Let $ABC$ be an acute triangle such that $AB\not=AC.$Let $M$ be the midpoint $BC,H$ the orthocenter of $\triangle ABC$$,O_1$ the midpoint of $AH$ and $O_2$ the circumcenter of $\triangle BCH$$.$ Prove that $O_1AMO_2$ is a parallelogram.

Let $ABC$ be an acute triangle and let $A_{1}, B_{1}, C_{1}$ be the feet of its altitudes. The incircle of the triangle $A_{1}B_{1}C_{1}$ touches its sides at the points $A_{2}, B_{2}, C_{2}$. Prove that the Euler lines of triangles $ABC$ and $A_{2}B_{2}C_{2}$ coincide.

$ABC$ is a right triangle with $\angle C=90$,$CD$ is perpendicular to $AB$,and $D$ is the foot,$\omega$ is the circumcircle of triangle $BCD$,$\omega_{1}$ is a circle inside triangle $ACD$,tangent to $AD$ and $AC$ at $M$ and $N$ respectively,and $\omega_{1}$ is also tangent to $\omega$.prove that: (1)$BD*CN+BC*DM=CD*BM$ (2)$BM=BC$

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ , and let $O$ , $H$ be the triangle's circumcenter and orthocenter respectively . Let also $A^{'}$ be the point where the angle bisector of the angle $BAC$ meets $\omega$ . If $A^{'}H=AH$ , then find the measure of the angle $BAC$.

Given a triangle $ABC$ with $\angle ACB = 90^{\circ}$. Let $\omega$ be the circumcircle of triangle $ABC$. The tangents of $\omega$ at $B$ and $C$ intersect at $P$. Let $M$ be the midpoint of $PB$. Line $CM$ intersects $\omega$ at $N$ and line $PN$ intersects $AB$ at $E$. Point $D$ is on $CM$ such that $ED \parallel BM$. Show that the circumcircle of $CDE$ is tangent to $\omega$.

Triangle $ABC$ with $AB=4$, $BC=5$, $CA=6$ has circumcircle $\Omega$ and incircle $\omega$. Let $\Gamma$ be the circle tangent to $\Omega$ and the sides $AB$, $BC$, and let $X=\Gamma \cap \Omega$. Let $Y$, $Z$ be distinct points on $\Omega$ such that $XY$, $YZ$ are tangent to $\omega$. Find $YZ^2$. [i]The following fact may be useful: if $\triangle{ABC}$ has incircle $w$ with incenter $I$ and radius $r$, and $\triangle{DEF}$ is the intouch triangle (i.e. $D$, $E$, $F$ are intersections of incircle with $BC$, $CA$, $AB$, respectively) and $H$ is the orthocenter of $\triangle{DEF}$, then the inversion of $X$ about $\omega$ (i.e. the point $X'$ on ray $IX$ such that $IX' \cdot IX=r^2$) is the midpoint of $DH$.[/i]

Let $\triangle ABC$ be an acute angled triangle and let $k$ be its circumscribed circle. A point $O$ is given in the interior of the triangle, such that $CE=CF$, where $E$ and $F$ are on $k$ and $E$ lies on $AO$ while $F$ lies on $BO$. Prove that $O$ is on the angle bisector of $\angle ACB$ if and only if $AC=BC$.

Let $I,\ O$ and $\Gamma$ be the incenter, circumcenter and the circumcircle of triangle $ABC$, respectively. Line $AI$ meets $\Gamma$ at $M$ $(M\neq A)$. The circumference $\omega$ is tangent internally to $\Gamma$ at $T$, and is tangent to the lines $AB$ and $AC$. The tangents through $A$ and $T$ to $\Gamma$ intersect at $P$. Lines $PI$ and $TM$ meet at $Q$. Prove that the lines $QA$ and $MO$ meet at a point on $\Gamma$.

$ ABC$ is an acute triangle. The points $ B'$ and $ C'$are the reflections of $ B$ and $ C$ across the lines $ AC$ and $ AB$ respectively. Suppose that the circumcircles of triangles$ ABB$' and $ ACC'$ meet at $ A$ and $ P$. Prove that the line $ PA$ passes through the circumcenter of triangle$ ABC.$

Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$. [asy] size(200); defaultpen(fontsize(10)); real x=22.61986495; pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x); draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F); pair point=(13,13); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F));[/asy]

Point $X$ is chosen inside a convex $ABCD$ so that $\angle XBC = \angle XAD, \angle XCB = \angle XDA$. Rays $AB, DC$ intersect at point $O$, circumcircles of triangles $BCO, ADO$ intersect at point $T$. Prove that line $TX$ and the line through $O$ perpendicular to $BC$ intersect on the circumcircle of $\triangle AOD$. [i]Proposed by Anton Trygub[/i]

Let $ABC$ be an acute triangle with circumcircle $\omega$. The altitudes $AD$, $BE$ and $CF$ of the triangle $ABC$ intersect at point $H$. A point $K$ is chosen on the line $EF$ such that $KH\parallel BC$. Prove that the reflection of $H$ in $KD$ lies on $\omega$.

Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.

Let $\triangle ABC$ be an acute-angled triangle with $AB<AC$ and let $(c)$ be its circumcircle with center $O$. Let $M$ be the midpoint of $BC$. The line $AM$ meets the circle $(c)$ again at the point $D$. The circumcircle $(c_1)$ of triangle $\triangle MDC$ intersects the line $AC$ at the points $C$ and $I$, and the circumcircle $(c_2)$ of $\triangle AMI$ intersects the line $AB$ at the points $A$ and $Z$. If $N$ is the foot of the perpendicular from $B$ on $AC$, and $P$ is the second point of intersection of $ZN$ with $(c_2)$, prove that the quadrilateral with vertices the points $N, P, I$ and $M$ is a parallelogram.

Let $ABC$ be an acute triangle and $D$ be the foot of the altiutude from $A$ on $BC$, $E$ and $F$ are the midpoints of $BD$ and $DC$ respectively. $O$ and $Q$ are the circumcenters of the triangles $\vartriangle BF$ and $\vartriangle ACE$ respectively. $P$ is the intersection point of $OE$ and $QF$, show that $PB = PC$.

Lei it \(ABCD\) be a non-cyclical, convex quadrilateral, with no parallel sides. The lines \(AB\) and \(CD\) meet in \(E\). Let it \(M \not= E\) be the intersection of circumcircles of \(ADE\) and \(BCE\). The internal angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(I\). The external angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(J\). Show that \(I,J,M\) are colinear.

The quadrilateral $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$ and $BD$ do not pass through $O$. If the circumcentre of triangle $AOC$ lies on the line $BD$, prove that the circumcentre of triangle $BOD$ lies on the line $AC$.

Let $\triangle P_1P_2P_3$ be an equilateral triangle. For each $n\ge 4$, [i]Mingmingsan[/i] can set $P_n$ as the circumcenter or orthocenter of $\triangle P_{n-3}P_{n-2}P_{n-1}$. Find all positive integer $n$ such that [i]Mingmingsan[/i] has a strategy to make $P_n$ equals to the circumcenter of $\triangle P_1P_2P_3$. [i]Proposed by Li4 and Untro368.[/i]

Let $ABC$ be a triangle such that $AB=6,BC=5,AC=7.$ Let the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $X.$ Let $Z$ be a point on the circumcircle of $ABC.$ Let $Y$ be the foot of the perpendicular from $X$ to $CZ.$ Let $K$ be the intersection of the circumcircle of $BCY$ with line $AB.$ Given that $Y$ is on the interior of segment $CZ$ and $YZ=3CY,$ compute $AK.$

Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that \[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3, \] denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel. [i]Nikolai Ivanov Beluhov, Bulgaria[/i]

Given are three pairwise externally tangent circles $ K_{1}$ , $ K_{2}$ and $ K_{3}$. denote by $ P_{1}$ tangent point of $ K_{2}$ and $ K_{3}$ and by $ P_{2}$ tangent point of $ K_{1}$ and $ K_{3}$. Let $ AB$ ($ A$ and $ B$ are different from tangency points) be a diameter of circle $ K_{3}$. Line $ AP_{2}$ intersects circle $ K_{1}$ (for second time) at point $ X$ and line $ BP_{1}$ intersects circle $ K_{2}$(for second time) at $ Y$. If $ Z$ is intersection point of lines $ AP_{1}$ and $ BP_{2}$ prove that points $ X$, $ Y$ and $ Z$ are collinear.

Let $ABC$ be an acute-angled triangle with $AC \neq BC$, and let $O$ be the circumcentre and $F$ the foot of the altitude through $C$. Furthermore, let $X$ and $Y$ be the feet of the perpendiculars dropped from $A$ and $B$ respectively to (the extension of) $CO$. The line $FO$ intersects the circumcircle of $FXY$ a second time at $P$. Prove that $OP<OF$.

Let $ABCD$ be a convex plane quadrilateral and let $A_1$ denote the circumcenter of $\triangle BCD$. Define $B_1, C_1,D_1$ in a corresponding way. (a) Prove that either all of $A_1,B_1, C_1,D_1$ coincide in one point, or they are all distinct. Assuming the latter case, show that $A_1$, C1 are on opposite sides of the line $B_1D_1$, and similarly,$ B_1,D_1$ are on opposite sides of the line $A_1C_1$. (This establishes the convexity of the quadrilateral $A_1B_1C_1D_1$.) (b) Denote by $A_2$ the circumcenter of $B_1C_1D_1$, and define $B_2, C_2,D_2$ in an analogous way. Show that the quadrilateral $A_2B_2C_2D_2$ is similar to the quadrilateral $ABCD.$

Let $ ABCD$ be a cyclic quadrilateral and $ O$ be the intersection of diagonal $ AC$ and $ BD$. The circumcircles of triangle $ ABO$ and the triangle $ CDO$ intersect at $ K$. Let $ L$ be a point such that the triangle $ BLC$ is similar to $ AKD$ (in that order). Prove that if $ BLCK$ is a convex quadrilateral, then it has an incircle.