Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic. [i]Author: Andrey Gavrilyuk, Russia[/i]

In a non isosceles triangle $ ABC$ let $ w$ be the angle bisector of the exterior angle at $ C$. Let $ D$ be the point of intersection of $ w$ with the extension of $ AB$. Let $ k_A$ be the circumcircle of the triangle $ ADC$ and analogy $ k_B$ the circumcircle of the triangle $ BDC$. Let $ t_A$ be the tangent line to $ k_A$ in A and $ t_B$ the tangent line to $ k_B$ in B. Let $ P$ be the point of intersection of $ t_A$ and $ t_B$. Given are the points $ A$ and $ B$. Determine the set of points $ P\equal{}P(C )$ over all points $ C$, so that $ ABC$ is a non isosceles, acute-angled triangle.

Let $ABC$ be a triangle and $D$ the foot of the altitude from $A$. Let $E$ and $F$ lie on a line passing through $D$ such that $AE$ is perpendicular to $BE$, $AF$ is perpendicular to $CF$, and $E$ and $F$ are different from $D$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $EF$, respectively. Prove that $AN$ is perpendicular to $NM$.

Let ABCD be a square inscribed in a circle k and P be an arbitrary point of that circle. Prove that at least one of the numbers PA, PB, PC and PD is not rational.

Let $ n\geqslant 3$ and $ P$ a convex $ n$-gon. Show that $ P$ can be, by $ n \minus{} 3$ non-intersecting diagonals, partitioned in triangles such that the circumcircle of each triangle contains the whole area of $ P$. Under which conditions is there exactly one such triangulation?

Let $U$ be the incenter of a triangle $ABC$ and $O_{1}, O_{2}, O_{3}$ be the circumcenters of the triangles $BCU, CAU, ABU$ , respectively. Prove that the circumcircles of the triangles $ABC$ and $O_{1}O_{2}O_{3}$ have the same center.

Let $ABC$ be an acute-angled triangle and let $AN$, $BM$ and $CP$ the altitudes with respect to the sides $BC$, $CA$ and $AB$, respectively. Let $R$, $S$ be the pojections of $N$ on the sides $AB$, $CA$, respectively, and let $Q$, $W$ be the projections of $N$ on the altitudes $BM$ and $CP$, respectively. (a) Show that $R$, $Q$, $W$, $S$ are collinear. (b) Show that $MP=RS-QW$.

In a triangle $ABC$ with $AB<AC<BC$, the perpendicular bisectors of $AC$ and $BC$ intersect $BC$ and $AC$ at $K$ and $L$, respectively. Let $O$, $O_1$, and $O_2$ be the circumcentres of triangles $ABC$, $CKL$, and $OAB$, respectively. Prove that $OCO_1O_2$ is a parallelogram.

Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$. Let $t_{1}$ and $t_{2}$ be the tangents to $\omega_{1}$ and $\omega_{2}$, respectively, at point $A$. Let the second intersection of $\omega_{1}$ and $t_{2}$ be $C$, and let the second intersection of $\omega_{2}$ and $t_{1}$ be $D$. Points $P$ and $E$ lie on the ray $AB$, such that $B$ lies between $A$ and $P$, $P$ lies between $A$ and $E$, and $AE = 2 \cdot AP$. The circumcircle to $\bigtriangleup BCE$ intersects $t_{2}$ again at point $Q$, whereas the circumcircle to $\bigtriangleup BDE$ intersects $t_{1}$ again at point $R$. Prove that points $P$, $Q$, and $R$ are collinear.

Let $ O$ be the circumcircle of a $ \Delta ABC$ and let $ I$ be its incenter, for a point $ P$ of the plane let $ f(P)$ be the point obtained by reflecting $ P'$ by the midpoint of $ OI$, with $ P'$ the homothety of $ P$ with center $ O$ and ratio $ \frac{R}{r}$ with $ r$ the inradii and $ R$ the circumradii,(understand it by $ \frac{OP}{OP'}\equal{}\frac{R}{r}$). Let $ A_1$, $ B_1$ and $ C_1$ the midpoints of $ BC$, $ AC$ and $ AB$, respectively. Show that the rays $ A_1f(A)$, $ B_1f(B)$ and $ C_1f(C)$ concur on the incircle.

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 $CF$.

The line $l$ intersects the extension of $AB$ in $D$ ($D$ is nearer to $B$ than $A$) and the extension of $AC$ in $E$ ($E$ is nearer to $C$ than $A$) of triangle $ABC$. Suppose that reflection of line $l$ to perpendicular bisector of side $BC$ intersects the mentioned extensions in $D'$ and $E'$ respectively. Prove that if $BD+CE=DE$, then $BD'+CE'=D'E'$.

Let $\odot O$ , $\odot I$ be the circumcircle and inscribed circles of triangle$ABC$ . Prove that : From every point $D$ on $\odot O$ ,we can construct a triangle $DEF$ such that $ABC$ and $DEF$ have the same circumcircle and inscribed circles

In triangle $ ABC$, $ AB \equal{} 10$, $ BC \equal{} 14$, and $ CA \equal{} 16$. Let $ D$ be a point in the interior of $ \overline{BC}$. Let $ I_B$ and $ I_C$ denote the incenters of triangles $ ABD$ and $ ACD$, respectively. The circumcircles of triangles $ BI_BD$ and $ CI_CD$ meet at distinct points $ P$ and $ D$. The maximum possible area of $ \triangle BPC$ can be expressed in the form $ a\minus{}b\sqrt{c}$, where $ a$, $ b$, and $ c$ are positive integers and $ c$ is not divisible by the square of any prime. Find $ a\plus{}b\plus{}c$.

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 $KL$.

Point $P$ is taken in the interior of the triangle $ABC$, so that \[\angle PAB = \angle PCB = \dfrac {1} {4} (\angle A + \angle C).\] Let $L$ be the foot of the angle bisector of $\angle B$. The line $PL$ meets the circumcircle of $\triangle APC$ at point $Q$. Prove that $QB$ is the angle bisector of $\angle AQC$. [i]Proposed by S. Berlov[/i]

Let $ABC$ be a scalene triangle, and let $I$ be its incenter. A point $D$ is chosen on line $BC$, such that the circumcircle of triangle $BID$ intersects $AB$ at $E\neq B$, and the circumcircle of triangle $CID$ intersects $AC$ at $F\neq C$. Circumcircle of triangle $EDF$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. Lines $FD$ and $IC$ intersect at $Q$, and lines $ED$ and $BI$ intersect at $P$. Prove that $EN\parallel MF\parallel PQ$.

Let $ABC$ be a triangle inscribed in a circle $(O)$. d is the tangent at $A$ of $(O), P$ is an arbitrary point in the plane. $D,E, F$ are the projections of $P$ on $BC,CA,AB$. Let $DE,DF$ intersect the line $d$ at $M,N$ respectively. The circumcircle of triangle $DEF$ meets $CA,AB$ at $K,L$ distinct from $E, F$. Prove that $KN$ meets $LM$ at a point on the circumcircle of triangle $DEF$. Trần Quang Hùng

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ intersect at $P$. Let $O$ be the circumcenter of triangle $APB$ and $H$ be the orthocenter of triangle $CPD$. Show that the points $H,P,O$ are collinear.

A circle passes through the vertices of a triangle with side-lengths of $7\tfrac{1}{2},10,12\tfrac{1}{2}$. The radius of the circle is: $\textbf{(A)}\ \dfrac{15}{4} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ \dfrac{25}{4} \qquad \textbf{(D)}\ \dfrac{35}{4} \qquad \textbf{(E)}\ \dfrac{15\sqrt2}{2} $

The triangle $ ABC$ is inscribed in a circle. The interior bisectors of the angles $ A,B$ and $ C$ meet the circle again at $ A', B'$ and $ C'$ respectively. Prove that the area of triangle $ A'B'C'$ is greater than or equal to the area of triangle $ ABC.$

In the equilateral $\bigtriangleup ABC$, $D$ is a point on side $BC$. $O_1$ and $I_1$ are the circumcenter and incenter of $\bigtriangleup ABD$ respectively, and $O_2$ and $I_2$ are the circumcenter and incenter of $\bigtriangleup ADC$ respectively. $O_1I_1$ intersects $O_2I_2$ at $P$. Find the locus of point $P$ as $D$ moves along $BC$.

Let the point $ B $ lie on the circle $ S_1 $ and let the point $ A $, other than the point $ B $, lie on the tangent to the circle $ S_1 $ passing through the point $ B $. Let a point $ C $ be chosen outside the circle $ S_1 $, so that the segment $ AC $ intersects $ S_1 $ at two different points. Let the circle $ S_2 $ touch the line $ AC $ at the point $ C $ and the circle $ S_1 $ at the point $ D $, on the opposite side from the point $ B $ with respect to the line $ AC $. Prove that the center of the circumcircle of triangle $ BCD $ lies on the circumcircle of triangle $ ABC $.

Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.

Let ${ABC}$ be a triangle inscribed in a circle. Point ${P}$ is the center of the arc ${BAC}$. The circle with the diameter ${CP}$ intersects the angle bisector of angle ${\angle BAC}$ at points ${K, L}$ ${(|AK| <|AL|)}$. Point ${M}$ is the reflection of ${L}$ with respect to line ${BC}$. Prove that the circumcircle of the triangle ${BKM}$ passes through the center of the segment ${BC}$ .