The points $A_1,B_1,C_1$ lie on the sides $BC,CA$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $O_A,O_B$ and $O_C$ be the circumcentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the incentre of triangle $O_AO_BO_C$ is the incentre of triangle $ABC$ too.

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 $\omega$.

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]

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 $\omega$.

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 $\omega$.

In $\triangle ABC$, $H$ is the orthocenter, and $AD,BE$ are arbitrary cevians. Let $\omega_1, \omega_2$ denote the circles with diameters $AD$ and $BE$, respectively. $HD,HE$ meet $\omega_1,\omega_2$ again at $F,G$. $DE$ meets $\omega_1,\omega_2$ again at $P_1,P_2$ respectively. $FG$ meets $\omega_1,\omega_2$ again $Q_1,Q_2$ respectively. $P_1H,Q_1H$ meet $\omega_1$ at $R_1,S_1$ respectively. $P_2H,Q_2H$ meet $\omega_2$ at $R_2,S_2$ respectively. Let $P_1Q_1\cap P_2Q_2 = X$, and $R_1S_1\cap R_2S_2=Y$. Prove that $X,Y,H$ are collinear. [i]Ray Li.[/i]

Let $ABC$ be a triangle with $ \widehat{BAC}\neq 90^\circ $. Let $M$ be the midpoint of $BC$. We choose a variable point $D$ on $AM$. Let $(O_1)$ and $(O_2)$ be two circle pass through $ D$ and tangent to $BC$ at $B$ and $C$. The line $BA$ and $CA$ intersect $(O_1),(O_2)$ at $ P,Q$ respectively. [b]a)[/b] Prove that tangent line at $P$ on $(O_1)$ and $Q$ on $(O_2)$ must intersect at $S$. [b]b)[/b] Prove that $S$ lies on a fix line.

Let $ABC$ be a triangle and $X,Y,Z$ be the tangency points of its inscribed circle with the sides $BC, CA, AB$, respectively. Suppose that $C_1, C_2, C_3$ are circle with chords $YZ, ZX, XY$, respectively, such that $C_1$ and $C_2$ intersect on the line $CZ$ and that $C_1$ and $C_3$ intersect on the line $BY$. Suppose that $C_1$ intersects the chords $XY$ and $ZX$ at $J$ and $M$, respectively; that $C_2$ intersects the chords $YZ$ and $XY$ at $L$ and $I$, respectively; and that $C_3$ intersects the chords $YZ$ and $ZX$ at $K$ and $N$, respectively. Show that $I, J, K, L, M, N$ lie on the same circle.

Let $ABC$ be an equilateral triangle. $D$ and $E$ are midpoints of $[AB]$ and $[AC]$. The ray $[DE$ cuts the circumcircle of $\triangle ABC$ at $F$. What is $\frac {|DE|}{|DF|}$? $ \textbf{(A)}\ \frac 12 \qquad\textbf{(B)}\ \frac {\sqrt 3}3 \qquad\textbf{(C)}\ \frac 23(\sqrt 3 - 1) \qquad\textbf{(D)}\ \frac 23 \qquad\textbf{(E)}\ \frac {\sqrt 5 - 1}2 $

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $K.$ The midpoints of diagonals $AC$ and $BD$ are $M$ and $N,$ respectively. The circumscribed circles $ADM$ and $BCM$ intersect at points $M$ and $L.$ Prove that the points $K ,L ,M,$ and $ N$ lie on a circle. (all points are supposed to be different.)

The lines \(AB\) and \(CD\), containing the lateral sides of the trapezoid \(ABCD\), intersect at point \(Q\). Inside the trapezoid \(ABCD\), a point \(P\) is chosen such that \(\angle APB = \angle CPD\). Prove that the circumcircles of triangles \(BPD\) and \(APC\) intersect again on the line \(PQ\). [i]Proposed by Mykhailo Shtandenko[/i]

Consider $\omega$ is circumcircle of an acute triangle $ABC$. $D$ is midpoint of arc $BAC$ and $I$ is incenter of triangle $ABC$. Let $DI$ intersect $BC$ in $E$ and $\omega$ for second time in $F$. Let $P$ be a point on line $AF$ such that $PE$ is parallel to $AI$. Prove that $PE$ is bisector of angle $BPC$. [i]Proposed by Mr.Etesami[/i]

Let $P$ be a point inside the triangle $ABC$, such that the angles $\angle CBP$ and $\angle PAC$ are equal. Denote the intersection of the line $AP$ and the segment $BC$ by $D$, and the intersection of the line $BP$ with the segment $AC$ by $E$. The circumcircles of the triangles $ADC$ and $BEC$ meet at $C$ and $F$. Show that the line $CP$ bisects the angle $DFE$. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

A circle $ \omega$ with center $ O$ is tangent to the rays of an angle $ BAC$ at $ B$ and $ C$. Point $ Q$ is taken inside the angle $ BAC$. Assume that point $ P$ on the segment $ AQ$ is such that $ AQ\perp OP$. The line $ OP$ intersects the circumcircles $ \omega_{1}$ and $ \omega_{2}$ of triangles $ BPQ$ and $ CPQ$ again at points $ M$ and $ N$. Prove that $ OM \equal{} ON$.

$ABC$ is an acute triangle. $P$(different from $B,C$) is a point on side $BC$. $H$ is an orthocenter, and $D$ is a foot of perpendicular from $H$ to $AP$. The circumcircle of the triangle $ABD$ and $ACD$ is $O _1$ and $O_2$, respectively. A line $l$ parallel to $BC$ passes $D$ and meet $O_1$ and $O_2$ again at $X$ and $Y$, respectively. $l$ meets $AB$ at $E$, and $AC$ at $F$. Two lines $XB$ and $YC$ intersect at $Z$. Prove that $ZE=ZF$ is a necessary and sufficient condition for $BP=CP$.

In a cyclic quadrilateral $ABCD$, the diagonals intersect at $E$. $F$ and $G$ are on chord $AC$ and chord $BD$ respectively such that $AF = BE$ and $DG = CE$. Prove that, $A, G, F, D$ lie on the same circle.

In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$

Let $ABC$ be a triangle and $m$ a line which intersects the sides $AB$ and $AC$ at interior points $D$ and $F$, respectively, and intersects the line $BC$ at a point $E$ such that $C$ lies between $B$ and $E$. The parallel lines from the points $A$, $B$, $C$ to the line $m$ intersect the circumcircle of triangle $ABC$ at the points $A_1$, $B_1$ and $C_1$, respectively (apart from $A$, $B$, $C$). Prove that the lines $A_1E$ , $B_1F$ and $C_1D$ pass through the same point. [i]Greece[/i]

We have $4$ circles in plane such that any two of them are tangent to each other. we connect the tangency point of two circles to the tangency point of two other circles. Prove that these three lines are concurrent. [i]proposed by Masoud Nourbakhsh[/i]

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

In the triangle $ABC$, $AM$ is median, $M \in BC$, $BB_{1}$ and $CC_{1}$ are altitudes, $C_{1} \in AB$, $B_{1} \in AC$. The line through $A$ which is perpendicular to $AM$ cuts the lines $BB_{1}$ and $CC_{1}$ at points $E$ and $F$, respectively. Let $k$ be the circumcircle of $\triangle EFM$. Suppose also that $k_{1}$ and $k_{2}$ are circles touching both $EF$ and the arc $EF$ of $k$ which does not contain $M$. If $P$ and $Q$ are the points at which $k_{1}$ intersects $k_{2}$, prove that $P$, $Q$, and $M$ are collinear.

Circles $\omega_1$ and $\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\omega$ is externally tangent to both $\omega_1$ and $\omega_2$. Suppose line $AB$ intersects $\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\widehat{PQ}$ is $120^{\circ}$. Find the distance between the centers of $\omega_1$ and $\omega_2$.

Let $ ABCDE$ be an arbitrary convex pentagon. Suppose that $ BD\cap CE \equal{} A'$, $ CE \cap DA \equal{} B'$, $ DA\cap EB \equal{} C'$, $ EB\cap AC \equal{} D'$ and $ AC \cap BD \equal{} E'$. Suppose also that $ (ABD')\cap (AC'E) \equal{} A''$, $ (BCE')\cap (BD'A) \equal{} B''$, $ (CDA')\cap (CE'B) \equal{} C''$, $ (DEB')\cap DA'C \equal{} D''$ and $ (EAC')\cap (EB'D) \equal{} E''$. Prove that $ AA''$, $ BB''$, $ CC''$, $ DD''$ and $ EE''$ are concurrent.

Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$. Prove that the lines $AP$ and $CS$ are parallel.

Let $ABCD$ be a trapezium $(AB \parallel CD)$ and $M,N$ be the intersection points of the circles of diameters $AD$ and $BC$. Prove that $O \in MN$, where $O \in AC \cap BD$.