Found problems: 114
Given a circle and a chord $AB$, different from the diameter. Point $C$ moves along the large arc $AB$. The circle passing through passing through points $A, C$ and point $H$ of intersection of altitudes of of the triangle $ABC$, re-intersects the line $BC$ at point $P$. Prove that line $PH$ passes through a fixed point independent of the position of point $C$.
Circles ${w}_{1},{w}_{2}$ intersect at points ${{A}_{1}} $ and ${{A}_{2}} $. Let $B$ be an arbitrary point on the circle ${{w}_{1}}$, and line $B{{A}_{2}}$ intersects circle ${{w}_{2}}$ at point $C$. Let $H$ be the orthocenter of $\Delta B{{A}_{1}}C$. Prove that for arbitrary choice of point $B$, the point $H$ lies on a certain fixed circle.
Given a circle $k$ with center $M$ and radius $r$, let $AB$ be a fixed diameter of $k$ and let $K$ be a fixed point on the segment $AM$. Denote by $t$ the tangent of $k$ at $A$. For any chord $CD$ through $K$ other than $AB$, denote by $P$ and Q the intersection points of $BC$ and $BD$ with $t$, respectively. Prove that $AP\cdot AQ$ does not depend on $CD$.
Two circles $\gamma_1, \gamma_2$ are given, with centers at points $O_1, O_2$ respectively. Select a point $K$ on circle $\gamma_2$ and construct two circles, one $\gamma_3$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $A$, and the other $\gamma_4$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $B$. Prove that, regardless of the choice of point K on circle $\gamma_2$, all lines $AB$ pass through a fixed point of the plane.
Let $ACBD$ be a asquare and $K,L,M,N$ be points of $AB,BC,CD,DA$ respectively. If $O$ is the center of the square , prove that the expression $$ \overrightarrow{OK}\cdot \overrightarrow{OL}+\overrightarrow{OL}\cdot\overrightarrow{OM}+\overrightarrow{OM}\cdot\overrightarrow{ON}+\overrightarrow{ON}\cdot\overrightarrow{OK}$$
is independent of positions of $K,L,M,N$, (i.e. is constant )
Let $C_1$ and $C_2$ be tangent circles internally at point $A$, with $C_2$ inside of $C_1$. Let $BC$ be a chord of $C_1$ that is tangent to $C_2$. Prove that the ratio between the length $BC$ and the perimeter of the triangle $ABC$ is constant, that is, it does not depend of the selection of the chord $BC$ that is chosen to construct the trangle.
For triangle $ABC, P$ and $Q$ satisfy $\angle BPA + \angle AQC = 90^o$. It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise (or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ($A \ne N$, however if $A$ is the only intersection $A = N$), and the midpoint of segment $BC$ be $M$. Show that the length of $MN$ does not depend on $P$ and $Q$.
Find all nonconvex quadrilaterals in which the sum of the distances to the lines containing the sides is the same for any interior point. Try to generalize the result in the case of an arbitrary non-convex polygon, polyhedron.
A plane intersects a sphere in a circle $C$. The points $A$ and $B$ lie on the sphere on opposite sides of the plane. The line joining $A$ to the center of the sphere is normal to the plane. Another plane $p$ intersects the segment $AB$ and meets $C$ at $P$ and $Q$. Show that $BP\cdot BQ$ is independent of the choice of $p$.
Fix a triangle $ABC$. The variable point $M$ in its interior is such that $\angle MAC = \angle MBC$ and $N$ is the reflection of $M$ with respect to the midpoint of $AB$. Prove that $|AM| \cdot |BM| + |CM| \cdot |CN|$ is independent of the choice of $M$.
A point $D$ is chosen in the interior of the side $BC$ of an acute triangle $ABC$, and another point $P$ in the interior of the segment $AD$, but not lying on the median through $C$. This median (through $C$) intersects the circumcircle of a triangle $CPD$ at $K(\ne C)$. Prove that the circumcircle of triangle $AKP$ always passes through a fixed point $M(\ne A)$ independent of the choices of the points $D$ and $P.$
Let $ABC$ be a triangle with two isogonal points $ P$ and $Q$ . Let $D, E$ be the projection of $P$ on $AB$, $AC$. $G$ is the projection of $Q$ on $BC$. $U$ is the projection of $G$ on $DE$, $ L$ is the projection of $P$ on $AQ$, $K$ is the symmetric of $L$ wrt $UG$.
Prove that $UK$ passes through a fixed point when $P$ and $Q$ vary.
The circle $\omega_0$ touches the line at point A. Let $R$ be a given positive number. We consider various circles $\omega$ of radius $R$ that touch a line $\ell$ and have two different points in common with the circle $\omega_0$. Let $D$ be the touchpoint of the circle $\omega_0$ with the line $\ell$, and the points of intersection of the circles $\omega$ and $\omega_0$ are denoted by $B$ and $C$ (Assume that the distance from point $B$ to the line $\ell$ is greater than the distance from point $C$ to this line). Find the locus of the centers of the circumscribed circles of all such triangles $ABD$.
Let $A$ be a point inside the acute angle $xOy$. An arbitrary circle $\omega$ passes through $O, A$, intersecting $Ox$ and $Oy$ at the second intersection $B$ and $C$, respectively. Let $M$ be the midpoint of $BC$. Prove that $M$ is always on a fixed line (when $\omega$ changes, but always goes through $O$ and $A$).