Let $ABC$ be a triangle. Let $x_1$ and $x_2$ be two congruent circles, which touch each other and the segment $BC$, and which both lie within triangle $ABC$, and for which it also holds that $x_1$ touches the segment $CA$, and that $x_2$ is the segment $AB$. Let $X$ be the contact point of these two circles $x_1$ and $x_2$. Let $y_1$ and $y_2$ two congruent circles that touch each other and the segment $CA$, and both within of triangle $ABC$, and for which it also holds that $y_1$ touches the segment $AB$, and that $y_2$ the segment $BC$. Let $Y$ be the contact point of these two circles $y_1$ and $y_2$. Let $z_1$ and $z_2$ be two congruent circles that touch each other and the segment $AB$, and both within triangle $ABC$, and for which it also holds that $z_1$ touches the segment $BC$, and that $z_2$ the segment $CA$. Let $Z$ be the contact point of these two circles $z_1$ and $z_2$. Prove that the straight lines $AX, BY$ and $CZ$ intersect at a point.

Three circles $C_1$, $C_2$, $C_3$ in the plane touch each other (in three different points). Connect the common point of $C_1$ and $C_2$ with the other two common points by straight lines. Show that these lines meet $C_3$ in diametrically opposite points.

On a line $\ell$ there exists $3$ points $A, B$, and $C$ where $B$ is located between $A$ and $C$. Let $\Gamma_1, \Gamma_2, \Gamma_3$ be circles with $AC, AB$, and $BC$ as diameter respectively; $BD$ is a segment, perpendicular to $\ell$ with $D$ on $\Gamma_1$. Circles $\Gamma_4, \Gamma_5, \Gamma_6$ and $\Gamma_7$ satisfies the following conditions: $\bullet$ $\Gamma_4$ touches $\Gamma_1, \Gamma_2$, and$ BD$. $\bullet$ $\Gamma_5$ touches $\Gamma_1, \Gamma_3$, and $BD$. $\bullet$ $\Gamma_6$ touches $\Gamma_1$ internally, and touches $\Gamma_2$ and $\Gamma_3$ externally. $\bullet$ $\Gamma_7$ passes through $B$ and the tangent points of $\Gamma_2$ with $\Gamma_6$, and $\Gamma_3$ with $\Gamma_6$. Show that the circles $\Gamma_4, \Gamma_5$, and $\Gamma_7$ are congruent.

Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that $\omega$ is also tangent to the circumcircle of the triangle $BOC$.

Two circles $c_1$ and $c_2$ with centres $O_1$ and $O_2$, respectively, are touching externally at $P$. On their common tangent at $P$, point $A$ is chosen, rays drawn from which touch the circles $c_1$ and $c_2$ at points $P_1$ and $P_2$ both different from $P$. It is known that $\angle P_1AP_2 = 120^o$ and angles $P_1AP$ and $P_2AP$ are both acute. Rays $AP_1$ and $AP_2$ intersect line $O_1O_2$ at points $G_1$ and $G_2$, respectively. The second intersection between ray $AO_1$ and $c_1$ is $H_1$, the second intersection between ray $AO_2$ and $c_2$ is $H_2$. Lines $G_1H_1$ and $AP$ intersect at $K$. Prove that if $G_1K$ is a tangent to circle $c_1$, then line $G_2A$ is tangent to circle $c_2$ with tangency point $H_2$.

Altitudes $BE$ and $CF$ of an acute-angled triangle $ABC$ meet at point $H$. The perpendicular from $H$ to $EF$ meets the line $\ell$ passing through $A$ and parallel to $BC$ at point $P$. The bisectors of two angles between $\ell$ and $HP$ meet $BC$ at points $S$ and $T$. Prove that the circumcircles of triangles $ABC$ and $PST$ are tangent.

Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order. Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other. Proposed by Ukraine

Let $ABCD$ be a cyclic quadrilateral. The point $P$ is chosen on the line $AB$ such that the circle passing through $C,D$ and $P$ touches the line $AB$. Similarly, the point $Q$ is chosen on the line $CD$ such that the circle passing through $A,B$ and $Q$ touches the line $CD$. Prove that the distance between $P$ and the line $CD$ equals the distance between $Q$ and $AB$.

Let $\Gamma$ and $\Gamma'$ be two circles with centres $O$ and $O'$, such that $O$ belongs to $\Gamma'$. Let $M$ be a point on $\Gamma'$, outside of $\Gamma$. The tangents to $\Gamma$ that go through $M$ touch $\Gamma$ in two points $A$ and $B$, and cross $\Gamma'$ again in two points $C$ and $D$. Finally, let $E$ be the crossing point of the lines $AB$ and $CD$. Prove that the circumcircles of the triangles $CEO'$ and $DEO'$ are tangent to $\Gamma'$.

It is known that a circle can be inscribed in a trapezium $ABCD$. Prove that the two circles, constructed on its oblique sides as diameters, touch each other. (D. Fomin, Leningrad)

Circles $A,B,C,D$ are given on the plane such that circles $A$ and $B$ are externally tangent at $P, B$ and $C$ at $Q, C$ and $D$ at $R$, and $D$ and $A$ at $S$. Circles $A$ and $C$ do not meet, and so do not $B$ and $D$. (a) Prove that the points $P,Q,R,S$ lie on a circle. (b) Suppose that $A$ and $C$ have radius $2, B$ and $D$ have radius $3$, and the distance between the centers of $A$ and $C$ is $6$. Compute the area of the quadrilateral $PQRS$.

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 $ABCD$ a quadrilateral inscribed in a circumference, $l_1$ the parallel to $BC$ through $A$, and $l_2$ the parallel to $AD$ through $B$. The line $DC$ intersects $l_1$ and $l_2$ at $E$ and $F$, respectively. The perpendicular to $l_1$ through $A$ intersects $BC$ at $P$, and the perpendicular to $l_2$ through $B$ cuts $AD$ at $Q$. Let $\Gamma_1$ and $\Gamma_2$ be the circumferences that pass through the vertex of triangles $ADE$ and $BFC$, respectively. Prove that $\Gamma_1$ and $\Gamma_2$ are tangent to each other if and only if $DP$ is perpendicular to $CQ$.

On a line $\ell$ there exists $3$ points $A, B$, and $C$ where $B$ is located between $A$ and $C$. Let $\Gamma_1, \Gamma_2, \Gamma_3$ be circles with $AC, AB$, and $BC$ as diameter respectively; $BD$ is a segment, perpendicular to $\ell$ with $D$ on $\Gamma_1$. Circles $\Gamma_4, \Gamma_5, \Gamma_6$ and $\Gamma_7$ satisfies the following conditions: $\bullet$ $\Gamma_4$ touches $\Gamma_1, \Gamma_2$, and$ BD$. $\bullet$ $\Gamma_5$ touches $\Gamma_1, \Gamma_3$, and $BD$. $\bullet$ $\Gamma_6$ touches $\Gamma_1$ internally, and touches $\Gamma_2$ and $\Gamma_3$ externally. $\bullet$ $\Gamma_7$ passes through $B$ and the tangent points of $\Gamma_2$ with $\Gamma_6$, and $\Gamma_3$ with $\Gamma_6$. Show that the circles $\Gamma_4, \Gamma_5$, and $\Gamma_7$ are congruent.

Given two circles with the radiuses $R$ and $r$, touching each other from the outer side. Consider all the trapezoids, such that its lateral sides touch both circles, and its bases touch different circles. Find the shortest possible lateral side.

Let $O$, $I$ be the circumcenter and the incenter of triangle $ABC$, respectively, and let the incircle tangents $BC$ at $D$. Furthermore, suppose that $H$ is the orthocenter of triangle $BIC$, $N$ is the midpoint of the arc $BAC$, and $X$ is the intersection of $OI$ and $NH$. If $P$ is the reflection of $A$ with respect to $OI$, show that $\odot(IDP)$ and $\odot(IHX)$ are tangent to each other.