In the non-isosceles triangle $ABC$ consider the points $X$ on $[AB]$ and $Y$ on $[AC]$ such that $[BX]=[CY]$, $M$ and $N$ are the midpoints of the segments $[BC]$, respectively $[XY]$, and the straight lines $XY$ and $BC$ meet in $K$. Prove that the circumcircle of triangle $KMN$ contains a point, different from $M$ , which is independent of the position of the points $X$ and $Y$.

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)

A circle $\omega$ with center $O$ and radius $r$ is constructed. A point $P$ is chosen on the circumference $\omega$ and a point A is taken inside it, such that is outside the line that passes through $P$ and $O$. Point $B$ is constructed, the reflection of $A$ wrt $O$. and $P'$ is another point on the circumference such that the chord $PP'$ is perpendicular to $PA$. Let $Q$ be the point on the line $PP'$ that minimizes the sum of distances from $A$ to $Q$ and from $Q$ to $B$. Show that the value of the sum of the lengths $AQ+QB$ does not depend on the choice of points $P$ or $A$

A circle $\Gamma$ and a line $\ell$ which does not intersect $\Gamma$ are given. Suppose $P, Q,R, S$ are variable points on circle $\Gamma$ such that the points $A = PQ\cap RS$ and $B = PS \cap QR$ lie on $\ell$. Prove that the circle on $AB$ as a diameter passes through two fixed points.

Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$. Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.

A circle and a point $P$ inside it are given. Two arbitrary perpendicular rays starting at point $P$ intersect the circle at points $A$ and $B$. Point $X$ is the projection of point $P$ onto line $AB, Y$ is the intersection point of tangents to the circle drawn through points $A$ and $B$. Prove that all lines $XY$ pass through the same point. (A. Zaslavsky)

The points $A \ne B$ are given on the plane. The point $C$ moves along the plane in such a way that $\angle ACB = \alpha$ , where $\alpha$ is the fixed angle from the interval ($0^o, 180^o$). The circle inscribed in triangle $ABC$ has center the point $I$ and touches the sides $AB, BC, CA$ at points $D, E, F$ accordingly. Rays $AI$ and $BI$ intersect the line $EF$ at points $M$ and $N$, respectively. Show that: a) the segment $MN$ has a constant length, b) all circles circumscribed around triangle $DMN$ have a common point

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 $Q$.

Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.

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

Two not necessarily equal non-intersecting wooden disks, one gray and one black, are glued to a plane. An infinite angle with one gray side and one black side can be moved along the plane so that the disks remain outside the angle, while the colored sides of the angle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that it is possible to draw a ray in the angle, starting from the vertex of the angle and such that no matter how the angle is positioned, the ray passes through some fixed point of the plane. (Egor Bakaev, Ilya Bogdanov, Pavel Kozhevnikov, Vladimir Rastorguev) (Junior version [url=https://artofproblemsolving.com/community/c6h2094701p15140671]here[/url]) [hide=note]There was a mistake in the text of the problem 3, we publish here the correct version. The solutions were estimated according to the text published originally.[/hide]

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 $BC$.

Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.

Let $A$ be a point and let k be a circle through $A$. Let $B$ and $C$ be two more points on $k$. Let $X$ be the intersection of the bisector of $\angle ABC$ with $k$. Let $Y$ be the reflection of $A$ wrt point $X$, and $D$ the intersection of the straight line $YC$ with $k$. Prove that point $D$ is independent of the choice of $B$ and $C$ on the circle $k$.

The circle divides each side of an equilateral triangle into three equal parts. Prove that the sum of the squares of the distances from any point of this circle to the vertices of the triangle is constant.

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)

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 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

An acute triangle $ABC$ is given. Let us denote $X$ for each of its inner points $X_a, X_b, X_c$ its images in axial symmetries sequentially along the lines $BC, CA, AB$. Prove that all $X_aX_bX_c$ triangles have a common interior point. (Josef Tkadlec)

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 $P.$

Two circles $\omega_1$ and $\omega_2$, which have centers $O_1$ and $O_2$, respectively, intersect at $A$ and $B$. A line $\ell$ that passes through $B$ cuts to $\omega_1$ again at $C$ and cuts to $\omega_2$ again in $D$, so that points $C, B, D$ appear in that order. The tangents of $\omega_1$ and $\omega_2$ in $C$ and $D$, respectively, intersect in $E$. Line $AE$ intersects again to the circumscribed circumference of the triangle $AO_1O_2$ in $F$. Try that the length of the $EF$ segment is constant, that is, it does not depend on the choice of $\ell$.

In the circle $\omega$, we draw a chord $BC$, which is not a diameter. Point $A$ moves in a circle $\omega$. $H$ is the orthocenter triangle $ABC$. Prove that for any location of point $A$, a circle constructed on $AH$ as on diameter, touches two fixed circles $\omega_1$ and $\omega_2$. (Dmitry Prokopenko)

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 acute triangle and $D \in (BC) , E \in (AD)$ be mobile points. The circumcircle of triangle $CDE$ meets the median from $C$ of the triangle $ABC$ at $F$ Prove that the circumcenter of triangle $AEF$ lies on a fixed line.

From point $A$ to circle $\omega$ tangent $AD$ and arbitrary a secant intersecting a circle at points $B$ and $C$ (B lies between points $A$ and $C$). Prove that the circle passing through points $C$ and $D$ and touching the straight line $BD$, passes through a fixed point (other than $D$).