Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

The circle, which is tangent to the circumcircle of isosceles triangle $ABC$ ($AB=AC$), is tangent $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that the midpoint $I$ of the segment $PQ$ is the center of the excircle (which is tangent to $BC$) of the triangle .

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$, let $T$ be the circumcenter of the triangle $AOC$, and let $M$ be the midpoint of the segment $AC$. We take a point $D$ on the side $AB$ and a point $E$ on the side $BC$ that satisfy $\angle BDM = \angle BEM = \angle ABC$. Show that the straight lines $BT$ and $DE$ are perpendicular.

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

In $\triangle ABC$ we have $BC>CA>AB$. The nine point circle is tangent to the incircle, $A$-excircle, $B$-excircle and $C$-excircle at the points $T,T_A,T_B,T_C$ respectively. Prove that the segments $TT_B$ and lines $T_AT_C$ intersect each other.

We're given two congruent, equilateral triangles $ABC$ and $PQR$ with parallel sides, but one has one vertex pointing up and the other one has the vertex pointing down. One is placed above the other so that the area of intersection is a hexagon $A_1A_2A_3A_4A_5A_6$ (labelled counterclockwise). Prove that $A_1A_4$, $A_2A_5$ and $A_3A_6$ are concurrent.

Let $ABCD$ be a cyclic quadrilateral and let $H_{A}, H_{B}, H_{C}, H_{D}$ be the orthocenters of the triangles $BCD$, $CDA$, $DAB$ and $ABC$ respectively. Show that the quadrilaterals $ABCD$ and $H_{A}H_{B}H_{C}H_{D}$ are congruent.

Prove that in triangle $ABC$, radical center of its excircles lies on line $GI$, which $G$ is Centroid of triangle $ABC$, and $I$ is the incenter.

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

One of the external common tangent lines of the two externally tangent circles with center $O_1$ and $O_2$ touches the circles at $B$ and $C$, respectively. Let $A$ be the common point of the circles. The line $BA$ meets the circle with center $O_2$ at $A$ and $D$. If $|BA|=5$ and $|AD|=4$, then what is $|CD|$? $ \textbf{(A)}\ \sqrt{20} \qquad\textbf{(B)}\ \sqrt{27} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ \frac{15}2 \qquad\textbf{(E)}\ 4\sqrt5 $

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

The point $E$ is the midpoint of the segment connecting the orthocentre of the scalene triangle $ABC$ and the point $A$. The incircle of triangle $ABC$ incircle is tangent to $AB$ and $AC$ at points $C'$ and $B'$ respectively. Prove that point $F$, the point symmetric to point $E$ with respect to line $B'C'$, lies on the line that passes through both the circumcentre and the incentre of triangle $ABC$.

Given a triangle $ ABC$. A circle passes through vertices $ B$ and $ C$ and intersects sides $ AB$ and $ AC$ at points $ D$ and $ E$, respectively. Segments $ CD$ and $ BE$ intersect at point $ O$. Denote the incenters of triangles $ ADE$ and $ ODE$ by $ M$ and $ N$, respectiely. Prove that the midpoint of the smaller arc $ DE$ lies on line $ MN$.

Let $\omega$ be incircle of $ABC$. $P$ and $Q$ are on $AB$ and $AC$, such that $PQ$ is parallel to $BC$ and is tangent to $\omega$. $AB,AC$ touch $\omega$ at $F,E$. Prove that if $M$ is midpoint of $PQ$, and $T$ is intersection point of $EF$ and $BC$, then $TM$ is tangent to $\omega$. [i]By Ali Khezeli[/i]

Let $ A^{\prime}$ be an arbitrary point on the side $ BC$ of a triangle $ ABC$. Denote by $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ the circles simultanously tangent to $ AA^{\prime}$, $ A^{\prime}B$, $ \Gamma$ and $ AA^{\prime}$, $ A^{\prime}C$, $ \Gamma$, respectively, where $ \Gamma$ is the circumcircle of $ ABC$. Prove that $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ are congruent if and only if $ AA^{\prime}$ passes through the Nagel point of triangle $ ABC$. ([i]If $ M,N,P$ are the points of tangency of the excircles of the triangle $ ABC$ with the sides of the triangle $ BC$, $ CA$ and $ AB$ respectively, then the Nagel point of the triangle is the intersection point of the lines $ AM$, $ BN$ and $ CP$[/i].)

Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ touch each outside at point $Q$. The other endpoints of the diameters through $Q$ are $P$ on $k_1$ and $R$ on $k_2$. We choose two points $A$ and $B$, one on each of the arcs $PQ$ of $k_1$. ($PBQA$ is a convex quadrangle.) Further, let $C$ be the second point of intersection of the line $AQ$ with $k_2$ and let $D$ be the second point of intersection of the line $BQ$ with $k_2$. The lines $PB$ and $RC$ intersect in $U$ and the lines $PA$ and $RD$ intersect in $V$ . Show that there is a point $Z$ that lies on all of these lines $UV$.

Consider an isosceles triangle $ABC$ with $AB=AC$, and a circle $\omega$ which is tangent to the sides $AB$ and $AC$ of this triangle and intersects the side $BC$ at the points $K$ and $L$. The segment $AK$ intersects the circle $\omega$ at a point $M$ (apart from $K$). Let $P$ and $Q$ be the reflections of the point $K$ in the points $B$ and $C$, respectively. Show that the circumcircle of triangle $PMQ$ is tangent to the circle $\omega$.

Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$. The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$. Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$, $A$ and $B$. Line $BM$ cuts $\Gamma_2$ at $C$, line $CA$ cuts $\Gamma_1$ at $D$, segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$, with $E$ in segment $PF$. Prove lines $AF$, $BP$, and $CE$ are concurrent.

The circle $\omega$ with center $O$ has given. From an arbitrary point $T$ outside of $\omega$ draw tangents $TB$ and $TC$ to it. $K$ and $H$ are on $TB$ and $TC$ respectively. [b]a)[/b] $B'$ and $C'$ are the second intersection point of $OB$ and $OC$ with $\omega$ respectively. $K'$ and $H'$ are on angle bisectors of $\angle BCO$ and $\angle CBO$ respectively such that $KK' \bot BC$ and $HH'\bot BC$. Prove that $K,H',B'$ are collinear if and only if $H,K',C'$ are collinear. [b]b)[/b] Consider there exist two circle in $TBC$ such that they are tangent two each other at $J$ and both of them are tangent to $\omega$.and one of them is tangent to $TB$ at $K$ and other one is tangent to $TC$ at $H$. Prove that two quadrilateral $BKJI$ and $CHJI$ are cyclic ($I$ is incenter of triangle $OBC$).

Two circles are externally tangent at the point $A$. A common tangent of the circles meets one circle at the point $B$ and another at the point $C$ ($B \ne C)$. Line segments $BD$ and $CE$ are diameters of the circles. Prove that the points $D, A$ and $C$ are collinear.

Let $ABC$ be a triangle, $D$ be a point on side $BC$, and let $\mathcal{O}$ be the circumcircle of triangle $ABC$. Show that the circles tangent to $\mathcal{O},AD,BD$ and to $\mathcal{O},AD,DC$ are tangent to each other if and only if $\angle BAD=\angle CAD$. [i]Dan Branzei[/i]

Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$

The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear. [i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]

Suppose $ABCD$ is a cyclic quadrilateral with $BC = CD$. Let $\omega$ be the circle with center $C$ tangential to the side $BD$. Let $I$ be the centre of the incircle of triangle $ABD$. Prove that the straight line passing through $I$, which is parallel to $AB$, touches the circle $\omega$.