.
a) Choose an arbitrary point $O$ on the line $l_{1}$ and consider a coordinate system with origin $O$ and the x-axis directed along the line $l_{1}$. The line $l_{2}$ is given in this coordinate system by the equation $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,780 |
17.23. On the plane, there are three lines $a, b, c$. Let $T=S_{a} \circ S_{b} \circ S_{c}$. Prove that $T \circ T$ is a parallel translation (or the identity transformation). | 17.23. Let's represent $T \circ T$ as a composition of three transformations:
$T \circ T=\left(S_{a} \circ S_{b} \circ S_{c}\right) \circ\left(S_{a} \circ S_{b} \circ S_{c}\right)=\left(S_{a} \circ S_{b}\right) \circ\left(S_{c} \circ S_{a}\right) \circ\left(S_{b} \circ S_{c}\right)$.
Here, $S_{a} \circ S_{b}, S_{c} \... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,781 |
17.24. Let $l_{3}=S_{l_{1}}\left(l_{2}\right)$. Prove that $S_{l_{3}}=S_{l_{1}} \circ S_{l_{2}} \circ S_{l_{1}}$. | 17.24. If points $X$ and $Y$ are symmetric with respect to line $l_{3}$, then points $S_{l_{1}}(X)$ and $S_{l_{1}}(Y)$ are symmetric with respect to line $l_{2}$, i.e., $S_{l_{1}}(X)=S_{l_{2}} \circ S_{l_{1}}(Y)$. Therefore, $S_{l_{1}} \circ S_{l_{3}}=S_{l_{2}} \circ S_{l_{1}}$ and $S_{l_{3}}=S_{l_{1}} \circ S_{l_{2}} ... | S_{l_{3}}=S_{l_{1}}\circS_{l_{2}}\circS_{l_{1}} | Geometry | proof | Yes | Yes | olympiads | false | 27,782 |
17.25. The incircle touches the sides of triangle $ABC$ at points $A_1, B_1$, and $C_1$; points $A_2, B_2$, and $C_2$ are symmetric to these points with respect to the bisectors of the corresponding angles of the triangle. Prove that $A_2 B_2 \parallel A B$ and the lines $A A_2, B B_2$, and $C C_2$ intersect at one poi... | 17.25. Let $O$ be the center of the inscribed circle; $a$ and $b$ be the lines $OA$ and $OB$. Then $S_{a} \circ S_{b}\left(C_{1}\right)=S_{a}\left(A_{1}\right)=A_{2}$ and $S_{b} \circ S_{a}\left(C_{1}\right)=S_{b}\left(B_{1}\right)=B_{2}$. Points $A_{2}$ and $B_{2}$ are obtained from point $C_{1}$ by rotations around $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,783 |
17.26*. Two lines intersect at an angle $\gamma$. A grasshopper jumps from one line to the other; the length of each jump is 1 m, and the grasshopper does not jump back unless it is impossible. Prove that the sequence of jumps is periodic if and only if $\gamma / \pi$ is a rational number. | 17.26. For each jump vector, there are exactly two positions of the grasshopper for which the jump is defined by this vector. Therefore, the sequence of jumps is periodic if and only if there is only a finite number of different jump vectors.
Let $a_{1}$ be the jump vector of the grasshopper from line $l_{2}$ to line ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,784 |
17.29*. Inscribe an $n$-sided polygon in the given circle, one of whose sides passes through the given point, and the other sides are parallel to the given lines.
## §5. Properties of Symmetries and Axes of Symmetry | 17.29. Under sequential symmetries with respect to lines $l_{1}, \ldots, l_{n-1}$, which are perpendicular to the given lines and pass through the center of the circle, vertex $A_{1}$ of the desired polygon is transformed into vertex $A_{n}$. If $n$ is odd, the composition of these symmetries is a rotation by a known a... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,787 |
17.30. Point $A$ is located 50 cm from the center of a circle with a radius of 1 cm. It is allowed to reflect the point symmetrically relative to any line intersecting the circle. Prove that: a) in 25 reflections, point $A$ can be "driven" inside the given circle; b) in 24 reflections, this cannot be done. | 17.30. Let $O$ be the center of a given circle, $D_{R}$ be the circle of radius $R$ centered at $O$. We will prove that the set of images of points of $D_{R}$ under reflections about lines passing through $D_{1}$ is the circle $D_{R+2}$. Indeed, the images of point $O$ under the specified reflections fill the circle $D... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,788 |
17.31. On a circle with center $O$, points $A_{1}, \ldots, A_{n}$ are given, dividing it into equal arcs, and a point $X$. Prove that the points symmetric to $X$ with respect to the lines $O A_{1}, \ldots, O A_{n}$ form a regular polygon. | 17.31. Let the symmetries with respect to the lines $O A_{1}, \ldots, O A_{n}$ be denoted by $S_{1}, \ldots, S_{n}$. Let $X_{k}=S_{k}(X)$ for $k=1, \ldots, n$. We need to prove that for some rotation about the point $O$, the system of points $X_{1}, \ldots, X_{n}$ maps onto itself. It is clear that $S_{k+1} \circ S_{k}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,789 |
17.32. Prove that if a plane figure has exactly two axes of symmetry, then these axes are perpendicular.
Prove that if a plane figure has exactly two axes of symmetry, then these axes are perpendicular. | 17.32. Let lines $l_{1}$ and $l_{2}$ be the axes of symmetry of a plane figure. This means that if point $X$ belongs to the figure, then points $S_{l_{1}}(X)$ and $S_{l_{2}}(X)$ belong to the figure. Consider the line $l_{3}=S_{l_{1}}\left(l_{2}\right)$. According to problem 17.24, $S_{l_{3}}(X)=S_{l_{1}} \circ S_{l_{2... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,790 |
17.33*. Prove that if a polygon has several (more than two) axes of symmetry, then all of them intersect at one point. | 17.33. Suppose a polygon has three axes of symmetry that do not intersect at a single point, i.e., they form a triangle. Let $X$ be the point of the polygon farthest from some interior point $M$ of this triangle. The points $X$ and $M$ lie on the same side of one of the considered axes of symmetry $l$. If $X^{\prime}$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,791 |
17.34*. Prove that if a polygon has an even number of axes of symmetry, then it has a center of symmetry.
## §6. Theorem of Sylow
A movement is a transformation that preserves the distances between points, i.e., if $A^{\prime}$ and $B^{\prime}$ are the images of points $A$ and $B$, then $A^{\prime} B^{\prime}=A B$. A... | 17.34. All axes of symmetry pass through one point $O$ (Problem 17.33). If $l_{1}$ and $l_{2}$ are axes of symmetry, then $l_{3}=S_{l_{1}}\left(l_{2}\right)$ is also an axis of symmetry (see Problem 17.24). Let us choose one of the axes of symmetry $l$ of our polygon. The other axes are divided into pairs of lines symm... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,792 |
17.35*. Prove that any motion of the plane is a composition of no more than three reflections with respect to lines.
A motion that is a composition of an even number of reflections with respect to lines is called a motion of the first kind or a motion that preserves the orientation of the plane. A motion that is a com... | 17.35. Let $F$ be a motion that maps point $A$ to $A^{\prime}$, where points $A$ and $A^{\prime}$ do not coincide; $S$ is the symmetry with respect to the perpendicular bisector $l$ of segment $A A^{\prime}$. Then $S \circ F(A)=A$, i.e., $A$ is a fixed point of the transformation $S \circ F$. Moreover, if $X$ is a fixe... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,793 |
17.36*. Prove that any motion of the first kind is a rotation or a parallel translation.
A glide reflection is defined as the composition of a reflection about some line $l$ and a translation by a vector parallel to $l$ (this vector can be zero). | 17.36. According to problem 17.35, any motion of the first kind is a composition of two symmetries with respect to lines. It remains to use the result of problem 17.22. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,794 |
17.37*. Prove that any motion of the second kind is a glide reflection.
## Problems for independent solving
| 17.37. According to problem 17.35, any motion of the second kind can be represented as $S_{3} \circ S_{2} \circ S_{1}$, where $S_{1}, S_{2}$, and $S_{3}$ are symmetries with respect to lines $l_{1}$, $l_{2}$, and $l_{3}$. Suppose first that the lines $l_{2}$ and $l_{3}$ are not parallel. Then, when the lines $l_{2}$ an... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,795 |
18.1. On the sides $B C$ and $C D$ of the square $A B C D$, points $M$ and $K$ are taken respectively, such that $\angle B A M = \angle M A K$. Prove that $B M + K D = A K$. | 18.1. Rotate the square $A B C D$ around point $A$ by $90^{\circ}$ so that point $B$ moves to point $D$. During this rotation, point $M$ moves to point $M^{\prime}$, and point $K$ moves to point $K^{\prime}$. It is clear that $\angle B M A=\angle D M^{\prime} A$. Since $\angle M A K=$ $=\angle M A B=\angle M^{\prime} A... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,796 |
18.2. In triangle $ABC$, the median $CM$ and the altitude $CH$ are drawn. Lines passing through an arbitrary point $P$ of the plane, perpendicular to $CA$, $CM$, and $CB$, intersect the line $CH$ at points $A_{1}$, $M_{1}$, and $B_{1}$. Prove that $A_{1} M_{1} = B_{1} M_{1}$. | 18.2. When rotated by $90^{\circ}$ about point $P$, the lines $P A_{1}, P B_{1}, P M_{1}$, and $C H$ transform into lines parallel to $C A, C B, C M$, and $A B$ respectively. Therefore, when the triangle $P A_{1} B_{1}$ is rotated, the segment $P M_{1}$ transforms into the median of the (rotated) triangle. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,797 |
18.3. Two squares $B C D A$ and $B K M N$ share a common vertex $B$. Prove that the median $B E$ of triangle $A B K$ and the altitude $B F$ of triangle $C B N$ lie on the same line. (The vertices of both squares are listed in a clockwise order.) | 18.3. Consider a $90^{\circ}$ rotation about point $B$, which maps vertex $K$ to vertex $N$, and vertex $C$ to $A$. Under this rotation, point $A$ is mapped to some point $A^{\prime}$, and point $E$ is mapped to $E^{\prime}$. Since $E^{\prime}$ and $B$ are the midpoints of sides $A^{\prime} N$ and $A^{\prime} C$ of tri... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,798 |
18.4. Inside the square $A_{1} A_{2} A_{3} A_{4}$, a point $P$ is taken. From vertex $A_{1}$, a perpendicular is dropped to $A_{2} P$, from $A_{2}$ - to $A_{3} P$, from $A_{3}$ - to $A_{4} P$, and from $A_{4}$ - to $A_{1} P$. Prove that all four perpendiculars (or their extensions) intersect at one point. | 18.4. When rotating around the center of the square by $90^{\circ}$, which maps point $A_{1}$ to point $A_{2}$, the perpendiculars dropped from points $A_{1}, A_{2}, A_{3}$, and $A_{4}$ transform into lines $A_{2} P, A_{3} P, A_{4} P$, and $A_{1} P$ respectively. Therefore, the point of their intersection is the image ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,799 |
18.5. On the sides $C B$ and $C D$ of the square $A B C D$, points $M$ and $K$ are taken such that the perimeter of triangle $C M K$ is equal to twice the side of the square. Find the measure of angle $M A K$. | 18.5. Rotate the given square around point $A$ by $90^{\circ}$ so that vertex $B$ moves to $D$. Let $M^{\prime}$ be the image of point $M$ under this rotation. Since by the condition $M K+M C+C K=(B M+M C)+(K D+C K)$, then $M K=B M+K D=D M^{\prime}+K D=K M^{\prime}$. Moreover, $A M=A M^{\prime}$, therefore $\triangle A... | 45 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,800 |
18.6. On the plane, there are three (equally oriented) squares: $A B C D, A B_{1} C_{1} D_{1}$, and $A_{2} B_{2} C D_{2}$; the first square shares the common vertices $A$ and $C$ with the other two. Prove that the median $B M$ of triangle $B B_{1} B_{2}$ is perpendicular to the segment $D_{1} D_{2}$. | 18.6. Let $R$ be a $90^{\circ}$ rotation that transforms the vector $\overrightarrow{B C}$ into $\overrightarrow{B A}$. Further, let $\overrightarrow{B C}=\boldsymbol{a}, \overrightarrow{C B_{2}}=\boldsymbol{b}$ and $\overrightarrow{A B_{1}}=\boldsymbol{c}$. Then $\overrightarrow{B A}=R \boldsymbol{a}, \overrightarrow{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,801 |
18.7*. Given a triangle $A B C$. On its sides $A B$ and $B C$, squares $A B M N$ and $B C P Q$ are constructed externally. Prove that the centers of these squares and the midpoints of segments $M Q$ and $A C$ form a square. | 18.7. Let's introduce the following notations: $\boldsymbol{a}=\overrightarrow{B M}, \boldsymbol{b}=\overrightarrow{B C} ; R \boldsymbol{a}$ and $R b$ - vectors obtained from vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ by a $90^{\circ}$ rotation: $R \boldsymbol{a}=\overrightarrow{B A}, R \boldsymbol{b}=\overrightarro... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,802 |
18.9. On the sides of triangle $A B C$, equilateral triangles $A_{1} B C, A B_{1} C$ and $A B C_{1}$ are constructed externally. Prove that $A A_{1}=B B_{1}=$ $=C C_{1}$. | 18.9. When rotating around point $C$ by $60^{\circ}$, point $A$ moves to $B_{1}$, and point $A_{1}$ moves to $B$. Therefore, segment $A A_{1}$ moves to segment $B_{1} B$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,804 |
18.10. On the segment $A E$, equilateral triangles $A B C$ and $C D E$ are constructed on the same side of it; $M$ and $P$ are the midpoints of segments $A D$ and $B E$. Prove that triangle $C P M$ is equilateral. | 18.10. Consider a rotation by $60^{\circ}$ about point $C$, which maps point $E$ to $D$. In this rotation, point $B$ maps to $A$, i.e., segment $B E$ maps to segment
Fig. 18.1
$A D$. There... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,805 |
18.11. Construct an equilateral triangle $ABC$ such that its vertices lie on three given parallel lines. | 18.11. Suppose we have constructed a triangle $A B C$ such that its vertices $A, B$, and $C$ lie on the lines $l_{1}, l_{2}$, and $l_{3}$ respectively. When rotated $60^{\circ}$ about $A$, point $B$ maps to point $C$, so $C$ is the intersection of line $l_{3}$ and the image of line $l_{2}$ under a $60^{\circ}$ rotation... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,806 |
18.13. On the sides $B C$ and $C D$ of parallelogram $A B C D$, regular triangles $B C P$ and $C D Q$ are constructed externally. Prove that triangle $A P Q$ is regular. | 18.13. When rotated by $60^{\circ}$, the vectors $\overrightarrow{Q C}$ and $\overrightarrow{C P}$ transform into $\overrightarrow{Q D}$ and $\overrightarrow{C B}=\overrightarrow{D A}$. Therefore, under this rotation, the vector $\overrightarrow{Q P}=\overrightarrow{Q C}+\overrightarrow{C P}$ transforms into the vector... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,808 |
18.14. Point $M$ lies on the arc $A B$ of the circumcircle of an equilateral triangle $A B C$. Prove that $M C=M A+M B$.
The point $M$ lies on the arc $A B$ of the circumcircle of an equilateral triangle $A B C$. We need to prove that $M C = M A + M B$. | 18.14. Let $M^{\prime}$ be the image of point $M$ after a $60^{\circ}$ rotation about point $B$, which maps $A$ to $C$. Then $\angle C M^{\prime} B=\angle A M B=120^{\circ}$. Triangle $M M^{\prime} B$ is equilateral, so $\angle B M^{\prime} M=60^{\circ}$. Since $\angle C M^{\prime} B+\angle B M^{\prime} M=180^{\circ}$,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 27,809 |
18.15. Find the geometric locus of points $M$, lying inside the equilateral triangle $ABC$, for which $M A^{2}=M B^{2}+M C^{2}$. | 18.15. When rotating by $60^{\circ}$ with center $A$, which maps $B$ to $C$, point $M$ transitions to some point $M^{\prime}$, and point $C$ - to point $D$. The equality $M A^{2} = M B^{2} + M C^{2}$ is equivalent to the equality $M^{\prime} M^{2} = M^{\prime} C^{2} + M C^{2}$, i.e., that $\angle M C M^{\prime} = 90^{\... | ThedesiredlocusofpointsisthearcofcirclelyinginsidethetrianglefromwhichthesegmentBCisseenatanangleof150 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 27,810 |
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