In scalene triangle $ABC$, let the feet of the perpendiculars from $A$ to $BC$, $B$ to $CA$, $C$ to $AB$ be $A_1, B_1, C_1$, respectively. Denote by $A_2$ the intersection of lines $BC$ and $B_1C_1$. Define $B_2$ and $C_2$ analogously. Let $D, E, F$ be the respective midpoints of sides $BC, CA, AB$. Show that the perpendiculars from $D$ to $AA_2$, $E$ to $BB_2$ and $F$ to $CC_2$ are concurrent.

Let $M$ be a point in the plane, distinct from the vertices of $\triangle ABC$. Consider $N,P,Q$ the reflections of $M$ with respect to lines $AB, BC$ and $CA$, in this order. a) Prove that $N, P ,Q$ are collinear if and only if $M$ lies on the circumcircle of $\triangle ABC$. b) If $M$ does not lie on the circumcircle of $\triangle ABC$ and the centroids of triangles $\triangle ABC$ and $\triangle NPQ$ coincide, prove that $\triangle ABC$ is equilateral.

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee, Korea[/i]

In the plane are given a circle with center $ O$ and radius $ r$ and a point $ A$ outside the circle. For any point $ M$ on the circle, let $ N$ be the diametrically opposite point. Find the locus of the circumcenter of triangle $ AMN$ when $ M$ describes the circle.

Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle $ABC$, and the circumcircle of the triangle $AEB$ cuts the side $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.

$ABC$ is an acute triangle with $AB> AC$. $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for $ \Gamma_C$. Let $D$ be the intersection $\Gamma_B$ and $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle $ABC$. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.

Let $ABC$ be an equilateral triangle , and $P$ be a point on the circumcircle of the triangle but distinct from $A$ ,$B$ and $C$. The lines through $P$ and parallel to $BC$ , $CA$ , $AB$ intersect the lines $CA$ , $AB$ , $BC$ at $M$ , $N$ and $Q$ respectively .Prove that $M$ , $N$ and $Q$ are collinear .

The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle. [img]https://1.bp.blogspot.com/-gojv6KfBC9I/XzT9ZMKrIeI/AAAAAAAAMVU/NB-vUldjULI7jvqiFWmBC_Sd8QFtwrc7wCLcBGAsYHQ/s0/2013%2BMohr%2Bp3.png[/img]

Given two circles intersecting at points $P$ and $Q$. Let C be an arbitrary point distinct from $P$ and $Q$ on the former circle. Let lines $CP$ and $CQ$ intersect again the latter circle at points A and B, respectively. Determine the locus of the circumcenters of triangles $ABC$.

Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.

In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.

Let $r,R$ and $r_a$ be the radii of the incircle, circumcircle and A-excircle of the triangle $ABC$ with $AC>AB$, respectively. $I,O$ and $J_A$ are the centers of these circles, respectively. Let incircle touches the $BC$ at $D$, for a point $E \in (BD)$ the condition $A(IEJ_A)=2A(IEO)$ holds. Prove that \[ED=AC-AB \iff R=2r+r_a.\]

Let $\Delta ABC$ be a triangle with orthocenter $H$ and $\Gamma$ be the circumcircle of $\Delta ABC$ with center $O$. Consider $N$ the center of the circle that passes through the feet of the heights of $\Delta ABC$ and $P$ the intersection of the line $AN$ with the circle $\Gamma$. Suppose that the line $AP$ is perpendicular to the line $OH$. Prove that $P$ belongs to the reflection of the line $OH$ by the line $BC$.

Let $ABC$ be an isosceles triangle ($AB = AC$) with its circumcenter $O$. Point $N$ is the midpoint of the segment $BC$ and point $M$ is the reflection of the point $N$ with respect to the side $AC$. Suppose that $T$ is a point so that $ANBT$ is a rectangle. Prove that $\angle OMT = \frac{1}{2} \angle BAC$. [i]Proposed by Ali Zamani[/i]

Given triangle $ABC$, $D$ is the foot of the external angle bisector of $A$, $I$ its incenter and $I_a$ its $A$-excenter. Perpendicular from $I$ to $DI_a$ intersects the circumcircle of triangle in $A'$. Define $B'$ and $C'$ similarly. Prove that $AA',BB'$ and $CC'$ are concurrent. [i]proposed by Amirhossein Zabeti[/i]

In a parallelogram $ABCD$, the bisector of $\angle A$ intersects $BC$ at $M$ and the extension of $DC$ at $N$. Let $O$ be the circumcircle of the triangle $MCN$. Prove that $\angle OBC = \angle ODC$

Let $ABC$ be an acute triangle such that $AB<AC$. Let $\Omega$ be its circumcircle, $O$ be its circumcenter, and $l$ be the internal angle bisector of $\angle BAC$. Suppose that the tangents to $\Omega$ at $B$ and $C$ intersect at $X$. Let $\omega$ be a circle whose center is $X$ and passes $B$, and $Y$ be the intersection of $l$ and $\omega$ which is chosen inside $\triangle ABC$. Let $D,E$ be the projections of $Y$ onto $AB,AC$, respectively. $OY$ meets $BC$ at $Z$. $ZD,ZE$ meet $l$ at $P,Q$, respectively. Prove that $BQ$ and $CP$ are parallel.

In an acute triangle $ABC$ with an angle $\angle ACB =75^o$, altitudes $AA_3,BB_3$ intersect the circumscribed circle at points $A_1,B_1$ respectively. On the lines $BC$ and $CA$ select points $A_2$ and $B_2$ respectively suchthat the line $B_1B_2$ is parallel to the line $BC$ and the line $A_1A_2$ is parallel to the line $AC$ . Let $M$ be the midpoint of the segment $A_2B_2$. Find in degrees the measure of the angle $\angle B_3MA_3$.

Let $ABC$ be a triangle and $L$ its circumscribed circle. The internal bisector of angle $A$ meets $BC$ at point $P$. Let $L_1$ be the circle tangent to $AP,BP$ and $L$. Similarly, let $L_2$ be the circle tangent to $AP,CP$ and $L$. Prove that the tangency points of $L_1$ and $L_2$ with $AP$ coincide.

Let $\triangle ABC$ be an acute triangle of circumcentre $O$. Let the tangents to the circumcircle of $\triangle ABC$ in points $B$ and $C$ meet at point $P$. The circle of centre $P$ and radius $PB=PC$ meets the internal angle bisector of $\angle BAC$ inside $\triangle ABC$ at point $S$, and $OS \cap BC = D$. The projections of $S$ on $AC$ and $AB$ respectively are $E$ and $F$. Prove that $AD$, $BE$ and $CF$ are concurrent. [i]Author: Cosmin Pohoata[/i]

In triangle $ABC$, $AB = AC$. Point $D$ is the midpoint of side $BC$. Point $E$ lies outside the triangle $ABC$ such that $CE \perp AB$ and $BE = BD$. Let $M$ be the midpoint of segment $BE$. Point $F$ lies on the minor arc $\widehat{AD}$ of the circumcircle of triangle $ABD$ such that $MF \perp BE$. Prove that $ED \perp FD.$ [asy] defaultpen(fontsize(10)); size(6cm); pair A = (3,10), B = (0,0), C = (6,0), D = (3,0), E = intersectionpoints( Circle(B, 3), C--(C+100*dir(B--A)*dir(90)) )[1], M = midpoint(B--E), F = intersectionpoints(M--(M+50*dir(E--B)*dir(90)), circumcircle(A,B,D))[0]; dot(A^^B^^C^^D^^E^^M^^F); draw(B--C--A--B--E--D--F--M^^circumcircle(A,B,D)); pair point = extension(M,F,A,D); pair[] p={A,B,C,D,E,F,M}; string s = "A,B,C,D,E,F,M"; int size = p.length; real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;} d[4] = -50; string[] k= split(s,","); for(int i = 0;i<p.length;++i) { label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i])); }[/asy]

Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE \equal{} PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$

Two regular polygons, a $7$-gon and a $17$-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal. (3)

$ I_a$ is the excenter of the triangle $ ABC$ with respect to $ A$, and $ AI_a$ intersects the circumcircle of $ ABC$ at $ T$. Let $ X$ be a point on $ TI_a$ such that $ XI_a^2\equal{}XA.XT$. Draw a perpendicular line from $ X$ to $ BC$ so that it intersects $ BC$ in $ A'$. Define $ B'$ and $ C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.

Let $\omega_1$ and $\omega_2$ be two circles that intersect at points $A$ and $B$. Let line $l$ be tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$ such that $A$ is closer to $PQ$ than $B$. Let points $R$ and $S$ lie along rays $PA$ and $QA$, respectively, so that $PQ = AR = AS$ and $R$ and $S$ are on opposite sides of $A$ as $P$ and $Q$. Let $O$ be the circumcenter of triangle $ASR$, and $C$ and $D$ be the midpoints of major arcs $AP$ and $AQ$, respectively. If $\angle APQ$ is $45$ degrees and $\angle AQP$ is $30$ degrees, determine $\angle COD$ in degrees.