An angle with vertex $O$ and a point $A$ inside it are placed on a plane. Points $M$ and $N$ are chosen on different sides of the angle so that the angles $CAM$ and $CAN$ are equal. Prove that the straight line $MN$ always passes through a fixed point (or is always parallel to a fixed line). (S Tokarev)

The lines $a$ and $b$ are parallel and the point $A$ lies on $a$. One chooses one circle $\gamma$ through A tangent to $b$ at $B$. $a$ intersects $\gamma$ for the second time at $T$. The tangent line at $T$ of $\gamma$ is called $t$. Prove that independently of the choice of $\gamma$, there is a fixed point $P$ such that $BT$ passes through $P$. Prove that independently of the choice of $\gamma$, there is a fixed circle $\delta$ such that $t$ is tangent to $\delta$.

The angle and the point $K$ inside it are given on the plane. Prove that there is a point $M$ with the following property: if an arbitrary line passing through intersects the sides of the angle at points $A$ and $B$, then $MK$ is the bisector of the angle $AMB$.

Let $ABCD$ a tetrahedron and $M$ a variable point on the face $BCD$. The line perpendicular to $(BCD)$ in $M$ . intersects the planes$ (ABC)$, $(ACD)$, and $(ADB)$ in $M_1$, $M_2$, and $M_3$. Show that the sum $MM_1 + MM_2 + MM_3$ is constant if and only if the perpendicular dropped from $A$ to $(BCD)$ passes through the centroid of triangle $BCD$.

Let $K$ be the midpoint of a fixed line segment $AB$, two circles $(O)$ and $(O')$ with variable radius each such that the straight line $OO'$ is throughK and $K$ is inside $(O)$, the two circles meet at $A$ and $C$, center $O'$ is on the circumference of $(O)$ and $O$ is interior to $(O')$. Assume that $M$ is the midpoint of $AC, H$ the projection of $C$ onto the perpendicular bisector of segment $AB$. Let $I$ be a variable point on the arc $AC$ of circle $(O')$ that is inside $(O), I$ is not on the line $OO'$ . Let $J$ be the reflection of $I$ about $O$. The tangent of $(O')$ at $I$ meets $AC$ at $N$. Circle $(O'JN)$ meets $IJ$ at $P$, distinct from $J$, circle $(OMP)$ intersects $MI$ at $Q$ distinct from $M$. Prove that (a) the intersection of $PQ$ and $O'I$ is on the circumference of $(O)$. (b) there exist a location of $I$ such that the line segment $AI$ meets $(O)$ at $R$ and the straight line $BI$ meets $(O')$ at $S$, then the lines $AS$ and $KR$ meets at a point on the circumference of $(O)$. (c) the intersection $G$ of lines $KC$ and $HB$ moves on a fixed line. Lê Phúc Lữ

On two intersecting lines $\ell_1$ and $\ell_2$, segments $AB$ and $CD$ of a given length are selected, respectively. Prove that the volume of the tetrahedron $ABCD$ does not depend on the position of the segments $AB$ and $CD$ on the lines $\ell_1$ and $\ell_2$.

Take an arbitrary point $D$ on side $BC$ of triangle $ABC$ and draw a circle through point $D$ and the centers of the circles inscribed in triangles $ABD$ and $ACD$. Prove that all circles obtained for different points $D$ of side $BC$ have a common point.

A point $P$ is fixed inside the circle. $C$ is an arbitrary point of the circle, $AB$ is a chord passing through point $B$ and perpendicular to the segment $BC$. Points $X$ and $Y$ are projections of point $B$ onto lines $AC$ and $BC$. Prove that all line segments $XY$ are tangent to the same circle. (A. Zaslavsky)

In an equilateral triangle $ ABC $, point $ O $ is chosen and perpendiculars $ OM $, $ ON $, $ OP $ are dropped to the sides $ BC $, $ CA $, $ AB $, respectively. Prove that the sum of the segments $ AP $, $ BM $, $ CN $ does not depend on the position of point $ O $.

The circle circumscribed about an acute triangle $ABC$ and the vertex $C$ are fixed. Orthocenter $H$ moves in a circle with center at point $C$. Find the locus of the midpoints of the segments connecting the feet of altitudes drawn from vertices $A$ and $B$.

Let $ABC$ be a fixed triangle. Point $D$ is an arbitrary point on the side $BC$. Point $P$ is fixed on $AD$. The circumcircle of triangle $BPD$ meets $AB$ at $E$ distinct from $B$. Point $Q$ varies on $AP$. Let $BQ$ and $CQ$ meet the circumcircles of triangles $BPD, CPD$ respectively at $F,Z$ distinct from $B,C$. Prove that the circumcircle $EFZ$ is through a fixed point distinct from $E$ and this fixed point is on the circumcircle of triangle $CPD$. Kostas Vittas

Two circles, one inside the other, are given in the plane. Construct a point $O$, inside the inner circle, such that if a ray from $O$ cuts the circles at $A$ and $B$ respectively, then the ratio $OA/OB$ is constant. (Folklore)

Let $BC$ be a chord not passing through the center of a circle $\omega$. Point $A$ varies on the major arc $BC$. Let $E$ and $F$ be the projection of $B$ onto $AC$, and of $C$ onto $AB$ respectively. The tangents to the circumcircle of $\vartriangle AEF$ at $E, F$ intersect at $P$. (a) Prove that $P$ is independent of the choice of $A$. (b) Let $H$ be the orthocenter of $\vartriangle ABC$, and let $T$ be the intersection of $EF$ and $BC$. Prove that $TH \perp AP$.

Two equal non-intersecting wooden disks, one gray and one black, are glued to a plane. A triangle with one gray side and one black side can be moved along the plane so that the disks remain outside the triangle, while the colored sides of the triangle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that the line that contains the bisector of the angle between the gray and black sides always passes through some fixed point of the plane. (Egor Bakaev, Pavel Kozhevnikov, Vladimir Rastorguev) (Senior version[url=https://artofproblemsolving.com/community/c6h2102856p15209040] here[/url])

Points $A, B$ move with equal speeds along two equal circles. Prove that the perpendicular bisector of $AB$ passes through a fixed point.

Draw a line passing through a point $M$ on the angle bisector of the angle $\angle AOB$, that intersects $OA$ and $OB$ at points $K$ and $L$ respectively. Prove that the valus of the sum $\frac{1}{|OK|}+\frac{1}{|OL|}$ does not depend on the choice of the straight line passing through $M$, i.e. is defined by the size of the angle AOB and the selection of the point $M$ only.

An arbitrary point $P$ is chosen on the lateral side $AB$ of the trapezoid $ABCD$. Straight lines passing through it parallel to the diagonals of the trapezoid intersect the bases at points $Q$ and $R$. Prove that the sides $QR$ of all possible triangles $PQR$ pass through a fixed point.

The trapezoid $ABCD$, of bases $AB$ and $CD$, is inscribed in a circumference $\Gamma$. Let $X$ a variable point of the arc $AB$ of $\Gamma$ that does not contain $C$ or $D$. We denote $Y$ to the point of intersection of $AB$ and $DX$, and let Z be the point of the segment $CX$ such that $\frac{XZ}{XC}=\frac{AY}{AB}$ . Prove that the measure of $\angle AZX$ does not depend on the choice of $X.$

Let $ABC$ be an equilateral triangle such that length of its altitude is $1$. Circle with center on the same side of line $AB$ as point $C$ and radius $1$ touches side $AB$. Circle rolls on the side $AB$. While the circle is rolling, it constantly intersects sides $AC$ and $BC$. Prove that length of an arc of the circle, which lies inside the triangle, is constant

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 $Q$.

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 $CD$.

Prove that the sum of the squares of the right-angled projections of the sides of a triangle onto the line $ p $ of the plane of this triangle does not depend on the position of the line $ p $ if and only if it the triangle is equilateral.

Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.

Points $A ,B ,C ,D$ lie on a line in this order. Draw parallel lines $a$ and $b$ through $A$ and $B$, respectively, and parallel lines $c$ and $d$ through $C$ and $D$, respectively, such that their points of intersection are vertices of a square. Prove that the side length of this square does not depend on the length of segment $BC$.

Let the point $D$ lie on the arc $AC$ of the circumcircle of the triangle $ABC$ ($AB < BC$), which does not contain the point $B$. On the side $AC$ are selected an arbitrary point $X$ and a point $X'$ for which $\angle ABX= \angle CBX'$. Prove that regardless of the choice of the point $X$, the circle circumscribed around $\vartriangle DXX'$, passes through a fixed point, which is different from point $D$. (Nikolaev Arseniy)