problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
89. A circle of radius $r$ touches internally a circle of radius $R$. $A$ is the point of tangency. A line perpendicular to the line of centers intersects one circle at point $B$, and the other at point $C$. Find the radius of the circle circumscribed about triangle $A B C$. | 89. Let $M$ be the point of intersection of the line $CB$ with the line of centers of the given circles. Denote $|AM|=x, \widehat{ACB}=\varphi$, $|AB|^2=2rx, \quad|AC|^2=2Rx, \quad \sin \varphi=\frac{x}{|AC|}$. If $\rho$ is the radius of the circumcircle of $\triangle ABC$, then $\rho=\frac{|AB|}{2 \sin \varphi}=\frac{... | \sqrt{Rr} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,897 |
90. Two circles with radii $R$ and $r$ intersect, $A$ is one of the points of intersection. $B C$ is a common tangent ( $B$ and $C$ are the points of tangency). Find the radius of the circle circumscribed about triangle $A B C$. | 90. Let (Fig. 21) $\widehat{O_{1} A O_{2}}=\varphi\left(O_{1}, O_{2}\right.$ be the centers of the circles, $A$ be the point of their intersection farthest from $B C$). We will show that $\widehat{B A C}=\frac{\varphi}{2} \cdot\left(\right.$ For the other point, the angle will be $\left.180^{\circ}-\frac{\varphi}{2} \c... | \sqrt{Rr} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,898 |
91. In quadrilateral $A B C D$, given are $|A B|=a$, $|A D|=b$; sides $B C, C D$ and $A D$ touch a certain circle, the center of which is at the midpoint of $A B$. Find the side $|B C|$. | 91. $D O$ and $C O$ are the bisectors of angles $A D C$ and $D C B$. Let $\alpha, \beta$ and $\gamma$ be the measures of the corresponding angles (Fig. 22). But $\alpha+2 \beta+2 \gamma+\alpha=2 \pi$, hence, $\alpha+\beta+\gamma=\pi$; from this it follows that $\widehat{D O A}=\gamma, \widehat{C O B}=\beta$ and $\trian... | \frac{^2}{4b} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,899 |
92. In the inscribed quadrilateral $A B C D$, given are $|A B|=a,|A D|=b, a>b$. Find the side $|B C|$, if it is known that $B C, C D$ and $A D$ touch a certain circle, the center of which lies on $A B$. | 92. From the condition of the problem, it follows that the bisectors of angles $C$ and $D$ intersect on side $A B$. Let's denote this point of intersection as $O$. We will describe a circle around $\triangle D O C$. Let $K$ be the second point of intersection of this circle with $A B$. We have
$$
\widehat{D K A}=\wide... | -b | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,900 |
93. Given an isosceles triangle \(ABC, |AB| = |BC|, AD\) is the angle bisector. A perpendicular line to \(AD\) at point \(D\) intersects the extension of \(AC\) at point \(E\); the feet of the perpendiculars dropped from \(B\) and \(D\) to \(AC\) are \(M\) and \(N\). Find \(|MN|\), if \(|AE| = a\). | 93. Let $P$ be the intersection point of line $DE$ with $AB$, and $K$ be a point on $AB$ such that $KD \| AC$. $\triangle AKD$ is isosceles with $\widehat{(KDA} = \widehat{DAC} = \widehat{DAK})$. Therefore, $KD$ is the median in a right triangle, and $|MN| = \frac{1}{2}|KD| = \frac{1}{4}|AP| = \frac{1}{4}|AE| = \frac{1... | \frac{1}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,901 |
94. From point $A$ at an angle $\alpha$, two rays emerge. On one ray, two points $B$ and $B_{1}$ are taken, and on the other $-C$ and $C_{1}$. Find the length of the common chord of the circumcircles of triangles $A B C$ and $A B_{1} C_{1}$, if $|A B|-|A C|=\left|A B_{1}\right|-\left|A C_{1}\right|=a$. | 94. Let (Fig. 23) the second intersection point of the circumcircles of $\triangle A B C$ and $\triangle A B_{1} C_{1}$ be $A_{1}$. From the condition, it follows that $\left|B B_{1}\right|=\left|C C_{1}\right|$, and $\widehat{A B A_{1}}=\widehat{A C A_{1}}$ and $\widehat{A B_{1} A_{1}}=$ $=\widehat{A C_{1} A_{1}}$. Th... | \frac{}{2\sin\frac{\alpha}{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,902 |
95. Let $O$ be the center of the circle, $C$ a point on the circle, and $M$ the midpoint of $OC$. $A$ and $B$ are points on the circle such that $\widehat{A M O}=\widehat{B M C}$; $A$ and $B$ lie on the same side of the line $O C$. Find $|A B|$, if $|A M| - |B M| = a$. | 95. Note that points $A, O, M, B$ lie on the same circle ( $\widehat{A M B}$ is measured as the half-sum of arc $\widehat{A B}$ and the arc symmetric to $\breve{A B}$ with respect to $O C$, i.e., $\widehat{A M B}=\widehat{A O B}$ ). Further, let $|M K|=|M B|$ on $A M$, then $\triangle A K B$ is similar to $\triangle O ... | |AB|=2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,903 |
97. In a circle of radius $R$, a chord $AB$ is given. Let $M$ be an arbitrary point on the circle. On the ray $MA$, we lay off the segment $MN, |MN| = R$, and on the ray $MB$ - the segment $MK$, equal to the distance from $M$ to the orthocenter of triangle $MAB$. Find $|NK|$, if the smaller arc subtended by $AB$ is $2\... | 97. Let $P$ be the foot of the perpendicular dropped from $N$ to the line $M B$; then $|M P'| = R \cos \alpha$, hence $|M P|$ is equal to the distance from the center $O$ of $A B$, but the distance from the vertex of the triangle to the orthocenter is twice the distance from the center of the circumscribed circle to th... | |MK|={\begin{pmatrix}R,\text{if}M\text{isonthelargerarcofthecircle,}\\R\sqrt{1+8\cos^{2}\alpha},\\\text{if}M\text{isonthesmallerarcofthecircle.}\end{pmatrix}.} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,905 |
98. The height dropped from the vertex of the right angle of a right-angled triangle to the hypotenuse divides the triangle into two triangles, each of which has an inscribed circle. Determine the angles and the area of the triangle formed by the legs of the original triangle and the line passing through the centers of... | 98. Let $ABC$ be a given triangle, $CD$ - the altitude, $O_{1}$ and $O_{2}$ the centers of the circles inscribed in $\triangle ACD$ and $\triangle BDC$, $K$ and $L$ the points of intersection of the lines $DO_{1}$ and $DO_{2}$ with $AC$ and $CB$. Since $\triangle ADC$ is similar to $\triangle CDB$, and $KD$ and $LD$ ar... | \frac{^{2}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,906 |
99. The height of a right triangle, dropped to the hypotenuse, is equal to $h$. Prove that the vertices of the acute angles of the triangle and the projections of the base of the height onto the legs lie on the same circle. Determine the length of the chord cut by the line containing the height on this circle, and the ... | 99. The notations are clear from Fig. 24. CKDL is a rectangle. Since $\widehat{L K A}=90^{\circ}+\alpha, \widehat{L B A}=90^{\circ}-\alpha$, the quadrilateral $B L K A$ is cyclic,
$$
\operatorname{tg} \varphi=\frac{|L C|}{|C A|}=\frac{h \cos \alpha}{\frac{h}{\sin \alpha}}=\frac{1}{2} \sin 2 \alpha
$$
If $R$ is the ra... | \frac{\sqrt{5}+1}{2},\frac{\sqrt{5}-1}{2} | Geometry | proof | Yes | Yes | olympiads | false | 24,907 |
100. A circle of radius $R$ touches the line $l$ at point $A$, $AB$ is a diameter of this circle, and $BC$ is an arbitrary chord. Let $D$ be the foot of the perpendicular dropped from $C$ to $AB$. Point $E$ lies on
2 I. $\Phi$, Sharygin
the extension of $CD$ beyond point $D$, such that $|ED|=|BC|$. The tangents to the... | 100. Let (Fig. 25) $P$ and $Q$ be the points of tangency of the tangents drawn from $E$. We will prove that $|E P|=|E Q|=|B D|$. Indeed,
\[
\begin{aligned}
|E P|^{2}=(|E D| + |D C|) & (|E D| - |D C|)= \\
& =|E D|^{2}-|D C|^{2}=|B C|^{2}-|D C|^{2}=|B D|^{2}
\end{aligned}
\]
(by the condition $|E D|=|B C|$). Let us den... | 2R | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,908 |
101. 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. | 101. Note that if we consider a system of $n$ 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 will not change, and on the other hand, the vector equal to their sum will rot... | \frac{n}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,909 |
102. Find the sum of the squares of the distances from the points of tangency of the inscribed circle with the sides of a given triangle to the center of the circumscribed circle, if the radius of the inscribed circle is $r$, and the radius of the circumscribed circle is $R$. | 102. 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, if $|BC|=a,|AC|=b,|AB|=c$, we have $|AN|=p-a$, $|CN|=p-c$ (see problem 18, section I) $|ON^{2}|=|OB_{1}^{2}+B_{1}N^{2}=$
\[
\begin{gathered}
=\left(\left.{ }^{... | 3R^{2}-4Rr-r^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,910 |
103. 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 radius of the given circle is $R$, the distance from its center ... | 103. Let (Fig. 26) $P$ be the point of intersection of the diagonals, and $K$, $L, M, N$ be the feet of the perpendiculars dropped from $P$ to $A B$,
Fig. 26. Since the quadrilateral $P K B... | \frac{R^{2}-^{2}}{2R} | Geometry | proof | Yes | Yes | olympiads | false | 24,911 |
104. The diagonals of an inscribed quadrilateral are perpendicular. Prove that the midpoints of its sides and the bases of the perpendiculars dropped from the point of intersection of the diagonals to the sides lie on one circle. Find the radius of this circle if the radius of the given circle is $R$, and the distance ... | 104. Let (Fig. 27) $ABCD$ 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 $\widehat{BPM} = \widehat{LPD} = \... | \frac{1}{2}\sqrt{2R^2-^2} | Geometry | proof | Yes | Yes | olympiads | false | 24,912 |
105. Prove that if a quadrilateral is inscribed in a circle of radius $R$, and is also circumscribed about a circle of radius $r$, and the distance between the centers of these circles is $d$, then the following relation holds:
$$
\frac{1}{(R+d)^{2}}+\frac{1}{(R-d)^{2}}=\frac{1}{r^{2}}
$$
There are infinitely many qu... | 105. 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 | 24,913 |
106. a) Two tangents are drawn to a given circle from point $\mathrm{K}$. 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 $l$, a... | 106. a) Let $l$ intersect $A C$ and $B C$ at points $K$ and $N$ and touch the circle at point $M$ (Fig. 28). Denote $|A C| = |B C| = a, \quad |A K| = |K M| = x, \quad |B N| = |N M| = y$. Clearly,
^{2} | Geometry | proof | Yes | Yes | olympiads | false | 24,914 |
107. In a convex quadrilateral $A B C D$, given are $|A B|=a,|A D|=b,|B C|=p-a,|D C|=p-b$. Let $O$ be the point of intersection of the diagonals. Denote by $\alpha$ the angle $\widehat{B} \widehat{A C}$. To what does the length $A O$ tend if $\alpha$ tends to zero?
## § 2. Proof Problems | 107. Let's first find $\lim _{\alpha \rightarrow 0} \frac{|A O|}{|O C|}$. Denote $\hat{C}=\beta$. We have
$$
\frac{|A O|}{|O C|}=\frac{S_{A B D}}{S_{B D C}}=\frac{\frac{1}{2} a b \sin \alpha}{\frac{1}{2}(p-a)(p-b) \sin \beta}
$$
By the cosine theorem,
$$
\begin{aligned}
& \quad \mid B D_{1^{2}}=a^{2}+b^{2}-2 a b \co... | \lim_{\alphaarrow0}|AO|=p\frac{\sqrt{}}{\sqrt{}+\sqrt{(p-)(p-b)}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,915 |
108. Prove that if one side of a triangle lies on a fixed straight line in a plane, and the point of intersection of the altitudes coincides with a fixed point, then the circle circumscribed around this triangle also passes through a fixed point. | 108. Prove that the point symmetric to the orthocenter of a triangle with respect to a side of the triangle lies on the circumcircle. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,916 |
110. In triangle $A B C$, the altitude $B D$ is drawn, $A N$ is perpendicular to $A B$, $C M$ is perpendicular to $B C$, and $|A N|=|D C|,|C M|=|A D|$. Prove that $M$ and $N$ are equidistant from vertex $B$. | 110. $\mid M B$ । $^{2}=a^{2}+c^{2} \cos ^{2} \hat{A}=a^{2}+c^{2}-c^{2} \sin ^{2} \hat{A}=a^{2}+c^{2}-a^{2} \times$ $\times \sin ^{2} \hat{C}=c^{2}+a^{2} \cos ^{2} \hat{C}=\mid N B{ }^{2}$.
110. $\mid M B$ । $^{2}=a^{2}+c^{2} \cos ^{2} \hat{A}=a^{2}+c^{2}-c^{2} \sin ^{2} \hat{A}=a^{2}+c^{2}-a^{2} \times$ $\times \sin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,918 |
111. Given a quadrilateral $A B C D$. On the lines $A C$ and $B D$, points $K$ and $M$ are taken such that $B K$ is parallel to $A D$, and $A M$ is parallel to $B C$. Prove that $K M$ is parallel to $C D$. | 111. If $O$ is the point of intersection of the diagonals $A C$ and $B D$, then, using the similarity of the corresponding triangles, we get
$$
\frac{|O K|}{|O C|}=\frac{|O K|}{|O B|} \cdot \frac{|O B|}{|O C|}=\frac{|O A|}{|O D|} \cdot \frac{|O M|}{|O A|}=\frac{|O M|}{|O D|}
$$
which is what was required. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,919 |
112. In $\triangle ABC$, the bisector of the internal angle $AD$ is drawn. Construct the tangent $l$ to the circumscribed circle
at point $A$. Prove that the line drawn through $D$ parallel to $l$ is tangent to the inscribed circle. | 112. Prove that $i$ forms the same angles with $A D$ as the line $B C$, which is tangent to our circle. It follows from this that the other tangent to the circle passing through $D$ will be parallel to $l$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,920 |
113. In triangle $ABC$, a line is drawn intersecting sides $AC$ and $BC$ at points $M$ and $N$ such that $|MN| = |AM| + |BM|$. Prove that all such lines are tangent to the same circle. | 113. Let's construct a circle (Fig. 29) that is tangent to the lines $M N$, $A C$, and $B C$ in such a way that the points of tangency $P$ and $Q$ with the lines $A C$ and $B C$ are outside the segments $C M$ and $C N$ (this will be the circle inscribed in the triangle $M C N$). If $R$ is the point of tangency with $M ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,921 |
114. Prove that the points symmetric to the center of the circle circumscribed around a triangle with respect to the midpoints of its medians lie on the heights of the triangle. | 114. If $O$ is the center of the circle circumscribed around $ABC$, $D$ is the midpoint of $CB$, $H$ is the point of intersection of the altitudes, $L$ is the midpoint of $AH$, then $|AL|=|OD|$, and since $AL \| OD$, it follows that $OL$ bisects $AD$, i.e., $L$ is symmetric to $O$ with respect to the midpoint of $AD$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,922 |
115. Prove that if the height of a triangle is $\sqrt{2}$ times the radius of the circumscribed circle, then the line connecting the bases of the perpendiculars dropped from the base of this height to the sides containing it passes through the center of the circumscribed circle. | 115. Let $BD$ be the height of the triangle, and $|BD| = R \sqrt{2}$, where $R$ is the radius of the circumscribed circle. $K$ and $M$ are the feet of the perpendiculars dropped from $D$ to $AB$ and $BC$, respectively, and $O$ is the center of the circumscribed circle. If angle $C$ is acute, then $\widehat{KBO} = 90^{\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,923 |
116. Let $ABC$ be a right triangle $\left(\hat{C}=90^{\circ}\right)$, $CD$ be the altitude, and $K$ be a point on the plane such that $|AK|=|AC|$. Prove that the diameter of the circumcircle of $\triangle ABK$, passing through vertex $A$, is perpendicular to the line $DK$. | 116. Note that $\triangle A D K$ is similar to $\triangle A B K$, since
$$
|A K|^{2}=|A C|^{2}=|A D| \cdot|A B|, \text{ i.e. } \frac{|A K|}{|A D|}=\frac{|A B|}{|A K|}
$$
If $O$ is the center of the circumcircle of $\triangle A B K$, then
$$
\widehat{O A D}+\widehat{A D K}=90^{\circ}-\widehat{A K B}+\widehat{A D K}=9... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,924 |
119. Let $E$ be an arbitrary point on the side $AC$ of triangle $ABC$. Through the vertex $B$, draw an arbitrary line $l$. The line passing through $E$ parallel to $BC$ intersects $l$ at point $N$, and the line parallel to $AB$ intersects $l$ at point $M$. Prove that $AN$ is parallel to $CM$. | 119. Let $F$ and $D$ be the points of intersection of $EN$ and $EM$ with $AB$ and $BC$ respectively. We will prove that $\triangle A F N$ and $\triangle M D C$ are similar. Using the similarity of various triangles and the equality of opposite sides of the parallelogram, we will have
$$
\begin{aligned}
\frac{|N F|}{|F... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,927 |
120. On the opposite sides $B C$ and $D A$ of a convex quadrilateral, points $M$ and $N$ are taken such that
$$
\frac{|B M|}{|M C|}=\frac{|A N|}{|N D|}=\frac{|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 $C D$. | 120. 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 $K M L$.
Consider the 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... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,928 |
121. 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. | 121. 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, and \(O\) the point of intersection of the diagonals. Suppose \(|BO| \leq |OD|, |AO| \leq |OC|\); consider \(\triangl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,929 |
123. 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 to each other. Prove that $A B C D$ is a rectangle. | 123. From the condition of the problem, it follows that $ABCD$ (Fig. 30) 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 ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,931 |
124. Given a right triangle $A B C$, angle $C$ is right, $O$ is the center of the inscribed circle, $M$ is the point of tangency of the inscribed circle with the hypotenuse, the circle with center at $M$, passing through $O$, intersects the angle bisectors of angles $A$ and $B$ at points $K$ and $L$, different from $O$... | 124. If $K N$ is the perpendicular from $K$ to $A B, \widehat{C A B}=\alpha$, then
\[
\begin{aligned}
& \frac{|K N|}{|O M|}=\frac{|A K|}{|A O|}=\frac{|A O|-|K O|}{|A O|}= \\
& =\frac{|A O|-2|O M| \sin \frac{\alpha}{2}}{|A O|}=\frac{|A O|-2|A O| \sin \frac{\alpha}{2} \sin \frac{\alpha}{2}}{|A O|}= \\
& \quad=1-2 \sin ^... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,932 |
125. 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. | 125. Prove that $\triangle A B P = \triangle A C Q$. To do this, we need 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 included angle):
$$
\begin{aligned}
\widehat{Q A P} & =\widehat{C A B}+\widehat{C A Q}+\widehat{B A P}=\widehat{C A B}+\widehat{C A Q}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,933 |
126. $A B C D$ is a rectangle, $E$ is a point on $B C$, $F$ is on $D C$, $E_{1}$ is the midpoint of $A E$, $F_{1}$ is the midpoint of $A F$. Prove that if $\triangle A E F$ is equilateral, then triangles $D E_{1} C$ and $B F_{1} C$ are also equilateral. | 126. Since $\widehat{F E_{1} E}=\widehat{F C E}=90^{\circ}$, the quadrilateral $F E_{1} E C$ is cyclic, $\widehat{F C E_{1}}=\widehat{F E E_{1}}=60^{\circ}$. Similarly, the quadrilateral $F E_{1} A D$ is cyclic and $\widehat{E_{1} D F}=\widehat{E_{1} A F}=60^{\circ}$, i.e., $\triangle D E_{1} C$ is equilateral. The sam... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,934 |
127. Quadrilateral $A B C D$ is inscribed in a circle. Let $O_{1}, O_{2}, O_{3}, O_{4}$ be 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}$, $\mathrm{H}_{4}$ be the points of intersection of the altitudes of the same triangles. Prove that $\mathrm{O}_{1} \mathrm{... | 127. 128. Note that if $O$ is the center of the circle inscribed in triangle $ABC$, then $\widehat{BOC}=90^{\circ}+\frac{1}{2} \hat{A}$. Indeed, $\widehat{BOC}=$ $=180^{\circ}-\frac{1}{2}(\hat{B}+\hat{C})=90^{\circ}+\frac{1}{2} \hat{A}$. Since $\widehat{BO_{1}A}=\widehat{BO_{4}A}$, the quadrilateral $ABO_{1}O_{4}$ is c... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,935 |
128. Given a triangle $ABC$, $D$ is an arbitrary point on the plane. Prove that the points of intersection of the altitudes of triangles $ABD$, $BCD$, and $CAD$ are the vertices of a triangle equal in area to the given one. | 128. If the sides of triangle $ABC$, opposite to vertices $A, B$, and $C$, are equal to $a, b$, and $c$ respectively, and the angles $\widehat{ADB}$, $\widehat{BDC}$, and $\widehat{CDA}$ are equal to $\alpha, \beta$, and $\gamma$ (assuming that $\alpha+\beta+\gamma=2 \pi$), then the distances from point $D$ to the poin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,936 |
129. Two circles intersect at points $A$ and $B$. An arbitrary line passes through $B$ and intersects the first circle again at $C$, and the second circle at $D$. The tangents to the first circle at $C$, and to the second circle at $D$ intersect at point $M$. A line through the intersection of $A M$ and $C D$ is parall... | 129. Let us denote (Fig. 32) by $O$ the intersection of $A M$ and $D C$. Draw a tangent through $B$ to the second circle and denote the intersection of this tangent with $A C$ by $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 a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,937 |
130. 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$. | 130. 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$. 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 of the correspo... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,938 |
131. The sides of a convex quadrilateral are divided into $(2 n+1)$ equal parts each. The corresponding division points of opposite sides are connected to each other. Prove that the area of the central quadrilateral is $1 /(2 n+1)^{2}$ of the area of the entire quadrilateral. | 131. The statement of our problem follows from the following two facts.
a)
, similarly $S_{BNM}=S_{DMN}$, from which the statement of the problem follows.
... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,940 |
133. In parallelogram $A B C D$, vertices $A, B, C$ and $D$ are connected to the midpoints of sides $C D, A D, A B$ and $B C$. Prove that the area of the quadrilateral formed by these lines is $1 / 5$ of the area of the parallelogram. | 133. If (Fig. 34) $M$ is the midpoint of $DC$, $N$ is the midpoint of $BC$, $K$ and $L$ are the points of intersection of $DN$ with $AM$ and $AB$ respectively, then $\frac{|KM|}{|AK|} = \frac{|DM|}{|AL|} = \frac{1}{4}$, i.e., $|AK| = \frac{4}{5}|AM|$, therefore,
$$
S_{ADK} = \frac{4}{5} S_{ADM} = \frac{4}{5} \cdot \fr... | \frac{1}{5}S | Geometry | proof | Yes | Yes | olympiads | false | 24,941 |
134. Prove that the area of the octagon formed by the lines connecting the vertices of a parallelogram to the midpoints of the opposite sides is $1 / 6$ of the area of the parallelogram. | 134. Let (Fig. 35) $Q$ be the midpoint of $A D$, $N$ be the midpoint of $B C$, $M$ be the midpoint of $D C$, and $K, P, R$ be the points of intersection of $D N$ and $A M$, $Q C$ and $D N$, $Q C$ and $A M$. Then $|D K|=\frac{2}{5}|D N|,\left|D P_{\mid}=\right| P N|, \quad Q P|=$
 \). Thus,
\[
s_{1} + x + s_{2} + s_{2} + y + s_{3} = x + y + s_{2} + Q \Rightarrow s_{1} + s_{2} + s_{3} = Q
\]
, $K$ is the intersection point of lines $E D$ and $F B$. Prove that quadrilaterals $A B K D$ and $C E K F$ are equal in area. | 137. If $S$ is the area of the parallelogram (Fig. 37), then $S_{A B K} + S_{K C D} = \frac{1}{2} S$. On the other hand, $S_{D E C} = S_{E K C} + S_{K C D} = \frac{1}{2} S$, which means $S_{A B K} + S_{K C D} = S_{E K C} + S_{K C D}$, i.e., $S_{A B K} = S_{E K C}$; analogously, $S_{A K D} = S_{K C F}$; adding the last ... | S_{ABKD}=S_{CEKF} | Geometry | proof | Yes | Yes | olympiads | false | 24,945 |
138. Given a triangle $A B C$. On the rays $A B$ and $C B$, segments $|A K|=|C M|=|A C|$ are laid off. Prove that the radius of the circumcircle of $\triangle B K M$ is equal to the distance between the centers of the inscribed and circumscribed circles of $\triangle A B C$, and that the line $K M$ is perpendicular to ... | 138. Let (Fig. $38, a$ ) $O$ be the center of the circumscribed circle, and $I$ be the center of the inscribed circle. Drop perpendiculars from $O$ and $I$ to $AB$ and $BC: ON, OP, IL, IQ$. If $a, b, c$ are the lengths of the sides $BC, AC$, and $AB$ of the triangle, it is easy to find $|BK|=|c-b|, |BM|=|a-b|$, $|BN|=\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,946 |
140. 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). | 140. 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 $c$. Then (see problem 87) $|Q P|^{2}=a^{2}+b^{2}-2 R^{2},|Q M|^{2}=b^{2}+c^{2}-2 R^{2},|P M|^{2}=c^{2}+a^{2}-2 R^{2}$. If $O$ is the center of the circle, then for $Q O$ to be perpendicular to $P M$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,948 |
141. Prove that if a circle can be inscribed in a quadrilateral, then: a) the circles inscribed in the two triangles into which the given quadrilateral is divided by a diagonal touch each other; b) the points of contact of these circles with the sides of the quadrilateral are the vertices of an inscribed quadrilateral. | 141. a) Consider the quadrilateral $A B C D$; let $K$ and $L$ be the points of tangency of the incircles of $\triangle A B C$ and $\triangle A C D$,
Fig. 39. with the line $A C$. Then (see ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,949 |
142. 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 $B D A$. | 142. Let (Fig. 40, a, 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 $\mathrm{O}_{1} \mathrm{O}_{2} \mathrm{O}_{3} \mathrm{O}_{4}$ is a rectangle (see problem 127),
$, and $B D$ is its height. A circle with 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. | 143. Let $\breve{K L}$ be an arc of a circle located inside triangle $A B C$. By extending sides $A B$ and $B C$ beyond point $B$, we obtain arc $\overline{M N}$, which is symmetric to $\breve{K L}$ with respect to the diameter parallel to $A C$. Since $\widehat{A B C}$ is measured by the arc $\frac{1}{2}$ ( $\breve{K ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,951 |
144. 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. | 144. Let (Fig. 41) $O$ be the point of intersection of the lines, $A$ and $A_{1}$ be two positions of a point on one line, and $B$ and $B_{1}$ be the positions of another point at the same moments. Construct perpendiculars to $A B$ and $A_{1} B_{1}$ at their midpoints and denote their point of intersection by $M$; $\tr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,952 |
145. 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... | 145. a) Let $A$ and $B$ be points
Fig. 41. points of intersection of the circles, $A$ being the point from which the cyclists started. $M$ and $N$ are the positions of the cyclists at some ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,953 |
146. 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. | 146. Let (Fig. 43) $A$ 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 enclosing $A_{k}, \alpha_{k}$ and $\beta_{k}$ be the angles formed by the line $A A_{k}$ with these
 the radius of the inscribed circle is equal to $1 / 3$ of the height dropped to the middle side;
b) the line connecting the centroid of the triangle with the center of the inscribed circle is parallel to the middle side... | 148. Let the lengths of the sides of a triangle be \(a, b, c\), and \(b = \frac{a+c}{2}\).
a) From the equality \(p r = \frac{1}{2} b h_{b}\) (where \(p\) is the semiperimeter, \(r\) is the radius of the inscribed circle, and \(h_{b}\) is the height dropped to side \(b\)), we get \(\frac{a+b+c}{2} r = \frac{1}{2} b h_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,956 |
149. Bretschneider's Theorem (Cosine Rule for Quadrilaterals). Let $a, b, c, d$ be the consecutive side lengths of a quadrilateral, $m$ and $n$ the lengths of its diagonals, $\hat{A}$ and $\hat{C}$ the measures of two opposite angles. Then the following relation holds:
$$
m^{2} n^{2}=a^{2} c^{2}+b^{2} d^{2}-2 a b c d ... | 149. Let in quadrilateral $A B C D$ (Fig. 15)
$$
\begin{array}{ll}
|A B|=a, & B C|=b, \quad| C D \mid=c \\
|D A|=d, & |A C|=m, \quad|B D|=n
\end{array}
$$
Construct triangle $A K B$ on side $A B$ outward, similar to triangle $A C D$, such that $\widehat{B A K}=\widehat{D C A}, \widehat{A B K}=\widehat{C A D}$. On sid... | ^{2}n^{2}=^{2}^{2}+b^{2}^{2}-2\cos(\hat{A}+\hat{C}) | Geometry | proof | Yes | Yes | olympiads | false | 24,957 |
151. Prove that if $A B C$ is an equilateral triangle, $M$ is an arbitrary point in the plane, not lying on the circumcircle of $A B C$, then there exists a triangle whose side lengths are $|M A|$, $|M B|$, and $|M C|$ (Pompeiu's theorem). Find the angle of this triangle opposite the side equal to $|M B|$, if $\widehat... | 151. If $|M B|$ is the greatest of the segments $|M A|,|M B|, |M C|$, then applying Bretschneider's theorem (problem 149) to the quadrilateral $A B C M$, we get that $|M B|^{2}=|M A|^{2}+|M C|^{2}-$ $-2|M A| \cdot|M C| \cos \left(\widehat{A M C}+60^{\circ}\right), \quad$ i.e. $\quad|M B|<|M A|+|M C|$, since $\cos \left... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,959 |
152. Consider a circle in which a regular $(2 n+1)$-gon $A_{1} A_{2} \ldots A_{2 n+1}$ is inscribed. Let $A$ be an arbitrary point on the arc $A_{1} A_{2 n+1}$.
a) Prove that the sum of the distances from $A$ to the vertices with even indices is equal to the sum of the distances from $A$ to the vertices with odd indic... | 152. a) Let $A$ be an arbitrary point on the circle (on the arc $\widehat{A_{2 n+1} A_{1}}$). Denote the side of the polygon by $a$, and the length of the diagonal connecting vertices separated by one vertex by $b$. By Ptolemy's theorem for the quadrilateral $A A_{k} A_{k+1} A_{k+2}$
$$
\left|A A_{k}\right| a+\left|A ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,960 |
153. Theorem of Leibniz. Let $M$ be an arbitrary point in the plane, and $G$ the centroid of triangle $ABC$. Then the following equality holds: $3|M G|^{2}=|M A|^{2}+|M B|^{2}+|M C|^{2}-$
$$
-1 / 3\left(|A B|^{2}+|B C|^{2}+|C A|^{2}\right)
$$ | 153. Let points $A, B, C$ and $D$ in the Cartesian coordinate system have coordinates respectively $(x_{1}, y_{1})$, $\left(x_{2}, y_{2}\right), \left(x_{3}, y_{3}\right), (x, y)$, and the coordinates of point $G-\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}\right)$. Then the validity of the statement ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,961 |
154. Let $A B C$ be an equilateral triangle with side $a$, and $M$ be some point on the plane, located at a distance $d$ from the center of the triangle $A B C$. Prove that the area of the triangle with sides equal to the segments $|M A|,|M B|,|M C|$, is expressed by the formula $S=\frac{\sqrt{\overline{3}}}{12}\left|a... | 154. Consider the case when point $M$ (Fig. 46) lies inside triangle $A B C$. Rotate triangle $A B M$ around $A$ by an angle of $60^{\circ}$ so that $B$ moves to $C$. We obtain triangle $A M_{1} C$, equal to $\triangle A B M$. Triangle $A M M_{1}$ is equilateral, hence the side lengths of $\triangle C M M_{1}$ are equa... | \frac{\sqrt{3}}{12}(^{2}-3^{2}) | Geometry | proof | Yes | Yes | olympiads | false | 24,962 |
155. 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 $|A B|+|C D|=$ $=|B C|+|A D|,|B K|+|B L|=|D K|+|D L|,|A K|+$... | 155. Show that each of these conditions is necessary and sufficient for the existence of a circle inscribed in a quadrilateral $A B C D$ (see also problem 19, section I). | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,963 |
156. The extensions of sides $A B$ and $D C$ of the convex quadrilateral $A E C D$ 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 $|A D|+|D C|=$ $=|A B|+|C B|, \quad|A K|+|C K|=|A L|+|C L... | 156. Show that each of these conditions is necessary and sufficient for there to exist a circle that is tangent to the lines $A B, B C, C D$ and $D A$, with its center located outside the quadrilateral $A B C D$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,964 |
159. On the sides of a quadrilateral, squares are constructed outwardly. Prove that the segments connecting the centers of opposite squares are equal in length and perpendicular to each other. | 159. Let $O_{1}, O_{2}, O_{3}, O_{4}$ be the consecutive centers of squares. We will perform sequential rotations in the same direction around the points $O_{1}, O_{2}, O_{3}, O_{4}$ by $90^{\circ}$. Reasoning as in the previous problem, we will show that the resulting transformation leaves all points of the plane fixe... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,966 |
160. 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... | 160. 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 $BC$), the sides of triangle $ABC$ are $a, b, c$ as usual.
a) The fact that triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equilateral follows, for ex... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,967 |
161. Given an arbitrary 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 line $B C$ than $A_{2}$).
Similarly, on the line perpendicular to $A C$ and p... | 161. 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, $\triangle C A A_{1}=\triangle C B B_{1}\left(B B_{1}{ }^{\prime}=A C, \quad B C==^{\prime} A A_{1}^{\prime}, \widehat{C B B_{1}}=\right.$ $=\widehat{C A ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,968 |
162. Prove that the midpoints of the sides of a triangle, the feet of the altitudes, and the midpoints of the segments of the altitudes from the vertices to their point of intersection lie on one circle, the "nine-point circle" (Euler). | 162. Let $A B C$ be a given triangle, $H$ the point of intersection of its altitudes, $A_{1}, B_{1}, C_{1}$ the midpoints of segments $A H, B H, C H$, $A A_{2}$ the altitude, and $A_{3}$ the midpoint of $B C$.
For convenience, let's assume that $\triangle A B C$ is an acute triangle. Since $\widehat{B_{1} A_{1} C_{1}}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,969 |
163. Let $H$ be the orthocenter of a triangle, $D$ the midpoint of one of its sides, and $K$ one of the intersection points of the line $H D$ with the circumcircle (with $D$ between $H$ and $K$). Prove that $D$ is the midpoint of the segment $H K$. | 163. Our statement follows from the fact that $D$ lies on the nine-point circle, and the nine-point circle is homothetic to the circumcircle with center $H$ and coefficient $1 / 2$ (see problem 162). | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,970 |
164. Let $M$ be the point of intersection of the medians of a triangle, $E$ be the foot of some altitude, and $F$ be one of the points of intersection of the line $M E$ with the circumscribed circle ($M$ is between $E$ and $F$). Prove that $|F M|=2|E M|$.
In problems 165-168, $a, b$ and $c$ denote the lengths of the s... | 164. Our statement follows from the fact that $E$ lies on the nine-point circle, and the nine-point circle is homothetic to the circumcircle with center $M$ and coefficient $-1 / 2$ (see problem 162). | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,971 |
166. Prove that the square of the distance between the centroid and the circumcenter of a triangle is equal to $R^{2}-\frac{a^{2}+b^{2}+c^{2}}{9}$. | 166. Use Leibniz's formula (problem 153), taking as $M$ the center of the circumscribed circle. | R^{2}-\frac{^{2}+b^{2}+^{2}}{9} | Geometry | proof | Yes | Yes | olympiads | false | 24,973 |
167. Prove that the square of the distance between the centroid of a triangle and the center of the inscribed circle is equal to $\frac{1}{9}\left(p^{2}+5 r^{2}-16 R r\right)$. | 167. Use Leibniz's formula (Problem 153), taking as \( M \) the center of the inscribed circle. To calculate, for example, \( |M A|^{2} \), drop a perpendicular \( M K \) to \( A B \); we have \( |M K|=r \), \( |A K|=p-a \), hence, \( |A M|^{2}=(p-a)^{2}+r^{2} \).
Similarly, \( |M B|^{2} \) and \( |M C|^{2} \) are cal... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,974 |
168. Let $d$ be the distance between the centers of the inscribed and circumscribed circles of a triangle. Prove that
$$
d^{2}=R^{2}-2 R r \quad(\text { Euler }) .
$$ | 168. Let $O$ be the center of the circumcircle of $\triangle ABC$, and $O_{1}$ be the center of the inscribed circle. $M$ is the intersection point of the angle bisector of $\angle B$ with the circumcircle (Fig. 49). Since the point $O_{1}$ is at a distance $d$ from the center $O$, then $\left|B O_{1}\right| \cdot \lef... | ^{2}=R^{2}-2Rr | Geometry | proof | Yes | Yes | olympiads | false | 24,975 |
169. Prove that the nine-point circle (see problem 162) is tangent to the incircle of the triangle (Feuerbach).
## § 3. Geometric Loci. Belonging of Points to Lines and Circles | 169. Let $G$ be the centroid of triangle $ABC$, $O$ the circumcenter, $I$ the incenter, and $O_{1}$ the center of the nine-point circle. $G$ lies on the segment $OO_{1}$, and $|OG|=2|GO_{1}|$. If $\widehat{OGI}=\varphi$, then
$$
\left\{\begin{aligned}
|O I_{1}|^{2} & =|OG|^{2}+|GI|^{2}-2|OG|\cdot|GI|\cos\varphi \\
|O_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,976 |
170. Given two points $A$ and $B$. Prove that the set of points $M$ such that $|A M|^{2}-|M B|^{2}=k$ (where $k$ is a given number), is a line perpendicular to $A B$. | 170. Prove that if $D$ is the projection of $M$ onto $A B$, then
$$
|A D|^{2}-|B D|^{2}=|A M|^{2}-|M B|^{2}.
$$ | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,977 |
172. Prove that for the perpendiculars dropped from points $A_{1}, B_{1}, C_{1}$ to the sides $B C, C A$, and $A B$ of triangle $A B C$ to intersect at one point, it is necessary and sufficient that
$\left|A_{1} B\right|^{2}-\left|B C_{1}\right|^{2}+\left|C_{1} A\right|^{2}-\left|A B_{1}\right|^{2}+\left|B_{1} C\right... | 172. If $M$ is the point of intersection of the perpendiculars dropped from $A_{1}$ and $B_{1}$ to $B C$ and $A C$, then (see problem 170)
$$
\begin{aligned}
& \left|M B^{2}-\right| M C^{2}=\left|A_{1} B^{2}-\right| A_{1} C^{2} \\
& \left.M C\right|^{2}-|M A|^{2}=\left.\left|B_{1} C^{2}-\right| B_{1} A\right|^{2}
\end... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,979 |
173. Prove that if the perpendiculars dropped from points $A_{1}, B_{1}, C_{1}$ to the lines $B C, C A$ and $A B$ respectively intersect at one point, then the perpendiculars dropped from points $A, B$ and $C$ to the lines $B_{1} C_{1}$, $C_{1} A_{1}$ and $A_{1} B_{1}$ also intersect at one point. | 173. From the result of problem 172, it follows that the condition for the perpendiculars dropped from $A_{1}, B_{1}, C_{1}$ to the sides $B C, C A$ and $A B$ to intersect at one point is the same as the condition for the perpendiculars dropped from $A, B$ and $C$ to $B_{1} C_{1}, C_{1} A_{1}$ and $A_{1} B_{1}$ to inte... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,980 |
176. Let $A_{1}, A_{2}, \ldots, A_{n}$ be fixed points, and $k_{1}$, $k_{2}, \ldots, k_{n}$ be given numbers. Then the set of points $M$ such that the sum $k_{1}\left|A_{1} M\right|^{2}+k_{2}\left|A_{2} M\right|^{2}+\ldots+k_{n}\left|A_{n} M\right|^{2}$ is constant, will be:
a) a circle, a point, or an empty set, if $... | 176. Let's introduce a rectangular coordinate system. If the coordinates of points $A_{1}, A_{2}, \ldots, A_{n}-\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{r}\right)$, and the point $M-(x, y)$, then the equation of the points of our set will have the form
$$
a x^{2}+a y^{2}+b x+c y+d=0
... | proof | Geometry | MCQ | Yes | Yes | olympiads | false | 24,983 |
177. Given a circle and a point $A$ outside it. Let the circle passing through $A$ touch the given one at an arbitrary point $B$, and the tangents to it, drawn from points $A$ and $B$, intersect at point $M$. Find the set of points $M$. | 177. If $B$ is the point of tangency, and $O$ is the center of the given circle, then
$$
\mid O M^{2}-A M^{2}=O M^{2}-B M^{2}=O B^{2}=R^{2}
$$
Thus, $M$ lies on the line perpendicular to $O A$ (see problem 170). | MliesonthelineperpendiculartoOA | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,984 |
178. Given two points $A$ and $B$. Find the set of points $M$ such that $\frac{|A M|}{|M B|}=k \neq 1$. | 178. The condition defining the set of points $M$ is equivalent to the condition $\left|A M^{2}-k^{2}\right| B M^{2}=0$, i.e., this is a circle (see problem 176). This circle is called the Apollonian circle; its center, as is easily verified, lies on the line $A B$. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,985 |
179. Three points $A, B$, and $C$ are located on the same line ($B$ is between $A$ and $C$). Take an arbitrary circle with center at $B$ and denote by $M$ the point of intersection of the tangents drawn from $A$ and $C$ to this circle. Find the set of points $M$ such that the points of tangency of $A M$ and $C M$ with ... | 179. Since $M B$ is the bisector of $\widehat{A M C}$, then $\frac{A M}{M C} = \frac{|A B|}{|B C|} \cdot$ Therefore, the bisector of the external angle relative to angle $A M C$ intersects the line $A C$ at a constant point $K$: $\frac{|A K|}{K C} = \frac{|A B|}{|B C|}$, and the set of points $M$ is an arc of a circle ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,986 |
180. Given two circles. Find the set of points $M$ such that the ratio of the lengths of the tangents drawn from $M$ to the given circles is a constant value $k$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 180. Let $O_{1}$ and $O_{2}$ be the centers of the given circles, $r_{1}$ and $r_{2}$ their radii, $M$ a point of the desired set, $M A_{1}$ and $M A_{2}$ the tangents. By the condition $M A_{1}=k \quad M A_{2}$. Therefore, $M O_{1}^{2}$ $-k^{2} M O_{2}^{2}=\left(M A_{1}^{2}+r_{1}^{2}\right)-k^{2}\left(M A_{2}^{2}+r_{2... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,987 | |
181. Let a line intersect one circle at points $A$ and $B$, and another circle at points $C$ and $D$. Prove that the points of intersection of the tangents to the first circle drawn at points $A$ and $B$ with the tangents drawn to the second circle at points $C$ and $D$ (considering the points where the tangents to dif... | 181. Let (Fig. 50) $K$ and $L$ be the points of intersection of the tangent to the second circle passing through $D$ with the tangents to the first circle passing through $B$ and $A$, and let $M$ and $N$ be the other two points. It is easy to see that $\widehat{D K B}=\widehat{C M A}$ (each of these angles is equal to ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,988 |
183. Given a triangle $ABC$. Consider all possible pairs of points $M_{1}$ and $M_{2}$ such that $\left|A M_{1}\right|:\left|B M_{1}\right|:\left|C M_{1}\right|=\left|A M_{2}\right|:\left|B M_{2}\right|:\left|C M_{2}\right|$. Prove that all lines $M_{1} M_{2}$ pass through a fixed point on the plane. | 183. Let $\left|A M_{1 \mid}:\right| B M_{1}\left|: C M_{1}\right|=p: q: r$. Then the set of points $M$ such that
$$
\left(r^{2}-q^{2}\right) A M{ }^{2}+\left(p^{2}-r^{2}\right) ;\left.B M\right|^{2}+\left(q^{2}-p^{2}\right) ; C M ;^{2}=0
$$
is a straight line passing through $M_{1}, M_{2}$ and the center of the circ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,990 |
185. Let $A_{1}, B_{1}, C_{1}$ be the feet of the perpendiculars dropped from the vertices $A, B$, and $C$ of triangle $ABC$ to the line $l$. Prove that the perpendiculars dropped from $A_{1}, B_{1}, C_{1}$ to $BC, CA$, and $AB$, respectively, intersect at one point. | 185. Let $\left|A A_{1}\right|=a,\left|B B_{1}\right|=b, \quad\left|C C_{1}\right|=c,\left|A_{1} B_{1}\right|=x$, $\left|B_{1} C_{1}\right|=y, \quad\left|C_{1} A_{1}\right|=z$. Then $\left|A B_{1}\right|^{2}=a^{2}+x^{2}, \mid B_{1} C^{2}=c^{2}+y^{2}$, $\left|C A_{1}{ }^{2}=c^{2}+z^{2},\right| A_{1} B=b^{2}+x^{2},\left|... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,992 |
186. Given an equilateral triangle $A B C$ and an arbitrary point $D$; $A_{1}, B_{1}$, and $C_{1}$ are the centers of the circles inscribed in triangles $B C D, A C D$, and $A B D$. Prove that the perpendiculars dropped from vertices $A, B$, and $C$ to $B_{1} C_{1}, C_{1} A_{1}$, and $A_{1} B_{1}$ respectively, interse... | 186. Let $|A D|=x,|B D|=y,|C D|=y,|A B|=a$. Denote by $A_{2}, B_{2}, C_{2}$ the points of tangency of the incircles of triangles $B C D, C A D, A B D$ with sides $B C, C A, A B$. The perpendiculars drawn through points $A_{1}, B_{1}, C_{1}$ to sides $B C, C A$, and $A B$ coincide with the perpendiculars erected to the ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,993 |
187. Given three pairwise intersecting circles. Prove that the three common chords of these circles pass through one point. | 187. Apply the condition proved in problem 173, taking points $A, B$ and $C$ as the centers of the circles, and points $A_{1}, B_{1}, C_{1}$ as one of the intersection points of the circles ( $A_{1}$ - one of the intersection points of the circles with centers $B$ and $C$ and so on.) | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,994 |
188. On the lines $A B$ and $A C$, points $M$ and $N$ are taken respectively. Prove that the common chord of the two circles with diameters $C M$ and $B N$ passes through the orthocenter of $\triangle A B C$. | 188. Let's take the third circle with diameter $B C$. The common chords of the 1st and 3rd, as well as the 2nd and 3rd circles, are the altitudes of the triangle dropped from vertices $B$ and $C$. Therefore (see problem 187), the common chord of these circles also passes through the orthocenter of triangle $A B C$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,995 |
189. On a plane, there is a circle and a point $N$. Let $A B$ be an arbitrary chord of the circle. Denote by $M$ the intersection point of the line $A B$ and the tangent at point $N$ to the circumcircle of $\triangle A B N$. Find the set of points $M$. | 189. Let $O$ be the center of the given circle, $R$ its radius, and $MC$ the tangent to it. We have: $|MO|^2 = |MN|^2 = |MO|^2 - |MB| \cdot |MA| = |MO|^2 - |MC|^2 = R^2$, i.e., the point $M$ lies on the line perpendicular to the line $ON$ (see problem 170). It is easy to show that all points of this line belong to our ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,996 |
190. Inside a circle, a point $A$ is taken. Find the set of intersection points of the tangents drawn to the circle at the ends of all possible chords passing through point $A$. | 190. Let $O$ be the center of the circle, $r$ the radius of the circle, $|OA| = a$, $BC$ a chord passing through $A$, and $M$ the point of intersection of the tangents. Then
$$
\left.\left|OM^{2}=\right| BM\right|^{2}+r^{2}
$$
$$
\left.\left|AM_{1}^{2}=\right| BM\right|^{2}-\frac{1}{4}|BC|^{2}+\left(\frac{1}{2}|BC|-|... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,997 |
191. Consider an arbitrary triangle $A B C$. Let $A_{1}, B_{1}, C_{1}$ be three points on the lines $B C, C A$, and $A B$ respectively. We introduce the following notations:
$$
\begin{aligned}
R & =\frac{\left|A C_{1}\right|}{\left|C_{1} B\right|} \cdot \frac{\left|B A_{1}\right|}{\left|A_{1} C\right|} \cdot \frac{\le... | 191. We have
$$
\frac{\left|A C_{1}\right|}{\left|C_{1} B\right|}=\frac{S_{A C C_{1}}}{S_{C C_{1} B}}=\frac{\frac{1}{2}|A C| \cdot\left|C C_{1}\right| \sin \widehat{A C C_{1}}}{\frac{1}{2}\left|C C_{1}\right| \cdot|C B| \sin \widehat{O_{1} C B}}=\frac{|A C|}{|B C|} \frac{\sin \widehat{A C C_{1}}}{\sin \widehat{C_{1} C... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,998 |
192. Ceva's Theorem. For the lines $A A_{1}, B B_{1}, C C_{1}$ to intersect at one point (or all three to be parallel), it is necessary and sufficient that $R=1$ (see problem 191) and at the same time, an odd number (i.e., one or all three) of the points $A_{1}, B_{1}, C_{1}$ lie on the sides of the triangle, and not o... | 192. Let's show that if the lines $A A_{1}, B B_{1}$, and $C C_{1}$ intersect at one point (denote it by $M$), then $R^{*}=1$ (and consequently, $R=1$; see problem 191). By the Law of Sines for $\triangle A M C$
$$
\frac{\sin \widehat{A C C_{1}}}{\sin \widehat{A_{1} A C}}=\frac{|A M|}{|M C|}
$$
Writing similar equali... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,999 |
193. Theorem of Menelaus. For points $A_{1}, B_{1}, C_{1}$ to lie on the same line, it is necessary and sufficient that $R=1$ (see problem 191) and at the same time, an even number (i.e., zero or two) of the three points $A_{1}, B_{1}, C_{1}$ lie on the sides of the triangle, and not on their extensions. | 193. Let $A_{1}, B_{1}, C_{1}$ lie on the same line. Draw a line through $C$ parallel to $A B$, and denote by $M$ the point of its intersection with the line $A_{1} B_{1}$. From the similarity of the corresponding triangles, we get $\frac{\left|B A_{1}\right|}{\left|A_{1} C\right|}=\frac{\left|B C_{1}\right|}{\mid C M}... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,000 |
194. Prove that if three lines passing through the vertices of a triangle intersect at one point, then the lines symmetric to them with respect to the corresponding angle bisectors of the triangle also intersect at one point or are parallel. | 194. Check that if for the given lines $R^{*}=1$, then it will also be the case for the symmetric ones. At the same time, if a line passing through, for example, vertex $A$ intersects side $B C$, then the line symmetric to it with respect to the bisector of angle $A$ will also intersect side $B C$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,001 |
195. Let $O$ be an arbitrary point in the plane, $M$ and $N$ be the feet of the perpendiculars dropped from point $O$ to the internal and external angle bisectors of $\angle A$ of $\triangle ABC$, $P$ and $Q$ are similarly defined for $\angle B$, and $R$ and $T$ are defined for $\angle C$. Prove that the lines $MN$, $P... | 195. If $A_{0}, B_{0}, C_{0}$ are the midpoints of segments $A O, B O, C O$ respectively, then the constructed lines turn out to be symmetric to the lines $A_{0} O, B_{0} O, C_{0} O$ with respect to the angle bisectors of triangle $A_{0} B_{0} C_{0}$ (see problem 194). | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,002 |
196. Given a triangle $A B C$. On the radii of the inscribed circle, drawn to the points of tangency, points are taken that are at equal distances from its center; these points are connected to the opposite vertices. Prove that the three resulting lines intersect at one point. | 196. Let $K$ be a point on the radius perpendicular to side $AC$, and $L$ be on the radius perpendicular to side $AB$. Line $BK$ intersects $AC$ at $B_{1}$, and line $CL$ intersects $AB$ at point $C_{1}$. Draw a line through $K$ parallel to $AC$, and let $M$ and $N$ be its points of intersection with $AB$ and $BC$. Cle... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,003 |
197. For the diagonals $A D, B E$ and $C F$ of a hexagon $A B C D E F$ inscribed in a circle to intersect at one point, it is necessary and sufficient that the equality $|A B| \cdot|C D| \cdot|E F|=$ $=|B C| \cdot|D E| \cdot|F A|$ holds. | 197. Consider the triangle $A C E$, through the vertices of which the lines $A D, C F$ and $E B$ are drawn. The sines of the angles formed by these lines with the sides of the triangle $A C E$ are proportional to the chords on which they stand (for example, $\sin \widehat{C A D}=\frac{C D}{2 R}$, where $R$ is the radiu... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,004 |
198. Let two tangents $A M$ and $A N$ (where $M$ and $N$ are the points of tangency) and two secants be drawn from a point $A$ outside a circle, and let $P$ and $Q$ be the points of intersection of the circle with the first secant, and $K$ and $L$ be the points of intersection with the second. Prove that the lines $P K... | 198. Note (Fig. $51, a$ ), that $\triangle A P M$ is similar to $\triangle A M Q$, $\triangle A P L$ is similar to $\triangle A K Q$, $\triangle A K N$ is similar to $\triangle A L N$; from these similarities we obtain
$$
\frac{P M}{|M Q|}=\frac{A M}{|\overline{A Q}|}, \quad \frac{Q K}{|P L|}=\frac{|A Q|}{|A L|}, \qua... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,005 |
200. A circle intersects side $A B$ of triangle $A B C$ at points $C_{1}$ and $C_{2}$, side $C A$ at points $B_{1}$ and $B_{2}$, and side $B C$ at points $A_{1}$ and $A_{2}$. Prove that if the lines $A A_{1}, B B_{1}$, and $C C_{1}$ intersect at one point, then the lines $A A_{2}, B B_{2}$, and $C C_{2}$ also intersect... | 200. By the property of secants drawn from an external point to a circle, or by the property of segments of chords of a circle passing through one point, we have \( \left|B C_{1}\right| \cdot \left|B C_{2}\right| = \left|B A_{1}\right| \cdot \left|B A_{2}\right| \), \( \left|C B_{1}\right| \cdot \left|C B_{2}\right| = ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,007 |
201. On the sides $AB, BC$, and $CA$ of triangle $ABC$, points $C_{1}, A_{1}$, and $B_{1}$ are taken. Let $C_{2}$ be the point of intersection of the lines $AB$ and $A_{1} B_{1}, A_{2}$ be the point of intersection of the lines $BC$ and $B_{1} C_{1}, B_{2}$ be the point of intersection of the lines $AC$ and $A_{1} C_{1... | 201. Writing the equality $R=1$ (according to Ceva's and Menelaus' theorems - see problems 192 and 193) for the points $A_{1}, B_{1}, C_{1} ; A_{1}, B_{1}, C_{2}$; $A_{2}, B_{1}, C_{1} ; A_{1}, B_{2}, C_{1}$, we obtain that for the points $A_{2}, B_{2}, C_{2}$ $R=1$ as well. Now it remains to prove that either all thre... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,008 |
203. Given a triangle $A B C$. Let's define a point $A_{1}$ on the side $B C$ as follows: $A_{1}$ is the midpoint of the side $K L$ of a regular pentagon $M K L N P$, where vertices $K$ and $L$ lie on $B C$, and vertices $M$ and $N$ lie on $A B$ and $A C$. Similarly, points $C_{1}$ and $B_{1}$ are defined on the sides ... | 203. If $a$ is the length of the side of pentagon $M K L N P$, $b$ is the length of the side of the pentagon with one side on $A B$, and $c$ is the length of the side of the pentagon, with one side on $A C$, then
$$
\frac{\left|B A_{1}\right|}{\mid C_{1} B}=\frac{a}{b}, \quad \frac{\left|A C_{1}\right|}{\left|A B_{1}\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,010 |
204. Through a fixed point $A$ inside a circle, two arbitrary chords $P Q$ and $K L$ are drawn. Find the set of points of intersection of the lines $P K$ and $Q L$. | 204. Use the results of problems 198 and 190. The obtained set coincides with the set of problem 190, i.e., this is the polar of point $A$ with respect to the given circle. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,011 |
205. $A, B$ and $C$ are three given points on a line. $D$ is an arbitrary point on the plane, not lying on this line. Draw lines through $C$, parallel to $A D$ and $B D$, until they intersect with the lines $B D$ and $A D$ at points $P$ and $Q$. Find the set of feet of the perpendiculars $M$, dropped from $C$ to $P Q$.... | 205. If $N$ is the intersection point of lines $P Q$ and $A B$, then $\frac{|C N|}{|A N|}=\frac{|P C|}{|A Q|}=\frac{|C B|}{|A C|}$, i.e., $N$ is a fixed point. The desired set is a circle with diameter $C N$.
If now $M$ is a fixed point, then $D$ lies on a line parallel to line $M N$ and passing through a fixed point ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,012 |
206. On the side $A C$ of triangle $A B C$, a point $K$ is taken, and on the median $B D$ - a point $P$ such that the area of triangle $A P K$ is equal to the area of triangle $B P C$. Find the set of points of intersection of the lines $A P$ and $B K$. | 206. Let (Fig. 52) $\varphi$ be the angle between $B D$ and $A C$; $S_{A P K}=\frac{1}{2}|A K| \cdot|P D| \sin \varphi, \quad S_{B P C}=-\frac{1}{2}|B P| \cdot|D C| \sin \varphi=$ $=\frac{1}{2}|B P| \cdot|A D| \sin \varphi$. Since $S_{A P K}=S_{B P C}$, then $|A K| \cdot|P D|=$ $=|B P| \cdot |A D|$ or $\frac{|A K|}{|A ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,013 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.