Let $ABC$ be a triangle such that $\measuredangle BAC = 60^{\circ}$. Let $D$ and $E$ be the feet of the perpendicular from $A$ to the bisectors of the external angles of $B$ and $C$ in triangle $ABC$, respectively. Let $O$ be the circumcenter of the triangle $ABC$. Prove that circumcircle of the triangle $BOC$ has exactly one point in common with the circumcircle of $ADE$.

Let $ABC$ be a triangle with right angle $ACB$. Denote by $F$ the projection of $C$ on $AB$. A circle $\omega$ touches $FB$ at point $P$, touches $CF$ at point $Q$, and the circumcircle of $ABC$ at point $R$. Prove that the points $A, Q, R$ all lie on the same line and $AP=AC$.

The incircle of $\triangle ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. The projections of $B$, $C$ to $AD$ are $U$, $V$, respectively; the projections of $C$, $A$ to $BE$ are $W$, $X$, respectively; and the projections of $A$, $B$ to $CF$ are $Y$, $Z$, respectively. Show that the circumcircle of the triangle formed by $UX$, $VY$, $WZ$ is tangent to the incircle of $\triangle ABC$.

Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $\omega_0$ be a circle tangent to chord $AB$ and arc $ACB$. For each $i = 1, 2$, let $\omega_i$ be a circle tangent to $AB$ at $T_i$ , to $\omega_0$ at $S_i$ , and to arc $ACB$. Suppose $\omega_1 \ne \omega_2$. Prove that there is a circle passing through $S_1, S_2, T_1$, and $T_2$, and tangent to $\Gamma$ if and only if $\angle ACB = 90^o$. .

Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ are tangent to each other externally at point $O$. Points $X$ and $Y$ on $k_1$ and $k_2$ respectively are such that rays $O_1X$ and $O_2Y$ are parallel and codirectional. Prove that two tangents from $X$ to $k_2$ and two tangents from $Y$ to $k_1$ touch the same circle passing through $O$. (V. Yasinsky)

The perpendicular bisector of the bisector $BL$ of the triangle $ABC$ intersects the bisectors of its external angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the circle circumscribed around the triangle $PBQ$ is tangent to the circle circumscribed around the triangle $ABC$.

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[ \angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.

Let $\omega$ be the circumcircle of the triangle $ABC$ with $\angle B = 3\angle C$. The internal angle bisector of $\angle A$, intersects $\omega$ and $BC$ at $M$ and $D$, respectively. Point $E$ lies on the extension of the line $MC$ from $M$ such that $ME$ is equal to the radius of $\omega$. Prove that circumcircles of triangles $ACE$ and $BDM$ are tangent. [i]Proposed by Mehran Talaei - Iran[/i]

Let $C (O)$ be a circle (with center $O$ ) and $A, B$ points on the circle with $\angle AOB = 90^o$. Circles $C_1 (O_1)$ and $C_2 (O_2)$ are tangent internally with circle $C$ at $A$ and $B$, respectively, and, also, are tangent to each other. Consider another circle $C_3 (O_3)$ tangent externally to the circles $C_1, C_2$ and tangent internally to circle $C$, located inside angle $\angle AOB$. Show that the points $O, O_1, O_2, O_3$ are the vertices of a rectangle.

In a circumference $\Gamma$ a chord $PQ$ is considered such that the segment that joins the midpoint of the smallest arc $PQ$ and the midpoint of the segment $PQ$ measures $1$. Let $\Gamma_1, \Gamma_2$ and $\Gamma_3$ be three tangent circumferences to the chord $PQ$ that are in the same half plane than the center of $\Gamma$ with respect to the line $PQ$. Furthermore, $\Gamma_1$ and $\Gamma_3$ are internally tangent to $\Gamma$ and externally tangent to$ \Gamma_2$, and the centers of $\Gamma_1$ and $\Gamma_3$ are on different halfplanes with respect to the line determined by the centers of $\Gamma$ and $\Gamma_2$. If the sum of the radii of $\Gamma_1, \Gamma_2$ and $\Gamma_3$ is equal to the radius of $\Gamma$, calculate the radius of $\Gamma_2$.

Given $\vartriangle ABC$ with circumcircle $\Omega$. Assume $\omega_a, \omega_b, \omega_c$ are circles which tangent internally to $\Omega$ at $T_a,T_b, T_c $ and tangent to $BC,CA,AB$ at $P_a, P_b, P_c$, respectively. If $AT_a,BT_b,CT_c$ are collinear, prove that $AP_a,BP_b,CP_c$ are collinear.

In parallelogram $ABCD$, angle $A$ is acute. Let $X$ be a point, symmetrical to point $C$ wrt to straight line $AD$, $Y$ is a point symmetrical to the point $C$ wrt point $D$, and $M$ is the intersection point of $AC$ and $BD$. It turned out, that the circumcircles of triangles $BMC$ and $AXY$ are tangent internally. Prove that $AM = AB$.

There is circle $\omega$ and $A, B$ on it. Circle $\gamma_1$ tangent to $\omega$ on $T$ and $AB$ on $D$. Circle $\gamma_2$ tangent to $\omega$ on $S$ and $AB$ on $E$. (like the figure below) Let $AB\cap TS=C$. Prove that $CA=CB$ iff $CD=CE$

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.

Let two circles be externally tangent at point $C$, with parallel diameters $A_1A_2, B_1B_2$ (i.e. the quadrilateral $A_1B_1B_2A_2$ is a trapezoid with bases $A_1A_2$ and $B_1B_2$ or parallelogram). Circle with the center on the common internal tangent to these two circles, passes through the intersection point of lines $A_1B_2$ and $A_2B_1$ as well intersects those lines at points $M, N$. Prove that the line $MN$ is perpendicular to the parallel diameters $A_1A_2, B_1B_2$. (Yuri Biletsky)

Let $r_1,r_2,r$, with $r_1 < r_2 < r$, be the radii of three circles $\Gamma_1,\Gamma_2,\Gamma$, respectively. The circles $\Gamma_1,\Gamma_2$ are internally tangent to $\Gamma$ at two distinct points $A,B$ and intersect in two distinct points. Prove that the segment $AB$ contains an intersection point of $\Gamma_1$ and $\Gamma_2$ if and only if $r_1 +r_2 = r$.

Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ are two points on different circles so that $RA$ is perpendicular to $RB$. Show that the line $AB$ passes through $P$.

Let $ABC$ be an acute triangle, and its altitudes $AX,BY,CZ$ concurrent at $H$. Construct circles $(K_a), (K_b), (K_c)$ circumscribing the triangles $AY Z, BZX, CXY$ . Construct a circle $(K)$ that is internally tangent to all the three circles $(Ka), (K_b), (K_c)$. Prove that $(K)$ is tangent to the circumcircle $(O)$ of the triangle $ABC$. Đỗ Thanh Sơn

Let $ABC$ be a triangle and $P$ be an interior point. Let $\omega_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ internally and tangent to the circumcircle of $ABC$ at $A_1$ internally and let $\Gamma_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ externally and tangent to the circumcircle of $ABC$ at $A_2$ internally. Define $B_1$, $B_2$, $C_1$, $C_2$ analogously. Let $O$ be the circumcentre of $ABC$. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ and $OP$ are concurrent.

Two circles $C_1$ and $C_2$ touch each other externally in a point $P$. At point $C_1$ there is a point $Q$ such that the tangent line in $Q$ at $C_1$ intersects the circle $C_2$ at points $A$ and $B$. The line $QP$ still intersects $C_2$ at point $C$. Prove that triangle $ABC$ is isosceles.

Given a convex quadrilateral $ABCD$.Let $\ell_A,\ell_B,\ell_C,\ell_D$ be exterior angle bisectors of quadrilateral $ABCD$. Let $\ell_A \cap \ell_B=K,\ell_B \cap \ell_C=L,\ell_C \cap \ell_D=M,\ell_D \cap \ell_A=N$.Prove that if circumcircles of triangles $ABK$ and $CDM$ be externally tangent to each other then circumcircles of the triangles $BCL$ and $DAN$ are externally tangent to each other.(L.Emelyanov)

Circle $k_1$ has radius $10$, externally touching circle $k_2$ with radius $18$. Circle $k_3$ touches both circles, as well as the line $e$ determined by their centres. Let $k_4$ be the circle touching $k_2$ and $k_3$ externally (other than $k_1$) whose center lies on line $e$. What is the radius of $k_4$?

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 $PQRS$.

Given are four circles $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$. Circles $\Gamma_1$ and $\Gamma_2$ are externally tangent at point $A$. Circles $\Gamma_2$ and $\Gamma_3$ are externally tangent at point $B$. Circles $\Gamma_3$ and $\Gamma_4$ are externally tangent at point $C$. Circles $\Gamma_4$ and $\Gamma_1$ are externally tangent at point $D$. Prove that $ABCD$ is cyclic.

Let $A_0$ be the midpoint of side $BC$ of triangle $ABC$, and $A'$ be the point of tangency with this side of the inscribed circle. Let's construct a circle $ \omega$ with center at $A_0$ and passing through $A'$. On other sides we will construct similar circles. Prove that if $ \omega$ is tangent to the cirucmscribed circle on arc $BC$ not containing $A$, then another one of the constructed circles touches the circumcircle.