Let $ABC$ be an acute triangle inscribed in a circle of center $O$. If the altitudes $BD,CE$ intersect at $H$ and the circumcenter of $\triangle BHC$ is $O_1$, prove that $AHO_1O$ is a parallelogram.

Let $D$ be a point on the side $BC$ of an equilateral triangle $ABC$ where $D$ is different than the vertices. Let $I$ be the excenter of the triangle $ABD$ opposite to the side $AB$ and $J$ be the excenter of the triangle $ACD$ opposite to the side $AC$. Let $E$ be the second intersection point of the circumcircles of triangles $AIB$ and $AJC$. Prove that $A$ is the incenter of the triangle $IEJ$.

In a $\triangle {ABC}$, consider a point $E$ in $BC$ such that $AE \perp BC$. Prove that $AE=\frac{bc}{2r}$, where $r$ is the radio of the circle circumscripte, $b=AC$ and $c=AB$.

Triangle $ABC$ has $AB=4$, $BC=6$, and $AC=5$. Let $O$ denote the circumcenter of $ABC$. The circle $\Gamma$ is tangent to and surrounds the circumcircles of triangle $AOB$, $BOC$, and $AOC$. Determine the diameter of $\Gamma$.

Let $O$ be a circumcenter of acute triangle $ABC$ and let $O_1$ and $O_2$ be circumcenters of triangles $OAB$ and $OAC$, respectively. Circumcircles of triangles $OAB$ and $OAC$ intersect side $BC$ in points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Perpendicular bisector of side $BC$ intersects side $AC$ in point $F$($F \neq A$). Prove that circumcenter of triangle $ADE$ lies on $AC$ iff $F$ lies on line $O_1O_2$

Let $ABC$ be triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies \[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\] Show that $AP \geq AI$, and that equality holds if and only if $P=I$.

Given an equilateral triangle $ABC$ of side $a$ in a plane, let $M$ be a point on the circumcircle of the triangle. Prove that the sum $s = MA^4 +MB^4 +MC^4$ is independent of the position of the point $M$ on the circle, and determine that constant value as a function of $a$.

Let $ABC$ be an acute triangle with circumcenter $O$, and let $D$, $E$, and $F$ be the feet of altitudes from $A$, $B$, and $C$ to sides $BC$, $CA$, and $AB$, respectively. Denote by $P$ the intersection of the tangents to the circumcircle of $ABC$ at $B$ and $C$. The line through $P$ perpendicular to $EF$ meets $AD$ at $Q$, and let $R$ be the foot of the perpendicular from $A$ to $EF$. Prove that $DR$ and $OQ$ are parallel.

Given triangle $ ABC$. Point $ O$ is the center of the excircle touching the side $ BC$. Point $ O_1$ is the reflection of $ O$ in $ BC$. Determine angle $ A$ if $ O_1$ lies on the circumcircle of $ ABC$.

Given an acute angled triangle $ ABC$ , $ O$ is the circumcenter and $ H$ is the orthocenter.Let $ A_1$,$ B_1$,$ C_1$ be the midpoints of the sides $ BC$,$ AC$ and $ AB$ respectively. Rays $ [HA_1$,$ [HB_1$,$ [HC_1$ cut the circumcircle of $ ABC$ at $ A_0$,$ B_0$ and $ C_0$ respectively.Prove that $ O$,$ H$ and $ H_0$ are collinear if $ H_0$ is the orthocenter of $ A_0B_0C_0$

Given a non-isosceles triangle $ABC$, let $D,E$, and $F$ denote the midpoints of the sides $BC,CA$, and $AB$ respectively. The circle $BCF$ and the line $BE$ meet again at $P$, and the circle $ABE$ and the line $AD$ meet again at $Q$. Finally, the lines $DP$ and $FQ$ meet at $R$. Prove that the centroid $G$ of the triangle $ABC$ lies on the circle $PQR$. [i](United Kingdom) David Monk[/i]

Consider triangles whose each side length squared is a rational number. Is it true that (a) the square of the circumradius of every such triangle is rational; (b) the square of the inradius of every such triangle is rational?

Scalene triangle $ABC$ has incenter $I$ and circumcircle $\Omega$ with center $O$. $H$ is the orthocenter of triangle $BIC$, and $T$ is a point on $\Omega$ for which $\angle ATI=90^\circ$. Circle $(AIO)$ intersects line $IH$ again at $X$. Show that the lines $AX, HT$ intersect on $\Omega$.

Let $ ABC$ be triangle, $ I$ its in-center; $ A_1,B_1,C_1$ be the reflections of $ I$ in $ BC, CA, AB$ respectively. Suppose the circum-circle of triangle $ A_1B_1C_1$ passes through $ A$. Prove that $ B_1,C_1,I,I_1$ are concylic, where $ I_1$ is the in-center of triangle $ A_1,B_1,C_1$.

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]

Given a triangle $ABC$ with circumcircle $\Gamma$. Points $E$ and $F$ are the foot of angle bisectors of $B$ and $C$, $I$ is incenter and $K$ is the intersection of $AI$ and $EF$. Suppose that $T$ be the midpoint of arc $BAC$. Circle $\Gamma$ intersects the $A$-median and circumcircle of $AEF$ for the second time at $X$ and $S$. Let $S'$ be the reflection of $S$ across $AI$ and $J$ be the second intersection of circumcircle of $AS'K$ and $AX$. Prove that quadrilateral $TJIX$ is cyclic. [i]Proposed by Alireza Dadgarnia and Amir Parsa Hosseini[/i]

Let $ ABC $ be an acute triangle having $ AB<AC, I $ be its incenter, $ D,E,F $ be intersection of the incircle with $ BC, CA, $ respectively, $ AB, X $ be the middle of the arc $ BAC, $ which is an arc of the circumcicle of it, $ P $ be the projection of $ D $ on $ EF $ and $ Q $ be the projection of $ A $ on $ ID. $ [b]a)[/b] Show that $ IX $ and $ PQ $ are parallel. [b]b)[/b] If the circle of diameter $ AI $ intersects the circumcircle of $ ABC $ at $ Y\neq A, $ prove that $ XQ $ intersects $ PI $ at $ Y. $

In the acute triangle $ABC$ the orthocenter $H$ and the center of the circumscribed circle $O$ were noted. The line $AO$ intersects the side $BC$ at the point $D$. A perpendicular drawn to the side $BC$ at the point $D$ intersects the heights from the vertices $B$ and $C$ of the triangle $ABC$ at the points $X$ and $Y$ respectively. Prove that the center of the circumscribed circle $\Delta HXY$ is equidistant from the points $B$ and $C$. (Danilo Hilko)

Let $ ABC$ be an isosceles triangle with $ AB \equal{} AC$. $ M$ is the midpoint of $ BC$ and $ O$ is the point on the line $ AM$ such that $ OB$ is perpendicular to $ AB$. $ Q$ is an arbitrary point on $ BC$ different from $ B$ and $ C$. $ E$ lies on the line $ AB$ and $ F$ lies on the line $ AC$ such that $ E, Q, F$ are distinct and collinear. Prove that $ OQ$ is perpendicular to $ EF$ if and only if $ QE \equal{} QF$.

Two circles $\gamma_1, \gamma_2$ in a plane, with centers $A$ and $B$ respectively, intersect at $C$ and $D$. Suppose that the circumcircle of $ABC$ intersects $\gamma_1$ in $E$ and $\gamma_2$ in $F$, where the arc $EF$ not containing $C$ lies outside $\gamma_1$ and $\gamma_2$. Prove that this arc $EF$ is bisected by the line $CD$.

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Let $\Omega$ be the circumcircle of triangle $ABC$. Suppose that $X$ is a point on the segment $AB$ with $XB=XC$, and the angle bisector of $\angle BAC$ intersects $BC$ and $\Omega$ at $D$, $M$, respectively. If $P$ is a point on $BC$ such that $AP$ is tangent to $\Omega$ and $Q$ is a point on $DX$ such that $CQ$ is tangent to $\Omega$, show that $AB$, $CM$, $PQ$ are concurrent.

Given an acute angled triangle $ABC$ with $M$ being the mid-point of $AB$ and $P$ and $Q$ are the feet of heights from $A$ to $BC$ and $B$ to $AC$ respectively. Show that if the line $AC$ is tangent to the circumcircle of $BMP$ then the line $BC$ is tangent to the circumcircle of $AMQ$.

Let $ \Omega$ is circle with radius $ R$ and center $ O$. Let $ \omega$ is a circle inside of the $ \Omega$ with center $ I$ radius $ r$. $ X$ is variable point of $ \omega$ and tangent line of $ \omega$ pass through $ X$ intersect the circle $ \Omega$ at points $ A,B$. A line pass through $ X$ perpendicular with $ AI$ intersect $ \omega$ at $ Y$ distinct with $ X$.Let point $ C$ is symmetric to the point $ I$ with respect to the line $ XY$.Find the locus of circumcenter of triangle $ ABC$ when $ X$ varies on $ \omega$

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.