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192. Prove that the three lines passing through the vertices of a triangle and dividing its perimeter in half intersect at one point $N$ (the Nagel point). Let $M$ be the centroid of the triangle, $I$ the incenter, and $S$ the center of the circle inscribed in the triangle with vertices at the midpoints of the sides of... | 192. We will prove that under a homothety with center at $M$ and coefficient $-1 / 2$, point $N$ transforms into $I$ (it is obvious that under this homothety, $I$ transforms into $S$). Let $A B C$ be the given triangle, $A_{0}, B_{0}$, and $C_{0}$ be the midpoints of sides $B C, C A$, and $A B$, respectively, and $A_{1... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,592 |
193. Let $a, b$, and $c$ be the sides of triangle $ABC$, $a+b+c=2p$; $G$ - the point of intersection of its medians, $O, I$, and $I_{a}$ - the centers of the circumscribed, inscribed, and exscribed circles (the exscribed circle touches side $BC$ and the extensions of sides $AB$ and $AC$), $R, r, r_{a}$ - their radii. P... | 193. a) Using the formulas for the right-hand side $r=\frac{S}{p}$, $R=\frac{a b c}{4 S}, S=\sqrt{p(p-a)(p-b)(p-c)}$, where $S$ is the area of triangle $ABC$, it is easy to prove the required ratio.
b) Use the formula of Leibniz (problem II.140), taking as $M$ the center of the circumscribed circle.
c) Use the formul... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,593 |
194. Let $B B_{1}$ and $C C_{1}$ be the angle bisectors of angles $B$ and $C$ of triangle $A B C$. Prove (using the notation of the previous problem) that $\left|B_{1} C_{1}\right|=\frac{a b c}{(b+a)(c+a) R}\left|O I_{a}\right|$. | 194. Draw lines through $O$ parallel to $A B$ and $A C$, and denote by $L$ and $K$ the points of intersection of these lines with the perpendiculars dropped from $I_{a}$ to $A B$ and $A C$, respectively. We will prove the similarity of triangles $A B_{1} C_{1}$ and $O L K$. We have: $\angle B_{1} A C_{1}=\angle L O K,\... | \frac{}{(b+)(+)R}|OI_{}| | Geometry | proof | Yes | Yes | olympiads | false | 23,594 |
196. Prove that the sum of the areas of three triangles, the vertices of each of which are the three points of tangency of the excircle with the corresponding side of the triangle and the extensions of the other two sides, is equal to twice the area of the triangle plus the area of the triangle whose vertices are the p... | 196. Let $O$ be the center of the circumcircle of $\triangle ABC$, $B_{1}$ the midpoint of $AC$, and $N$ the point of tangency of the incircle with $AC$; then $|AN|=p-a,|CN|=p-c$ (see problem I.18), $|ON|^{2}=|OB_{1}|^{2}+|BB_{1}N|^{2}=$ $=|AO|^{2}-|AB_{1}|^{2}+|B_{1}N|^{2}=R^{2}-\frac{b^{2}}{4}+\left(p-a-\frac{b}{2}\r... | 3R^{2}-4Rr-r^{2} | Geometry | proof | Yes | Yes | olympiads | false | 23,596 |
198. Through the bases of the bisectors of triangle $ABC$, a circle is drawn. Prove that one of the chords formed by the intersection of this circle with the sides of the triangle is equal to the sum of the other two. | 198. Let $K_{1}$ and $L_{1}$ be points on $B C$ and $B A$ such that $K_{1} K \parallel L_{1} L \parallel B_{1} B$. It is sufficient to prove that triangles $B K_{1} K$ and $B L_{1} L$ are similar, i.e., $\frac{\left|B K_{1}\right|}{\left|K_{1} K\right|}=\frac{\left|B L_{1}\right|}{\left|L_{1} L\right|}$. We have: $\fra... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,598 |
199. Let $A A_{1}, B B_{1}$, and $C C_{1}$ be the angle bisectors of triangle $A B C$, $L$ be the point of intersection of lines $A A_{1}$ and $B_{1} C_{1}$, and $K$ be the point of intersection of $C C_{1}$ and $A_{1} B_{1}$. Prove that $B B_{1}$ is the angle bisector of angle $L B K$. | 199. Let $\angle K A L = \angle K L A = \varphi, \angle K C L = \angle L K C = \psi$. Then $\angle B K L = 2 \varphi, \angle B L K = 2 \psi, 2 \varphi + 2 \psi = 180^{\circ} - \angle B$. If $Q$ is the intersection point of $A L$ and $K C$, then $\angle A Q C = 180^{\circ} - (\varphi + \psi) = 90^{\circ} + \frac{1}{2} \... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,599 |
200. In triangle $ABC$, points $K$ and $L$ are taken on sides $AB$ and $BC$ such that $|AK|=|KL|=|LC|$. A line is drawn through the intersection of lines $AL$ and $CK$, parallel to the bisector of angle $B$, intersecting line $AB$ at point $M$. Prove that $|AM|=|BC|$. | 200. Let $B_{1}$ be the midpoint of $A C$. Extend the bisector to intersect at point $B_{2}$ with the perpendicular erected to $A C$ at point $B_{1}$. Point $B_{2}$ lies on the circumscribed circle. Draw a perpendicular through $M$ to $A C$; let $L$ be the point of intersection with $A C$, and $K$ be the point of inter... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,600 |
201. In triangle $ABC$, the bisector of angle $B$ intersects the line passing through the midpoint of $AC$ and the midpoint of the height dropped to $AC$ at point $M$; $N$ is the midpoint of the bisector of angle $B$. Prove that the bisector of angle $C$ is also the bisector of angle $MCN$. | 201. a) This well-known problem has many different proofs. Let's present one of them, based on the following criterion for the equality of triangles. Two triangles are equal if they have respectively equal sides, opposite angles, and the bisectors of these angles. Let's prove this criterion. Consider two triangles $A C... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,601 |
204. Let $A B C D E F$ be a cyclic hexagon. Denote by $K$ the intersection of $A C$ and $B F$, and by $L$ the intersection of $C E$ and $F D$. Prove that the diagonals $A D, B E$ and the line $K L$ intersect at one point (Pascal). | 204. Let $M$ be the intersection point of $A D$ and $K L$:
$$
\frac{|K M|}{|M L|}=\frac{S_{A K D}}{S_{A L D}}=\frac{\frac{1}{2}|A K| \cdot|A D| \sin \angle K A D}{\frac{1}{2}|D L| \cdot|A D| \sin \angle A D L}=\frac{|A K| \cdot|C D|}{|D L| \cdot|A F|}
$$
(We used the fact that the sines of inscribed angles are propor... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,603 |
205. Given a triangle $A B C$ and a point $M$. A line passing through $M$ intersects the lines $A B, B C$, and $C A$ at points $C_{1}, A_{1}$, and $B_{1}$, respectively. The lines $A M, B M$, and $C M$ intersect the circumcircle of triangle $A B C$ at points $A_{2}, B_{2}$, and $C_{2}$, respectively. Prove that the lin... | 205. Let $N$ be the point of intersection of the line $A_{2} A_{1}$ with the circle, different from $A_{2}$. Apply Pascal's theorem (problem II.204) to the hexagon $A B C C_{2} N A_{2}$ (which may be self-intersecting). The points of intersection of the lines $A B$ and $C_{2} N$, $B C$ and $N A_{2}$ (point $A_{1}$), $C... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,604 |
206. Through the point of intersection of the altitudes of a triangle, two mutually perpendicular lines are drawn. Prove that the midpoints of the segments cut off by these lines on the sides of the triangle (on the lines forming the triangle) lie on one straight line.
$$
\text { * } \quad *
$$ | 206. Let the mutually perpendicular lines be the $x$ and $y$ axes of a rectangular coordinate system. Then the altitudes of the triangle lie on the lines $y=k_{i} x \quad(i=1,2,3)$; the sides of the triangle must have slopes $-\frac{1}{k_{i}}$, and from the condition that the vertices $\left(x_{i}, y_{i}\right)$ belong... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,605 |
207. Given a triangle $A B C$ and an arbitrary point $P$. The bases of the perpendiculars dropped from $P$ to the sides of triangle $A B C$ serve as the vertices of triangle $A_{1} B_{1} C_{1}$. The vertices of triangle $A_{2} B_{2} C_{2}$ are the points of intersection of the lines $A P$, $B P$, and $C P$ with the cir... | 207. To determine the angles of triangle $A_{1} B_{1} C_{1}$, use the fact that points $P, A_{1}, B_{1}, C_{1}$ lie on the same circle (the same applies to other sets of four points). If $P$ is inside triangle $A B C$, then $\angle A_{1} C_{1} B_{1}=\angle A_{2} C_{2} B_{2}=\angle A P B-\angle A C B$. For a scalene tri... | 8 | Geometry | proof | Yes | Yes | olympiads | false | 23,606 |
209. Let $S$ be the area of a given triangle, and $R$ the radius of the circle circumscribed about it. Further, let $S_{1}$ be the area of the triangle formed by the bases of the perpendiculars dropped from a point, which is at a distance $d$ from the center of the circumscribed circle, onto the sides of the given tria... | 209. Notations: $A B C$ - the given triangle, $M$ - a point at a distance $d$ from the center of the circumcircle of $A B C$, $A_{1}, B_{1}, C_{1}$ - the feet of the perpendiculars dropped from $M$ to $B C, C A$, $A B, A_{2}, B_{2}, C_{2}$ - the points of intersection of $A M, B M, C M$ with the circumcircle of $\trian... | S_{1}=\frac{S}{4}|1-\frac{^{2}}{R^{2}}| | Geometry | proof | Yes | Yes | olympiads | false | 23,608 |
210. Prove that if $A, B, C$ and $D$ are arbitrary points on a plane, then the four circles, each passing through three points: the midpoints of segments $A B, A C$ and $A D ; B A, B C$ and $B D ; C A, C B$ and $C D ; D A, D B$ and $D C$, have a common point. | 210. The statement follows from a more general fact: if circles are constructed on the sides of a triangle such that the arcs located outside the triangle sum up to $4 \pi$ or $2 \pi$, then these circles have a common point (in our case, we can take the triangle with vertices at the midpoints of $\triangle A B C$ and p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,609 |
211. Let $A B C$ be a triangle, and $D$ be an arbitrary point on the plane. The triangle formed by the bases of the perpendiculars dropped from $D$ to the sides of triangle $A B C$ will be called the pedal triangle of point $D$ with respect to triangle $A B C$, and the circle circumscribed around the pedal triangle wil... | 211. The statement follows from the following fact. Let an arbitrary circle intersect the sides of an angle with vertex $N$ at points $A, B$ and $C, D$; the perpendiculars erected to the sides of the angle at points $A$ and $D$ intersect at point $K$, and the perpendiculars erected at points $B$ and $C$ intersect at po... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,610 |
212. Consider four points on a plane, no three of which lie on the same line. Prove that the four pedal circles, each corresponding to one of the considered points relative to the triangle whose vertices are the three remaining points, have a common point. | 212. Let $A, B, C, D$ be given points, and $D_{1}$ be the point of intersection of the lines symmetric to the lines $A D, B D$, and $C D$ with respect to the corresponding angle bisectors of $\triangle A B C$. In the previous problem, it was proven that the pedal circles of points $D$ and $D_{1}$ with respect to $\tria... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,611 |
213. A line passing through the center of the circumcircle of triangle $ABC$ intersects $AB$ and $AC$ at points $C_{1}$ and $B_{1}$, respectively. Prove that the circles constructed on $BB_{1}$ and $CC_{1}$ as diameters intersect at two points, one of which lies on the circumcircle of $ABC$, and the other on the nine-p... | 213. Let $B_{2}$ and $C_{2}$ be the points diametrically opposite to points $B$ and $C$, $M$ be the second intersection point of $B_{2} B_{1}$ with the circumcircle of $\triangle A B C$, and $C_{1}^{\prime}$ be the intersection point of $A B$ and $C_{2} M$. By Pascal's theorem (problem II.204), applied to the hexagon $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,612 |
218. Let $2 \varphi$ be the sum of two opposite angles of a cyclic quadrilateral, $a, b, c$ and $d$ be its sides, $S$ be its area. Prove that $S=\sqrt{a b c d} \sin \varphi$. | 218. Let the radius of the circle be $r$, and the angles between adjacent radii, drawn to the points of tangency, in the order of traversal, be $2 \alpha$, $2 \beta, 2 \gamma, 2 \delta (\alpha+\beta+\gamma+\delta=\pi)$. Then
$$
S=r^{2}(\operatorname{tg} \alpha+\operatorname{tg} \beta+\operatorname{tg} \gamma+\operator... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,613 |
219. On the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$, points $M$ and $N$ are taken, dividing them in the same ratio (counting from vertices $A$ and $C$). These points are connected to all vertices of the quadrilateral, as a result of which $ABCD$ is divided into six triangles and one quadrilateral. Prove th... | 219. We will prove that $S_{B N A}=S_{B M C}+S_{A M D}$. If $\frac{|A M|}{|A B|}=\frac{|C N|}{|N D|}=\lambda$, then $S_{B M C}=(1-\lambda) S_{B A C}, S_{A M D}=\lambda S_{B A D}$. On the other hand, denoting by $h_{1}, h_{2}$ and $h$ the distances from $C, D$ and $N$ to $A B$, we find that $h=$
$$
\begin{aligned}
& =\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,614 |
221. Given a convex quadrilateral $Q_{1}$. Lines perpendicular to its sides and passing through the midpoints of the sides form a quadrilateral $Q_{2}$. Similarly, for quadrilateral $Q_{2}$, a quadrilateral $Q_{3}$ is formed. Prove that quadrilateral $Q_{3}$ is similar to the original quadrilateral $Q_{1}$. | 221. The angles between the sides, as well as between the sides and diagonals of quadrilateral $Q_{2}$, are expressed in terms of the angles between the sides and between the sides and diagonals of quadrilateral $Q_{1}$. (The diagonals of quadrilateral $Q_{2}$ are perpendicular to the corresponding diagonals of quadril... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,615 |
222. On the opposite sides $B C$ and $D A$ of a convex quadrilateral, points $M$ and $N$ are taken such that $|B M|:|M C| = |A N|:|N D| = |A B|:|C D|$. Prove that the line $M N$ is parallel to the bisector of the angle formed by the sides $A B$ and $C D$. | 222. Consider parallelograms $A B M K$ and $D C M L$ and prove that $K L$ divides $D A$ in the same ratio as point $N$, and that line $M N$ is the bisector of angle $K M L$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,616 |
223. The diagonals divide a convex quadrilateral into four triangles. The radii of the circles inscribed in these triangles are equal. Prove that the given quadrilateral is a rhombus. | 223. First, let's prove that the diagonals of the given quadrilateral are bisected at the point of intersection, i.e., that the quadrilateral is a parallelogram. Let \(ABCD\) be the given quadrilateral, \(O\) the point of intersection of the diagonals. Suppose \(|BO| < |OD|\), \(|AO| \leq |OC|\); consider \(\triangle O... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,617 |
225. For quadrilateral $A B C D$, it is known that the radii of the circles inscribed in triangles $A B C, B C D, C D A, D A B$ are equal. Prove that $A B C D$ is a rectangle. | 225. From the condition of the problem, it follows that $ABCD$ (Fig. 38) is a convex quadrilateral. Consider the parallelogram $AC C_{1} A_{1}$, in which sides $A A_{1}$ and $C C_{1}$ are equal and parallel to the diagonal $BD$. Triangles $A D A_{1}, C D C_{1}$, and $C_{1} D A_{1}$ are equal to triangles $A B D, B C D$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,619 |
226. A quadrilateral $A B C D$ is inscribed in a circle. Let $M$ be the point of intersection of the tangents to the circle passing through $A$ and $C$, $N$ be the point of intersection of the tangents drawn through $B$ and $D$, $K$ be the point of intersection of the angle bisectors of angles $A$ and $C$ of the quadri... | 226. A necessary and sufficient condition for the fulfillment of all four points is the equality $|A B| \cdot|C D|=|A D| \cdot|B C|$. For points a) and b), this follows from the theorem about the bisector of the interior angle of a triangle, for points c) and d) - from the result of problem I.234. | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,620 |
227. Prove that four lines, each of which passes through the bases of two perpendiculars dropped from a vertex of an inscribed quadrilateral to the sides not containing it, intersect at one point. | 227. Let $ABCD$ be the given quadrilateral. We will assume that angles $A$ and $D$ are obtuse, and $B$ and $C$ are acute. Let the feet of the perpendiculars dropped from vertex $A$ be $M$ and $N$, and from vertex $C$ be $K$ and $L$ (Fig. $39, a$), and let $R$ be the intersection point of $MN$ and $LK$. Note that
, triangles $C D M$ and $B A M_{1}$ are equal, i.e., the radius of the circumcircle of $\triang... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,623 |
230. Let $A B C D$ be a parallelogram. A circle is tangent to the lines $A B$ and $A D$ and intersects $B D$ at points $M$ and $N$. Prove that there exists a circle passing through $M$ and $N$ and tangent to the lines $C B$ and $C D$. | 230. Let $K$ and $L$ be the points of tangency of a given circle with the lines $A B$ and $A D$. Suppose, for definiteness, that $K$ and $L$ are inside the segments $A B$ and $A D$. Take a point $P$ on the line $C B$ such that $|B P|=|B K|$, with $B$ between $P$ and $C$, and a point $Q$ on the line $C D$ such that $|D ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,624 |
231. Let $A B C D$ be a parallelogram. Construct a circle with diameter $A C$ and denote by $M$ and $N$ the points of intersection of this circle with the lines $A B$ and $A D$. Prove that the lines $B D, M N$ and the tangent to the circle at point $C$ intersect at one point. | 231. For definiteness, let's assume that $B$ and $D$ are inside the circle. Let $P$ and $Q$ be the points of intersection of the line $BD$ with the circle (with $P$ being the closest to $B$), and let $L$ be the point of intersection of $CB$ with the circle, and $l$ be the tangent to the circle passing through $C$.
Con... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,625 |
232. Quadrilateral $A B C D$ is inscribed in a circle, $O_{1}, O_{2}$, $O_{3}, O_{4}$ are the centers of the circles inscribed in triangles $A B C$, $B C D, C D A, D A B$, and $H_{1}, H_{2}, H_{3}, H_{4}$ are the points of intersection of the altitudes of the same triangles. Prove that $O_{1} O_{2} O_{3} O_{4}$ is a re... | 232. 233. Since $O_{1}$ is the center of the circle inscribed in triangle $ABC$, then $\angle B O_{1} A=90^{\circ}+\frac{1}{2} \angle B C A$ (Problem I.46). Therefore, $\angle B O_{1} A=\angle B O_{4} A$ and quadrilateral $A B O_{1} O_{4}$ is cyclic (Fig. $40, a$), hence the adjacent angle to $\angle B O_{1} O_{4}$ is ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,626 |
235. Prove that if $A B C D$ is a cyclic quadrilateral, then the sum of the radii of the circles inscribed in triangles $A B C$ and $A C D$ is equal to the sum of the radii of the circles inscribed in triangles $B C D$ and $B D A$.
```
* * *
``` | 235. Let (Fig. 42a,b) $O_{1}, O_{2}, O_{3}, O_{4}$ be the centers of the circles inscribed in $\triangle A B C, \triangle B C D, \triangle C D A$ and $\triangle D A B$. Since $O_{1} O_{2} O_{3} O_{4}$ is a rectangle (see problem II.232), then $\left|O_{1} O_{3}\right|=\left|O_{2} O_{4}\right|$.
. Let $a, b, c, d$ be the consecutive sides of a quadrilateral, $m$ and $n$ its diagonals, and $A$ and $C$ two opposite angles. Then the following relation holds:
$$
m^{2} n^{2}=a^{2} c^{2}+b^{2} d^{2}-2 a b c d \cos (A+C)
$$ | 236. Let in quadrilateral $A B C D$ (Fig. 43) $|A B|=a,|B C|=b$, $|C D|=c,|D A|=d,|A C|=m,|B D|=n$. Construct triangle $A K B$ on side $A B$ outwardly, similar to triangle $A C D$, such that $\angle B A K=\angle D C A, \angle A B K=\angle C A D$, and on side $A D$ construct $\triangle A M D$, similar to $\triangle A B ... | ^{2}n^{2}=^{2}^{2}+b^{2}^{2}-2\cos(A+C) | Geometry | proof | Yes | Yes | olympiads | false | 23,628 |
238. Prove that if $A B C$ is an equilateral triangle, $M$ is an arbitrary point in the plane, not lying on the circumcircle of triangle $A B C$, then there exists a triangle with sides equal to $|M A|,|M B|$, and $|M C|$ (Pompeiu's theorem). Find the angle of this triangle opposite the side equal to $|M B|$, if $\angl... | 238. If $M B$ is the greatest of the segments $|M A|,|M B|,|M C|$, then, applying Bretschneider's theorem (problem II.236) to the quadrilateral $A B C M$, we get that $|M B|^{2}=|M A|^{2}+|M C|^{2}-$ $-2|M A| \cdot|M C| \cos \left(\angle A M C+60^{\circ}\right)$, i.e., $|M B|<|M A|+|M C|$, since $\angle A M C \neq 120^... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,630 |
239. Let $A B C D$ be a cyclic quadrilateral. Four circles $\alpha, \beta, \gamma$, and $\delta$ touch the circumcircle of quadrilateral $A B C D$ at points $A, B, C$, and $D$, respectively. Denote by $t_{\alpha \beta}$ the segment of the tangent to circles $\alpha$ and $\beta$, where $t_{\alpha \beta}$ is the segment ... | 239. By substituting the tangent segments in the expression
$$
t_{\alpha \beta} t_{\gamma \delta} + t_{\beta \gamma} t_{\delta \alpha} = t_{\alpha \gamma} t_{\beta \delta}
$$
using the formulas obtained when solving problem I.201, we can verify that if relation (1) holds for some circles $\alpha, \beta, \gamma$, and ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,631 |
241. The extensions of sides $A B$ and $D C$ of a convex quadrilateral $A B C D$ intersect at point $K$, and the extensions of sides $A D$ and $B C$ intersect at point $L$, such that segments $B L$ and $D K$ intersect. Prove that if one of the three relations holds: $|A B|+|C D|=|B C|+|A D|,|B K|+|B L|=|D K|+|D L|,|A K... | 241. Show that each of these conditions is necessary and sufficient for the existence of a circle inscribed in quadrilateral $A B C D$ (see also problem I.19). | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,633 |
243. Prove that if there exists a circle tangent to the lines $A B, B C, C D$ and $D A$, then its center and the midpoints of $A C$ and $B D$ lie on the same line. | 243. Let $A B C D$ be a circumscribed quadrilateral, $O$ be the center of the inscribed circle, $M_{1}$ be the midpoint of $A C$, $M_{2}$ be the midpoint of $B D$, $r$ be the radius of the circle (the distances from $O$ to the sides are equal to $r$), $x_{1}, y_{1}, z_{1}, u_{1}$ be the distances from $M_{1}$ to $A B, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,635 |
244. Let $A B C D$ be a cyclic quadrilateral. The perpendicular to $B A$, erected at point $A$, intersects line $C D$ at point $M$, and the perpendicular to $D A$, erected at point $A$, intersects line $B C$ at point $N$. Prove that $M N$ passes through the center of the circle circumscribed around quadrilateral $A B C... | 244. Let $L$ and $P$ be the points of intersection of the lines $A M$ and $A N$ with the circle. As follows from problem II.204, the lines $B L, D P$ and $M N$ intersect at one point. But $B L$ and $D P$ are diameters - they intersect at the center of the circle, therefore, $M N$ passes through the center of the circle... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,636 |
246. Prove that the bases of the perpendiculars dropped from the intersection point of the diagonals of an inscribed quadrilateral to its sides are the vertices of a quadrilateral into which a circle can be inscribed. Find the radius of this circle if the diagonals of the inscribed quadrilateral are perpendicular, the ... | 246. Let (Fig. 44) $P$ be the point of intersection of the diagonals, and $K, L, M$, $N$ be the feet of the perpendiculars dropped from $P$ to $A B, B C, C D$ and $D A$ respectively. Since the quadrilateral $P K B L$ is cyclic, then
 $ABCD$ be the given quadrilateral, $P$ the point of intersection of the diagonals, $K$ the midpoint of $BC$, and $L$ the midpoint of $AD$. We will prove that the line $LP$ is perpendicular to $BC$. Denoting by $M$ the point of intersection of $LP$ with $BC$, we have: $\angle BPM = \angle LPD = \angle... | \frac{1}{2}\sqrt{2R^2-^2} | Geometry | proof | Yes | Yes | olympiads | false | 23,639 |
248. Prove that if a quadrilateral is inscribed in a circle of radius $R$ and simultaneously circumscribed about a circle of radius $r$, and the distance between the centers of these circles is $d$, then the relation $\frac{1}{(R+d)^{2}}+\frac{1}{(R-d)^{2}}=$ $=1 / r^{2}$ holds; moreover, there are infinitely many quad... | 248. From the two previous problems, it follows that if the diagonals of an inscribed quadrilateral are perpendicular, then the projections of the intersection point of the diagonals of this quadrilateral onto its sides serve as the vertices of a quadrilateral that can be inscribed in a circle and circumscribed around ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,640 |
249. A convex quadrilateral is divided into four triangles by its diagonals. Prove that the line connecting the centroids of two opposite triangles is perpendicular to the line connecting the orthocenters of the other two triangles. | 249. The midpoints of the sides of a quadrilateral form a parallelogram, the diagonals of which are parallel to the segments connecting the centroids of opposite triangles. Another parallelogram is formed by the four heights of the considered triangles, emanating from the vertices of the quadrilateral. The sides of the... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,641 |
250. Let $A B C D$ be a cyclic quadrilateral, $M$ and $N$ be the midpoints of $A C$ and $B D$. Prove that if $B D$ is the bisector of angle $A N C$, then $A C$ is also the bisector of angle $B M D$. | 250. We will prove that both statements ($B D$ is the bisector of angle $A N C$, $A C$ is the bisector of angle $B M D$) are equivalent to the equality $|A B| \cdot|C D| = |A D| \cdot|B C|$. Take a point $A_{1}$ on the arc $B A D$ such that $\left|D A_{1}\right| = |A B|$. The condition of the problem is equivalent to t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,642 |
251. Let $A B C D$ be a cyclic quadrilateral. The opposite sides $A B$ and $C D$ intersect at point $K$ when extended, and the sides $B C$ and $A D$ intersect at point $L$. Prove that the bisectors of angles $B K C$ and $B L A$ are perpendicular and intersect on the line connecting the midpoints of $A C$ and $B D$. | 251. Perpendicularity of the bisectors is easily proved. Let us prove the second statement. Let $M$ be the midpoint of $A C$, and $N$ be the midpoint of $B D$. From the similarity of triangles $A K C$ and $B K D$, it follows that $\angle M K A = \angle N K D$ and $\frac{|M K|}{|K N|} = \frac{|A C|}{|B D|}$, i.e., the b... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,643 |
252. The diagonals of a quadrilateral are perpendicular. Prove that the four lines, each connecting one of the vertices of the quadrilateral and the center of the circle passing through that vertex and the two adjacent vertices of the quadrilateral, intersect at one point. | 252. Let $A B C D$ be the given quadrilateral, $O$ the center of the circle circumscribed around triangle $A B C$, $O_{1}$ and $O_{2}$ the centers of the circles circumscribed around triangles $D A B$ and $B C D$, $K$ and $L$ the midpoints of sides $A B$ and $B C$. The points $O_{1}$ and $O_{2}$ lie on $O K$ and $O L$,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,644 |
253. Let $P, Q$ and $M$ be the points of intersection of the diagonals and the extensions of the opposite sides of a cyclic quadrilateral, respectively. Prove that the orthocenter of triangle $P Q M$ coincides with the center of the circle circumscribed around the given quadrilateral (Brocard). | 253. Let the radius of the circle be denoted by $R$, and the distances from $P$, $Q$, and $M$ to the center by $a$, $b$, and $c$ respectively. Then (Problem I.272) $|Q P|^{2}=a^{2}+b^{2}-2 R^{2}, \quad|Q M|^{2}=b^{2}+c^{2}-2 R^{2}, \quad|P M|^{2}=c^{2}+a^{2}-2 R^{2}$. If $O$ is the center of the circle, then for $Q O$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,645 |
254. Let $A B C D$ be a circumscribed quadrilateral, $K$ be the intersection point of lines $A B$ and $C D$, $L$ be the intersection point of lines $A D$ and $B C$. Prove that the orthocenter of the triangle formed by lines $K L$, $A C$, and $B D$ coincides with the center of the circle inscribed in quadrilateral $A B ... | 254. If $M, N, P$ and $Q$ are the points of tangency of the sides $A B$, $B C, C D$ and $D A$ with the circle, then, as follows from the solution of problem I.236, $M P$ and $N Q$ intersect at the point of intersection of $A C$ and $B D$. Similarly, we will prove that the lines $M N$ and $P Q$ intersect at the same poi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,646 |
255. Let $A B C D$ be a convex quadrilateral, $\angle A B C=$ $=\angle A D C, M$ and $N-$ the feet of the perpendiculars dropped from $A$ to $B C$ and $C D$ respectively, $K$ - the intersection point of the lines $M D$ and $N B$. Prove that the lines $A K$ and $M N$ are perpendicular. | 255. Let: $\angle D A N=\angle M A B=\varphi$. Let $L$ be the intersection point of $A M$ and $N B$, $P$ be the intersection point of $A N$ and $D M$, and $Q$ be the intersection point of $A K$ and $M N$. By Ceva's theorem (problem II.44) for $\triangle A M N$, we have:
$\frac{|N Q|}{|Q M|}=\frac{|A L|}{|L M|} \cdot \... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,647 |
257. Prove that the centers of the four circles circumscribed around the four triangles formed by four intersecting lines in a plane lie on one circle. | 257. First, prove the following auxiliary statement: if $A, B$ and $C$ are points on a straight line, and $M$ is an arbitrary point in the plane, then the centers of the circumcircles of triangles $M A C$, $M B C$, $M C A$, and the point $M$ lie on one circle. Then use the result of problem II. 256. | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,648 |
258. Given four pairwise intersecting lines. Let $M$ be the Miquel point corresponding to these lines (see problem II. 256). Prove that if four of the six points of pairwise intersection of the given lines lie on a circle with center $O$, then the line passing through the two remaining points contains the point $M$ and... | 258. Let the points of intersection of the lines be denoted by $A, B, C, D, P$, and $Q$ (the points are arranged as in the solution to problem I.271); $O$ is the center of the circle passing through $A, B, C$, and $D$, $R$ is its radius, $a$ and $b$ are the tangents drawn to the circle from $P$ and $Q$, respectively. I... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,649 |
259. Four pairwise intersecting lines form four triangles. Prove that if one line is parallel to the Euler line (see problem II. 147) of the triangle formed by the other three lines, then any other line also has this property. | 259. If we move one straight line parallel to itself, then the Euler line of the triangle, one of whose sides is the moving line, will move parallel to itself.
Considering this, the proble... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,650 |
260. Given a triangle $A B C$. A line intersects the lines $A B$, $B C$, and $C A$ at points $D, E$, and $F$ respectively. The lines $D C, A E$, and $B F$ form a triangle $K L M$. Prove that the circles constructed on $D C, A E$, and $B F$ as diameters intersect at two points $P$ and $N$ (assuming that these circles in... | 260. From the result of problem II.19, it follows that the common chord of the circles with diameters $A E$ and $D C$ (as well as $D C$ and $B F$, $B F$ and $A E$) contains the points of intersection of the altitudes of triangles $A B C, B D E, D A F$, and $C E F$. Let $K$ be the point of intersection of $A E$ and $D C... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,651 |
262. Prove that the perpendicular bisectors, erected to the segments connecting the points of intersection of the altitudes and the centers of the circumscribed circles of the four triangles formed by four arbitrary lines in a plane, intersect at one point (the Ehrhart point). | 262. Let $l(A B C)$ denote the perpendicular bisector of the segment connecting the orthocenter and the circumcenter of triangle $A B C$. Let the line intersect the sides $B C, C A$ and $A B$ of triangle $A B C$ at points $D, E$ and $F$ respectively. First, we will prove that when the line $D E F$ is moved parallel to ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,653 |
263. Consider sixteen points, which are the centers of all possible inscribed and exscribed circles for four triangles formed by four intersecting lines on a plane. Prove that these sixteen points can be divided into four quartets in two ways such that each quartet lies on one circle. The centers of these circles, when... | 263. Let $l, m, n$ and $p$ be lines forming our triangles (Fig. 47, a). We introduce the following notation: $P$ is the center of the circle inscribed in the triangle formed by the lines $l, m$ and $n$, $P_{l}$ is the center of the exscribed circle of the same triangle, which touches the side lying on the line $l$. The... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,654 |
266. Three circles pass through two given points on the plane each. Let $O_{1}, O_{2}, O_{3}$ be their centers. A line passing through one of the points, common to all three circles, intersects them again at points $A_{1}, A_{2}, A_{3}$ respectively. Prove that $\left|A_{1} A_{2}\right|:\left|A_{2} A_{3}\right|=\left|O... | 266. Let one of the intersection points through which the line passes be denoted as $C$. Let $B_{1}, B_{2}, B_{3}$ be the feet of the perpendiculars dropped from $O_{1}, O_{2}, O_{3}$ to the line, and let $K$ and $M$ be the points of intersection of the lines parallel to $A_{1} A_{2}$, passing through $O_{1}$ and $O_{2... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,655 |
267. Given two non-intersecting circles. Prove that the four points of tangency of the common external tangents to these circles lie on one circle; similarly, the four points of tangency of the common internal tangents lie on one circle, and the four points of intersection of the common internal tangents with the commo... | 267. Let $O_{1}$ and $O_{2}$ be the centers of the circles, $R_{1}$ and $R_{2}$ their radii, $\left|O_{1} O_{2}\right|=a, M$ the point of intersection of the common internal tangents. The circle with diameter $O_{1} O_{2}$ passes through the points of intersection of the common external tangents with the common interna... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,656 |
273. Given a circle and two points $A$ and $B$ on it. Let $N$ be an arbitrary point on the line $A B$. Construct two circles, each passing through the point $N$ and tangent to the given circle: one at point $A$, and the other at point $B$. Denote by $M$ the second point of intersection of these circles. Find the geomet... | 273. Angles $A M N$ and $B M N$ can be expressed in terms of the central angle corresponding to the arc $A B$ of the given circle (various cases of the position of point $N$ need to be considered); after this, $\angle A M B$ can be determined. The required geometric locus of points is a circle. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,660 |
274. Through a fixed point $A$ inside a circle, two arbitrary chords $P Q$ and $K L$ are drawn. Find the geometric locus of the points of intersection of the lines $P K$ and $Q L$. | 274. Use the results of problems II. 271 and II.21. The obtained locus of points coincides with the locus of points of problem II.21, i.e., it is the polar of point $A$ with respect to the given circle. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,661 |
275. Two circles intersect at points $A$ and $B$. An arbitrary line passes through $B$ and intersects the first circle again at point $C$, and the second circle at point $D$. The tangents to the first circle at $C$ and to the second circle at $D$ intersect at point $M$. A line passing through the intersection of $A M$ ... | 275. Let (Fig. 49) $O$ be the point of intersection of $A M$ and $D C$. Draw a tangent through point $B$ to the second circle and denote the point of intersection with $A C$ as $K$ (as in the problem statement). It is clear that the statement of the problem is equivalent to the statement that $K O \| C M$. Let the angl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,662 |
276. Given a circle and a tangent line $l$ to it. Let $N$ be the point of tangency, $NM$ be a diameter. A fixed point $A$ is taken on the line $NM$. Consider an arbitrary circle passing through $A$, with its center on $l$. Let $C$ and $D$ be the points of intersection of this circle with $l$, and $P$ and $Q$ be the poi... | 276. Since the circle with diameter $C D$ passes through a fixed point $A$ on $M N (M N \perp C D)$, then
$$
|C N| \cdot|N D|=|N A|^{2}
$$
is a constant value. Let $K$ be the point of intersection of $P Q$ with $M N$. We will show that $\frac{|M K|}{|K N|}$ is a constant value. Note that $\angle P N Q=180^{\circ}-\an... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,663 |
278. In a circle, a diameter $A B$ is drawn, and $C D$ is a chord perpendicular to $A B$. An arbitrary circle is tangent to the chord $C D$ and the arc $C B D$. Prove that the tangent from point $A$ to this circle is equal to $A C$. | 278. Let $O$ and $O_{1}$ be the centers of the two considered circles ($O$ is the midpoint of $AB$), $K$ be the point of tangency of the circles ($K$ lies on the line $OO_{1}$), $N$ be the point of tangency of the circle $O_{1}$ with the line $CD$, and $M$ be the point of intersection of $AB$ and $CD$. Since $O_{1}N$ i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,665 |
279. Given a circular segment. Two arbitrary circles touch the chord and the arc of this segment and intersect at points $M$ and $N$. Prove that the line $M N$ passes through a fixed point on the plane.
```
* * *
``` | 279. Let $A$ be the midpoint of the arc of a given circle, not included in the segment, the tangents from $A$ to the circles inscribed in the segment are equal (Problem II.278). From this, it follows that $A$ lies on the line $M N$, since $\left|A O_{1}\right|^{2}-\left|A O_{2}\right|^{2}=\left|O_{1} M\right|^{2}-\left... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,666 |
280. Two equal non-intersecting circles are given. On two common internal tangents, take two arbitrary points $F$ and $F^{\prime}$. From both points, one more tangent can be drawn to each circle. Let the tangents drawn from points $F$ and $F^{\prime}$ to one circle meet at point $A$, and to the other at point $B$. It i... | 280. Let us consider the general case of arbitrary circles. Suppose points $F$ and $F^{\prime}$ are located as shown in Fig. 50. The notation is clear from the figure. We will prove that there exists a circle inscribed in the quadrilateral $A K B M$, after which we will use the result of problem II.55. For this, it is ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,667 |
281. Given three circles $\alpha, \beta$ and $\gamma$. Let $l_{1}$ and $l_{2}$ be the common internal tangents to circles $\alpha$ and $\beta, m_{1}$ and $m_{2}$ be the common internal tangents to circles $\beta$ and $\gamma, n_{1}$ and $n_{2}$ be the common internal tangents to circles $\gamma$ and $\alpha$. Prove tha... | 281. Let (Fig. 51) $M$ be the point of intersection of the tangents $l_{1}, m_{1}$ and $n_{1}$, $N$ be the point of intersection of $l_{2}$ and $m_{2}$. Draw through $N$ a line $n_{2}^{\prime}$, tangent to $\alpha$, different from $l_{2}$. As was done in problem II.280, it can be proven that the lines $m_{1}, n_{1}, m_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,668 |
282. Arc $AB$ of a circle is divided into three equal parts by points $C$ and $D$ ($C$ is the point closest to $A$). After rotating around $A$ by an angle of $\pi / 3$, points $B, C$, and $D$ will move to $B_{1}, C_{1}$, and $D_{1}$, respectively. $F$ is the intersection point of lines $A B_{1}$ and $D C_{1}$, and $E$ ... | 282. We will prove that the line $D_{1} C$ passes through $O$ - the center of the arc $A B$, and the line $D C_{1}$ - through $O_{1}$ - the center of the arc $A B_{1}$ (Fig. 52). Triangle $D A D_{1}$ is equilateral, $|D C|=|A C|$, therefore, $D_{1} C \perp D A$ and $D_{1} C$
 $\triangle A B C$ is circumscribed around a given circle; 2) the given circle touches the extensions of sides $A B$ and $A C$.
In the first case, consider a circle that touches the sides of the angle at points $M$ and $N$ and is internally tangent to the circumcircle of $\triangle A B... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,670 |
284. In triangle $ABC$, a point $D$ is taken on side $AC$. Consider a circle that touches segment $AD$ at point $M$, segment $BD$, and the circumcircle of triangle $ABC$. Prove that the line passing through $M$ parallel to $BD$ is tangent to the incircle of triangle $ABC$. | 284. Let the sides of $\triangle ABC$ be denoted as usual: $a, b, c$; let $|BD| = d$, $|AD| = b_1$, and $|AM| = x$. We will use the generalized Ptolemy's theorem (problem II.239): $x a + (d - b_1 + x) b = (b - x) c$, from which we get
$$
x = \frac{b (c + b_1 - d)}{a + b + c}
$$
Take a point $N$ on $AB$ such that $MN$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,671 |
285. In triangle $A B C$, a point $D$ is taken on side $A C$. Let $\dot{O}_{1}$ be the center of the circle tangent to segments $A D, B D$ and the circumcircle of triangle $A B C$, and $O_{2}$ be the center of the circle tangent to segments $C D, B D$ and the circumcircle. Prove that the line $O_{1} O_{2}$ passes throu... | 285. Let $M$ and $K$ be the points of tangency of the circles with centers $O_{1}$ and $O_{2}$ with $A C$. From the result of the previous problem, it follows that,
$$
\begin{aligned}
& \text { that } \angle O_{1} D M=\angle O K D=\frac{\varphi}{2} \\
& \angle O_{2} D K=\angle O M D=90^{\circ}-\frac{\varphi}{2}
\end{a... | \operatorname{tg}^{2}\frac{\varphi}{2} | Geometry | proof | Yes | Yes | olympiads | false | 23,672 |
288. Let $H$ be the orthocenter of triangle $ABC$. Prove that the nine-point circle is tangent to all the incircles and excircles of triangles $AHB$, $BHC$, $CHA$. | 288. The statement of this problem follows from Feuerbach's theorem (see problem II.287) and from the fact that the triangles $A B C, A H B, B H C, C H A$ have the same nine-point circle (prove it). | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,675 |
289. Prove that the point of intersection of the diagonals of a quadrilateral with vertices at the points of tangency of the nine-point circle of triangle $ABC$ with the inscribed and excircles of this triangle lies on its midline. | 289. Let in $\triangle A B C$, for definiteness, $a \leqslant b \leqslant c$. Denote by $A_{1}, B_{1}, C_{1}$ the midpoints of sides $B C, C A, A B$, and by $F, F_{a}, F_{b}, F_{c}$ the points of tangency of the incircle and the excircles with the nine-point circle of $\triangle A B C$. We need to prove that in the hex... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,676 |
290. Let $F, F_{a}, F_{b}$, and $F_{c}$ be the points of tangency of the nine-point circle of triangle $ABC$ with the incircle and the three excircles (where $F_{a}$ is the point of tangency with the circle centered at $I_{a}$, and so on). Let $A_{1}$ and $A_{2}, B_{1}$ and $B_{2}, C_{1}$ and $C_{2}$ be the points of i... | 290. Using the formulas of problems II.193, II.194, II.289 (see the solution of the last problem), we find $\frac{\left|F_{b} F_{c}\right|}{\left|B_{1} C_{1}\right|}=\frac{(a+b)(b+c)(c+a) R^{3}}{a b c \cdot\left|O I_{a}\right| \cdot\left|O I_{b}\right| \cdot\left|O I_{c}\right|}$. The ratios of the other corresponding ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,677 |
291. On the sides $BC$, $CA$, and $AB$ of triangle $ABC$, squares $BCDE$, $ACFG$, and $BAHK$ are constructed outwardly. Let $FCDQ$ and $EBKP$ be parallelograms. Prove that triangle $APQ$ is an isosceles right triangle. | 291. Prove that $\triangle A B P = \triangle A C Q$. For this, it is sufficient to prove that $\triangle K B P = \triangle A B C$ and $\triangle F C Q = \triangle A B C$ (by two sides and the angle between them): $\angle Q A P = \angle C A B + \angle C A Q + \angle B A P = \angle C A B + \angle C A Q + \angle C Q A = \... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,678 |
292. Let $A B C D$ be a rectangle, $E$ a point on $B C$, $F$ on $D C$, $E_{1}$ the midpoint of $A E$, and $F_{1}$ the midpoint of $A F$. Prove that if $\triangle A E F$ is equilateral, then the triangles $D E_{1} C$ and $B F_{1} C$ are also equilateral. | 292. Since $\angle F E_{1} E=\angle F C E=90^{\circ}$, the quadrilateral $F E_{1} E C$ is cyclic, $\angle F C E_{1}=\angle F E E_{1}=60^{\circ}$. Similarly, the quadrilateral $F E_{1} A D$ is cyclic and $\angle E_{1} D F=\angle E_{1} A F=60^{\circ}$, i.e., $\triangle D E_{1} C$ is equilateral. In the same way, it can b... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,679 |
293. On the legs $AC$ and $BC$ of a right triangle, squares $ACKL$ and $BCMN$ are constructed outward. Prove that the quadrilateral bounded by the legs and the lines $LB$ and $NA$ is equal in area to the triangle formed by the lines $LB$, $NA$, and the hypotenuse $AB$. | 293. Let $P, Q$, and $R$ be the points of intersection of $L B$ and $A C$, $A N$ and $B C$, $L B$ and $A N$, respectively. Let $|B C|=a,|A C|=b$. It is sufficient to show that $S_{A C Q}=S_{A P B}$ (both these areas differ from the considered ones by the addition of the area of $\triangle A P R$). From the similarity o... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,680 |
295. Prove that if the centers of squares constructed on the sides of a given triangle outwardly serve as the vertices of a triangle, the area of which is twice the area of the given one, then the centers of the squares constructed on the sides of the triangle inwardly lie on one straight line. | 295. Let: $\angle A_{1} B C=\alpha, \angle A_{1} C B=\beta$; then $A A_{1}$ divides $B C$ in the ratio equal to
$$
\frac{S_{A B A_{1}}}{S_{A C A_{1}}}=\frac{\frac{1}{2}|A B| \cdot\left|B A_{1}\right| \sin (\angle B+\alpha)}{\frac{1}{2}|A C| \cdot\left|C A_{1}\right| \sin (\angle C+\beta)}=
$$
$=\frac{c}{b} \frac{\sin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,682 |
297. Let $A B C$ be an isosceles triangle $(|A B| = |B C|)$; $B D$ is its height. A circle of radius $B D$ rolls along the line $A C$. Prove that while the vertex $B$ is inside the circle, the arc of the circle located inside the triangle has a constant length. | 297. Let $K L$ be an arc of a circle located inside triangle $A B C$. Extending sides $A B$ and $B C$ beyond point $B$, we obtain arc
Fig. 54 $M N$, which is symmetric to arc $K L$ with re... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,683 |
298. Two points move along two intersecting straight lines with equal speeds. Prove that there exists a fixed point in the plane that is equidistant from them at all times. | 298. Let $O$ (Fig. 54) - the point of intersection of the lines, $A$ and $A_{1}$ - two positions of a point on one line, $B$ and $B_{1}$ - positions of another point at the same moments in time. Construct perpendiculars to $A B$ and $A_{1} B_{1}$ at their midpoints and denote their point of intersection by $M$; $\trian... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,684 |
299. Two cyclists are riding along two intersecting circles. Each is riding along their own circle at a constant speed. Starting simultaneously from one point where the circles intersect and completing one revolution, the cyclists meet again at this point. Prove that there exists a fixed point such that the distances f... | 299. a) Let $A$ and $B$ be the points of intersection of the circles, $A$ be the point from which the cyclists started, and $M$ and $N$ be the positions of the cyclists at some moment in time. If $M$ and $N$ are on the same side of $AB$, then $\angle ABM = \angle ABN$. If they are on opposite sides, then $\angle ABM + ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,685 |
300. Prove that: a) a rotation around point $O$ by an angle $\alpha$ is equivalent to the sequential application of two axial symmetries, the axes of which pass through point $O$, and the angle between the axes is $\alpha / 2$; a parallel translation is equivalent to two axial symmetries with parallel axes; b) two cons... | 300. b) Use the result from part a). Replace the rotation around $O_{1}$ with two axial symmetries, taking the line $O_{1} O_{2}$ as the axis of the second symmetry, and the rotation around point $O_{2}$ with two symmetries, taking the line $O_{1} O_{2}$ as the axis of the first symmetry. Remark. If $\alpha+\beta=2 \pi... | if\alpha+\beta=2\pi,thentheangles\pi-\frac{\alpha}{2},\pi-\frac{\beta}{2},\frac{\alpha+\beta}{2} | Geometry | proof | Yes | Yes | olympiads | false | 23,686 |
301. Given an arbitrary triangle $A B C$. On its sides as bases, three isosceles triangles $A K B, B L C, C M A$ are constructed with angles at the vertices $K, L$, and $M$ equal to $\alpha, \beta$, and $\gamma$, respectively, where $\alpha + \beta + \gamma = 2 \pi$. Moreover, all three triangles are either outside tri... | 301. Let us perform three consecutive rotations in the same direction around points $K, L$ and $M$ (or around $K_{1}, L_{1}$ and $M_{1}$) by angles $\alpha, \beta$ and $\gamma$. Since $\alpha+\beta+\gamma=2 \pi$, the resulting transformation is a parallel translation (see problem II.300). But since one of the vertices ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,687 |
302. Let $A B C D E F$ be an inscribed hexagon, in which $|A B|=|C D|=|E F|=R$, where $R$ is the radius of the circle, and $O$ is its center. Prove that the points of pairwise intersections of the circumcircles of triangles ВОС, DOE, FOA, distinct from $O$, are the vertices of an equilateral triangle with side $R$. | 302. Let's denote: $\angle B O C=2 \alpha, \angle D O E=2 \beta, \angle F O A=2 \gamma$. Let $K$, $M$, and $L$ be the points of intersection of the circumcircles of triangles $B O C$ and $A O F$, $B O C$ and $D O E$, $A O F$ and $D O E$. Point $K$ is inside triangle $A O B$, and $\angle B K O=180^{\circ}-\angle B C O=9... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,688 |
303. On the sides of a convex quadrilateral, rhombi are constructed outwardly, with each having an acute angle of $\alpha$. Moreover, the angles of two rhombi adjacent to one vertex of the quadrilateral are equal. Prove that the segments connecting the centers of opposite rhombi are equal, and the acute angle between t... | 303. Notations: $A B C D$ - the given quadrilateral, $O_{1}, O_{2}, O_{3}$, $O_{4}$ - the centers of rhombuses constructed respectively on $A B, B C, C D, D A$; $K$ and $L$ - midpoints of sides $A B$ and $B C, M$ - midpoint of diagonal $A C$. Triangles $O_{1} K M$ and $O_{2} L M$ are equal $\left(\left|O_{1} K\right|=\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,689 |
304. Given an arbitrary triangle. On its sides, equilateral triangles are constructed outward, the centers of which serve as the vertices of triangle $\Delta$. The centers of the equilateral triangles constructed on the sides of the original triangle inward serve as the vertices of another triangle $\delta$. Prove that... | 304. Let $ABC$ be a given triangle, $A_1B_1C_1$ be triangle $\Delta$, and $A_2B_2C_2$ be triangle $\delta$ (where $A_1$ and $A_2$ are the centers of triangles constructed on $BC$), and the sides of triangle $ABC$ are, as usual, $a, b, c$.
a) The fact that triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equilateral follows, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,690 |
305. On the plane, there are three points. Three lines are drawn through these points, forming an equilateral triangle. Find the geometric locus of the centers of these triangles. | 305. Let three given points form a triangle $ABC$. There are two families of equilateral triangles circumscribed about $\triangle ABC$. The first family is constructed as follows. Construct circles on the sides of the triangle such that the arcs of these circles, located outside the triangle, measure an angle of $4\pi ... | \frac{1}{3}\sqrt{\frac{1}{2}(^{2}+b^{2}+^{2})\2S\sqrt{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,691 |
306. Given a triangle $A B C$. On the line passing through vertex $A$ and perpendicular to side $B C$, two points $A_{1}$ and $A_{2}$ are taken such that $\left|A A_{1}\right|=\left|A A_{2}\right|=|B C|$ ($A_{1}$ is closer to the line $B C$ than $A_{2}$). Similarly, on the line perpendicular to $A C$ and passing throug... | 306. We will prove that triangles $C B_{1} A_{2}$ and $C A_{1} B_{2}$ are obtained from each other by a rotation about point $C$ by an angle of $90^{\circ}$. Indeed, $\triangle C A A_{1}=\triangle C B B_{1}\left(\left|B B_{1}\right|=|A C|,|B C|=\left|A A_{1}\right|, \angle C B B_{1}=\angle C A A_{1}\right)$, and since ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,692 |
307. Prove that a circumscribed polygon, all sides of which are equal, is regular if the number of sides is odd. | 307. Prove that the tangents to the circle, drawn from the vertices between which one vertex of the polygon is located, are equal. It follows from this that for a polygon with an odd number of sides, the points of tangency are the midpoints of the sides. | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,693 |
308. A line is drawn through the center of a regular $n$-sided polygon inscribed in a unit circle. Find the sum of the squares of the distances from the vertices of the $n$-sided polygon to this line. | 308. Note that if we consider a system of vectors originating from the center of a regular $n$-gon and ending at its vertices, the sum of these vectors is zero. Indeed, if we rotate all these vectors by an angle of $2 \pi / n$, their sum does not change, but on the other hand, the vector equal to their sum will rotate ... | \frac{n}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 23,694 |
309. Prove that the sum of the distances from an arbitrary point inside a convex polygon to its sides is constant if: a) all sides of the polygon are equal; b) all angles of the polygon are equal. | 309. a) If the side of a polygon is $a$, $S$ is its area, and $x_{1}, x_{2}, \ldots, x_{n}$ are the distances from some point inside it to the sides, then the statement of the problem follows from the equality $S=\left(a x_{1}+a x_{2}+\ldots+\right.$ $\left.+a x_{n}\right) / 2$.
b) Consider a regular polygon containing... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,695 |
310. A semicircle is divided into $2 n+1$ equal arcs by points $A_{0}, A_{1}, \ldots, A_{2 n+1}$ ( $A_{0}$ and $A_{2 n+1}$ are the endpoints of the semicircle), $O$ is the center of the semicircle. Prove that the lines $A_{1} A_{2 n}$, $A_{2} A_{2 n-1}, \ldots, A_{n} A_{n+1}$ form segments with the lines $O A_{n}$ and ... | 310. Let $B_{1}, B_{2}, \ldots, B_{n+1}$ be the points symmetric to $A_{1}$, $A_{2}, \ldots, A_{n+1}$ with respect to the diameter $A_{0} A_{2 n+1}, C_{k}$ and $C_{k}^{\prime}$ - the points of intersection of the line $A_{k} A_{2 n+1-k}$ with $O A_{n}$ and $O A_{n+1}$. Let $D_{k-1}$ and $D_{k}$ be the points of interse... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,696 |
311. Prove that if perpendiculars are dropped from an arbitrary point on a circle to the sides of an inscribed $2 n$-gon, then the products of the lengths of these perpendiculars taken alternately will be equal. | 311. Let $A$ (Fig. 56) be a given point, $A_{k}$ be some vertex of a $2n$-gon, $B_{k-1}$ and $B_{k}$ be the feet of the perpendiculars dropped from $A$ to the sides containing $A_{k}$, $\alpha_{k}$ and $\beta_{k}$ be the angles formed by the line $A A_{k}$ with these sides $\left(\beta_{k}=\right.$ $\left.=\angle A A_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,697 |
312. Let $A_{1} A_{2} \ldots A_{n}$ be an inscribed polygon; the center of the circle is inside the polygon. A system of circles touches the given one from the inside at points $A_{1}, A_{2}, \ldots, A_{n}$, and one of the intersection points of two adjacent circles lies on the corresponding side of the polygon. Prove ... | 312. Prove that if $O_{k}$ and $O_{k+1}$ are the centers of circles touching a given circle at points $A_{k}$ and $A_{k+1}, B$ is their intersection point lying on the chord $A_{k} A_{k+1}; r_{k}, r_{k+1}$ are their radii, then $r_{k}+r_{k+1}=r$, $\angle A_{k} O_{k} B=\angle A_{k+1} O_{k+1} B=\angle A_{k} O A_{k+1}$ ( ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,698 |
313. Consider a circle in which a regular $(2 n+1)$-gon $A_{1} A_{2} \ldots A_{2 n+1}$ is inscribed. Let $A$ be an arbitrary point on the arc $A_{1} A_{2 n+1}$.
a) Prove that the sum of the distances from $A$ to the vertices with even indices is equal to the sum of the distances from $A$ to the vertices with odd indice... | 313. a) Let $A$ be an arbitrary point on the circle ( $A$ is on the arc $A_{1} A_{2 n+1}$ ). Denote the side of the polygon by $a$, and the length of the diagonal connecting vertices through one, by $b$. By Ptolemy's theorem (problem II.237) for the quadrilateral $A A_{k} A_{k+1} A_{k+2}$ we have: $\left|A A_{k}\right|... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,699 |
314. a) Two tangents are drawn to a given circle. Let $A$ and $B$ be the points of tangency, and $C$ be the point of intersection of the tangents. Draw an arbitrary line $l$ tangent to the given circle, not passing through $A$ and $B$. Let $u$ and $v$ be the distances from $A$ and $B$ to $l$, and $w$ be the distance fr... | 314. a) Let $l$ intersect $A C$ and $B C$ at points $K$ and $N$ respectively and touch the circle at point $M$ (Fig. 57). Denote:
Fig. 57 $|A C|=|B C|=a,|A K|=|K M|=x$, $|B N|=|N M|=y$. Cle... | \sin^{2}\frac{\alpha}{2} | Geometry | proof | Yes | Yes | olympiads | false | 23,700 |
315. In a cyclic polygon, non-intersecting diagonals are drawn, dividing it into triangles. Prove that the sum of the radii of the circles inscribed in these triangles does not depend on how the diagonals are drawn. | 315. The statement of the problem can be proved by induction. The base case, $n=4$, is considered in problem II.235.
However, another approach can be suggested, based on the following equality. Let in triangle $ABC$ angle $A$ be the largest, $r$ and $R$ be the radii of the inscribed and circumscribed circles, respecti... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,701 |
316. Let $A_{1} A_{2} \ldots A_{n}$ be a polygon with perimeter $2 p$, circumscribed around a circle of radius $r$, and let $B_{1}, B_{2}, \ldots, B_{n}$ be the points of tangency of the sides $A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{n} A_{1}$ with the circle. Let $M$ be a point at a distance $d$ from the center of the ci... | 316. Let's consider, for definiteness, the case when point $M$ is inside the polygon. Denote by $u$ and $v$ the distances from $M$ to $A_{1} A_{2}$ and $A_{1} A_{n}$, respectively, and by $x$ and $y$ the projections of $A_{1} M$ onto $A_{1} A_{2}$ and $A_{1} A_{n}$ (the values of $x$ and $y$ should be considered positi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,702 |
317. Let $A B C D$ be a cyclic quadrilateral, and $M$ be an arbitrary point on the circle. Prove that the projections of point $M$ onto the Simson lines (see problem II.153) corresponding to point $M$ relative to triangles $A B C, B C D, C D A$, and $D A B$, lie on one line (the Simson line of the quadrilateral).
Furt... | 317. Consider three triangles - $A B C, A C D$ and $A D B$, having a common vertex $A$. Let $B_{1}, C_{1}$ and $D_{1}$ be the projections of $M$ onto $A B, A C$ and $A D$, respectively. The lines $B_{1} C_{1}, C_{1} D_{1}$ and $D_{1} B_{1}$ are the Simson lines of point $M$ with respect to triangles $A B C, A C D$ and ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,703 |
318. Inside the circle $\alpha$ there is a circle $\beta$. On the circle $\alpha$ there are two sequences of points: $A_{1}, A_{2}$, $A_{3} \ldots$ and $B_{1}, B_{2}, B_{3} \ldots$, following in the same direction, such that the lines $A_{1} A_{2}, A_{2} A_{3}, A_{3} A_{4} \ldots$ and $B_{1} B_{2}, B_{2} B_{3}, B_{3} B... | 318. Let, for definiteness, $B_{1}$ be on the arc $A_{1} A_{2}$, bounding a segment that does not contain the circle $\beta$. Denote by $C_{1}, C_{2}, \ldots$ the points of tangency of $A_{1} A_{2}, A_{2} A_{3}, \ldots$ with the circle $\beta$, and by $D_{1}, D_{2}, \ldots$ the points of tangency of $B_{1} B_{2}, B_{2}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,704 |
319. Using the result of the previous problem, prove the following statement (Poncelet's theorem). If there exists one $n$-gon inscribed in some circle $\alpha$ and circumscribed about another circle $\beta$, then there exist infinitely many $n$-gons inscribed in circle $\alpha$ and circumscribed about circle $\beta$, ... | 319. In the notation of the previous problem, the statement reduces to the following: if $A_{n+1}$ coincides with $A_{1}$, then $B_{n+1}$ also coincides with $B_{1}$. Suppose this is not the case. Then $A_{1} B_{1}$ and $A_{1} B_{n+1}$ are tangent to the circle $\gamma$, $A_{1} A_{2}$ intersects $\gamma$, and $B_{1}$ a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,705 |
320. On the sides of an equilateral triangle $P Q R$, isosceles triangles $P X Q, Q Y R$, and $R Z P$ are constructed outward with respect to triangle $P Q R$, such that $\angle P X Q=\frac{1}{3}(\pi+2 \angle A), \angle Q Y R=\frac{1}{3}(\pi+2 \angle B)$, and $\angle R Z P=\frac{1}{3}(\pi+2 \angle C)$, where $A, B, C$ ... | 320. Consider $\triangle B_{0} X C_{0}$. The line $X R$ is the bisector of angle $C_{0} X B_{0}$. It is easy to verify that $\angle C_{0} R B_{0}=\frac{\pi}{2}+\frac{1}{2} \angle C_{0} X B_{0}$. From this, it follows that $C_{0} R$ and $B_{0} R$ are the bisectors of angles $X C_{0} B_{0}$ and $X B_{0} C_{0}$ (see probl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 23,706 |
322. At the beginning of the 19th century, the Italian geometer Malfatti posed the following problem: to cut three circles from a given triangle so that the sum of their areas is the largest. In later research, the Malfatti circles came to be understood as three circles that are pairwise tangent to each other, each of ... | 322. For an equilateral triangle with side 1, the radius of each of the Malfatti circles is $\frac{\sqrt{3}-1}{4}$. The sum of the areas of the corresponding circles is $\frac{3 \pi(2-\sqrt{3})}{8}$. The sum of the areas of three circles, one of which is inscribed in this triangle, and the other two each touch the tria... | \frac{11\pi}{108}>\frac{3\pi(2-\sqrt{3})}{8} | Geometry | proof | Yes | Yes | olympiads | false | 23,708 |
323. Prove that $p \geqslant \frac{3}{2} \sqrt{6 R r}$, where $p$ is the semiperimeter, and $r$ and $R$ are the radii of the inscribed and circumscribed circles of the triangle. | 323. Use the equality $R r=\frac{a b c}{4 p}$ and the inequality $2 p=$ $=a+b+c \geqslant 3 \sqrt[3]{a b c}$ (the theorem of means). | proof | Inequalities | proof | Yes | Yes | olympiads | false | 23,709 |
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