A point $ P$ inside a circle is such that there are three chords of the same length passing through $ P$. Prove that $ P$ is the center of the circle.

Let $ABCD$ be a parallelogram and $H$ the Orthocentre of $\triangle{ABC}$. The line parallel to $AB$ through $H$ intersects $BC$ at $P$ and $AD$ at $Q$ while the line parallel to $BC$ through $H$ intersects $AB$ at $R$ and $CD$ at $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic. [i](Swiss Mathematical Olympiad 2011, Final round, problem 8)[/i]

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

[b]Problem 1.[/b] Points $D$ and $E$ are chosen on the sides $AB$ and $AC$, respectively, of a triangle $\triangle ABC$ such that $DE\parallel BC$. The circumcircle $k$ of triangle $\triangle ADE$ intersects the lines $BE$ and $CD$ at the points $M$ and $N$ (different from $E$ and $D$). The lines $AM$ and $AN$ intersect the side $BC$ at points $P$ and $Q$ such that $BC=2\cdot PQ$ and the point $P$ lies between $B$ and $Q$. Prove that the circle $k$ passes through the point of intersection of the side $BC$ and the angle bisector of $\angle BAC$. [i]Nikolai Nikolov[/i]

Let $ABC$ be a triangle with $|AB|=|AC|$. $D$ is the midpoint of $[BC]$. $E$ is the foot of the altitude from $D$ to $AC$. $BE$ cuts the circumcircle of triangle $ABD$ at $B$ and $F$. $DE$ and $AF$ meet at $G$. Prove that $|DG|=|GE|$

Let $ k$ be a positive real number. Triangle ABC is acute and not isosceles, O is its circumcenter and AD,BE,CF are the internal bisectors. On the rays AD,BE,CF, respectively, let points L,M,N such that $ \frac {AL}{AD} \equal{} \frac {BM}{BE} \equal{} \frac {CN}{CF} \equal{} k$. Denote $ (O_1),(O_2),(O_3)$ be respectively the circle through L and touches OA at A, the circle through M and touches OB at B, the circle through N and touches OC at C. 1) Prove that when $ k \equal{} \frac{1}{2}$, three circles $ (O_1),(O_2),(O_3)$ have exactly two common points, the centroid G of triangle ABC lies on that common chord of these circles. 2) Find all values of k such that three circles $ (O_1),(O_2),(O_3)$ have exactly two common points

The circles $\mathcal C_1$ and $\mathcal C_2$ touch each other externally at $D$, and touch a circle $\omega$ internally at $B$ and $C$, respectively. Let $A$ be an intersection point of $\omega$ and the common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $D$. Lines $AB$ and $AC$ meet $\mathcal C_1$ and $\mathcal C_2$ again at $K$ and $L$, respectively, and the line $BC$ meets $\mathcal C_1$ again at $M$ and $\mathcal C_2$ again at $N$. Prove that the lines $AD$, $KM$, $LN$ are concurrent. [i]Greece[/i]

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.

Let $C$ and $D$ be two points on the semicricle with diameter $AB$ such that $B$ and $C$ are on distinct sides of the line $AD$. Denote by $M$, $N$ and $P$ the midpoints of $AC$, $BD$ and $CD$ respectively. Let $O_A$ and $O_B$ the circumcentres of the triangles $ACP$ and $BDP$. Show that the lines $O_AO_B$ and $MN$ are parallel.

A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$

Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.

Let $I_a$ be the $A$-excenter of $\Delta ABC$ and the $A$-excircle of $\Delta ABC$ be tangent to the lines $AB,AC$ at $B',C'$, respectively. $ I_aB,I_aC$ meet $B'C'$ at $P,Q$, respectively. $M$ is the meet point of $BQ,CP$. Prove that the length of the perpendicular from $M$ to $BC$ is equal to $r$ where $r$ is the radius of incircle of $\Delta ABC$.

Let $H$ be the orthocenter of $\triangle ABC$. The points $A_{1} \not= A$, $B_{1} \not= B$ and $C_{1} \not= C$ lie, respectively, on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$ and satisfy $A_{1}H=B_{1}H=C_{1}H$. Denote by $H_{1}$, $H_{2}$ and $H_{3}$ the orthocenters of $\triangle A_{1}BC$, $\triangle B_{1}CA$ and $\triangle C_{1}AB$, respectively. Prove that $\triangle A_{1}B_{1}C_{1}$ and $\triangle H_{1}H_{2}H_{3}$ have the same orthocenter.

Let $ABC$ be a triangle, let ${A}'$, ${B}'$, ${C}'$ be the orthogonal projections of the vertices $A$ ,$B$ ,$C$ on the lines $BC$, $CA$ and $AB$, respectively, and let $X$ be a point on the line $A{A}'$.Let $\gamma_{B}$ be the circle through $B$ and $X$, centred on the line $BC$, and let $\gamma_{C}$ be the circle through $C$ and $X$, centred on the line $BC$.The circle $\gamma_{B}$ meets the lines $AB$ and $B{B}'$ again at $M$ and ${M}'$, respectively, and the circle $\gamma_{C}$ meets the lines $AC$ and $C{C}'$ again at $N$ and ${N}'$, respectively.Show that the points $M$, ${M}'$, $N$ and ${N}'$ are collinear.

Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$ are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is equal to area of $O_1CO_2$, then find all possible values for $\angle CAA'$

In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If $AG=2$, $GF=13$, $FC=1$, and $HJ=7$, then $DE$ equals [asy] size(200); defaultpen(fontsize(10)); real r=sqrt(22); pair B=origin, A=16*dir(60), C=(16,0), D=(10-r,0), E=(10+r,0), F=C+1*dir(120), G=C+14*dir(120), H=13*dir(60), J=6*dir(60), O=circumcenter(G,H,J); dot(A^^B^^C^^D^^E^^F^^G^^H^^J); draw(Circle(O, abs(O-D))^^A--B--C--cycle, linewidth(0.7)); label("$A$", A, N); label("$B$", B, dir(210)); label("$C$", C, dir(330)); label("$D$", D, SW); label("$E$", E, SE); label("$F$", F, dir(170)); label("$G$", G, dir(250)); label("$H$", H, SE); label("$J$", J, dir(0)); label("2", A--G, dir(30)); label("13", F--G, dir(180+30)); label("1", F--C, dir(30)); label("7", H--J, dir(-30));[/asy] $\textbf {(A) } 2\sqrt{22} \qquad \textbf {(B) } 7\sqrt{3} \qquad \textbf {(C) } 9 \qquad \textbf {(D) } 10 \qquad \textbf {(E) } 13$

Let $ABCD$ be a cyclic quadrilateral with center $O$. Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$. Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$. Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$. Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$. [i]Proposed by Yang Liu[/i]

The quadrilateral $ABCD$ is circumscribed around a circle $k$ with center $I$ and $DA\cap CB=E$, $AB\cap DC=F$. In $\Delta EAF$ and $\Delta ECF$ are inscribed circles $k_1 (I_1,r_1)$ and $k_2 (I_2,r_2)$ respectively. Prove that the middle point $M$ of $AC$ lies on the radical axis of $k_1$ and $k_2$.

$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$. Let $AB \cap CD = E$, $AD \cap BC = F$. Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$, respectively. Let $\omega \cap \omega_1 = G$, $\omega \cap \omega_2 = H$. Show that $AC, BD, GH$ are concurrent. [i]Proposed by Yang Liu[/i]

Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.

The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.

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 $\omega$.

Given a non-degenerate triangle $ABC$, let $A_{1}, B_{1}, C_{1}$ be the point of tangency of the incircle with the sides $BC, AC, AB$. Let $Q$ and $L$ be the intersection of the segment $AA_{1}$ with the incircle and the segment $B_{1}C_{1}$ respectively. Let $M$ be the midpoint of $B_{1}C_{1}$. Let $T$ be the point of intersection of $BC$ and $B_{1}C_{1}$. Let $P$ be the foot of the perpendicular from the point $L$ on the line $AT$. Prove that the points $A_{1}, M, Q, P$ lie on a circle.

The length of the altitude of equilateral triangle $ ABC$ is $3$. A circle with radius $2$, which is tangent to $ \left[BC\right]$ at its midpoint, meets other two sides. If the circle meets $ AB$ and $ AC$ at $ D$ and $ E$, at the outer of $\triangle ABC$ , find the ratio $ \frac {Area\, \left(ABC\right)}{Area\, \left(ADE\right)}$. $\textbf{(A)}\ 2\left(5 \plus{} \sqrt {3} \right) \qquad\textbf{(B)}\ 7\sqrt {2} \qquad\textbf{(C)}\ 5\sqrt {3} \\ \qquad\textbf{(D)}\ 2\left(3 \plus{} \sqrt {5} \right) \qquad\textbf{(E)}\ 2\left(\sqrt {3} \plus{} \sqrt {5} \right)$

Let $C_1$ be a circle inside the circle $C_2$ and let $P$ in the interior of $C_1$, $Q$ in the exterior of $C_2$. One draws variable lines $l_i$ through $P$, not passing through $Q$. Let $l_i$ intersect $C_1$ in $A_i,B_i$, and let the circumcircle of $QA_iB_i$ intersect $C_2$ in $M_i,N_i$. Show that all lines $M_i,N_i$ are concurrent.