Let $s$ be an arbitrary straight line passing through the incenter $I$ of the triangle $ABC$ . Line $s$ intersects lines $AB$ and $BC$ at points $D$ and $E$, respectively. Points $P$ and $Q$ are the centers of the circumscribed circles of triangles $DAI$ and $CEI$, respectively, and point $F$ is the second intersection point of these circles. Prove that the circumcircle of the triangle $PQF$ is always passes through a fixed point on the plane regardless of the position of the straight line $s$. (Matvii Kurskyi)

Let $ABC$ be a triangle with $AB\neq CB$. Let $C^{\prime}$ be a point on the ray $[AB$ such that $AC^{\prime}=CB$. Let $A^{\prime}$ be a point on the ray $[CB$ such that $CA^{\prime}=AB$. Let the circumcircles of triangles $ABA^{\prime}$ and $CBC^{\prime}$ intersect at a point $Q$ (apart from $B$). Prove that the line $BQ$ bisects the segment $CA$. Darij

Let $ABC$ be an acute angled triangle with orthocenter $H$ and $AB<AC$. The point $T$ lies on line $BC$ so that $AT$ is a tangent to the circumcircle of $ABC$. Let lines $AH$ and $BC$ meet at point $D$ and let $M$ be the midpoint of $HC$. Let the circumcircle of $AHT$ meets $CH$ in $P \not=H$ and the circumcircle of $PDM$ meet $BC$ in $Q \not=D$. Prove that $QT=QA$.

Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$. Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.

Let $H$ be the orthocenter of the triangle $ABC$, $M$ is the midpoint of the segment $BC$. Let $X$ be the point of the intersection of the line $HM$ with arc $BC$(without $A$) of the circumcircle of $ABC$, let $Y$ be the point of intersection of the line $BH$ with the circle, show that $XY = BC$.

Let $BM$ be a median in an acute-angled triangle $ABC$. A point $K$ is chosen on the line through $C$ tangent to the circumcircle of $\vartriangle BMC$ so that $\angle KBC = 90^\circ$. The segments $AK$ and $BM$ meet at $J$. Prove that the circumcenter of $\triangle BJK$ lies on the line $AC$. Aleksandr Kuznetsov, Russia

Let $ ABCD$ be a cyclic convex quadrilateral. Let $ E $ be the intersection of lines $ AB, CD $. $ P $ is the intersection of line passing $ B $ and perpendicular to $ AC $, and line passing $ C $ and perpendicular to $ BD$. $ Q $ is the intersection of line passing $ D $ and perpendicular to $ AC $, and line passing $ A $ and perpendicular to $ BD $. Prove that three points $ E, P, Q $ are collinear.

In a triangle $ABC$, $I$ is the incentre and $D$ the intersection point of $AI$ and $BC$. Show that $AI+CD=AC$ if and only if $\angle B=60^{\circ}+\frac{_1}{^3}\angle C$.

Let $ABC$ be a triangle with incenter $I$. Denote the midpoints of $AI$, $AC$ and $CI$ by $L$, $M$ and $N$ respectively. Point $D$ lies on segment $AM$ such that $BC= BD$. Let the incircle of triangle $ABD$ be tangent to $AD$ and $BD$ at $E$ and $F$ respectively. Denote the circumcenter of triangle $AIC$ by $J$, and the circumcircle of triangle $JMD$ by $\omega$. Lines $MN$ and $JL$ meet $\omega$ again at $P$ and $Q$ respectively. Prove that $PQ$, $LN$ and $EF$ are concurrent.

In triangle $ ABC$ points $ E$ and $ F$ lie on sides $ AC$ and $ BC$ such that segments $ AE$ and $ BF$ have equal length, and circles formed by $ A,C,F$ and by $ B,C,E,$ respectively, intersect at point $ C$ and another point $ D.$ Prove that that the line $ CD$ bisects $ \angle ACB.$

Given an acute triangle $ABC$, let $D$, $E$ and $F$ be points in the lines $BC$, $AC$ and $AB$ respectively. If the lines $AD$, $BE$ and $CF$ pass through $O$ the centre of the circumcircle of the triangle $ABC$, whose radius is $R$, show that: \[\frac{1}{AD}\plus{}\frac{1}{BE}\plus{}\frac{1}{CF}\equal{}\frac{2}{R}\]

Let $ ABCD$ be a (not self-intersecting) quadrilateral satisfying $ \measuredangle DAB \equal{} \measuredangle BCD\neq 90^{\circ}$. Let $ X$ and $ Y$ be the orthogonal projections of the point $ D$ on the lines $ AB$ and $ BC$, and let $ Z$ and $ W$ be the orthogonal projections of the point $ B$ on the lines $ CD$ and $ DA$. Establish the following facts: [b]a)[/b] The quadrilateral $ XYZW$ is an isosceles trapezoid such that $ XY\parallel ZW$. [b]b)[/b] Let $ M$ be the midpoint of the segment $ AC$. Then, the lines $ XZ$ and $ YW$ pass through the point $ M$. [b]c)[/b] Let $ N$ be the midpoint of the segment $ BD$, and let $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$, $ W^{\prime}$ be the midpoints of the segments $ AB$, $ BC$, $ CD$, $ DA$. Then, the point $ M$ lies on the circumcircles of the triangles $ W^{\prime}X^{\prime}N$ and $ Y^{\prime}Z^{\prime}N$. [hide="Notice"][i]Notice.[/i] This problem has been discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=172417 .[/hide]

Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

Five distinct points $A, B, C, D$ and $E$ lie in this order on a circle of radius $r$ and satisfy $AC = BD = CE = r$. Prove that the orthocentres of the triangles $ACD, BCD$ and $BCE$ are the vertices of a right-angled triangle.

Let $ A$, $ B$ be two fixed points and $ C$ is a variable point on the plane such that $ \angle ACB\equal{}\alpha$ (constant) ($ 0^{\circ}\le \alpha\le 180^{\circ}$). Let $ D$, $ E$, $ F$ be the projections of the incenter $ I$ of triangle $ ABC$ to its sides $ BC$, $ CA$, $ AB$, respectively. Denoted by $ M$, $ N$ the intersections of $ AI$, $ BI$ with $ EF$, respectively. Prove that the length of the segment $ MN$ is constant and the circumcircle of triangle $ DMN$ always passes through a fixed point.

Given a triangle $ABC$. Point $M$ moves along the side $BA$ and point $N$ moves along the side $AC$ beyond point $C$ such that $BM=CN$. Find the geometric locus of the centers of the circles circumscribed around the triangle $AMN$.

Let $P$ be a point inside a triangle $ABC$ such that $\angle PBC = \angle PCA < \angle PAB$. The line $PB$ meets the circumcircle of triangle $ABC$ at a point $E$ (apart from $B$). The line $CE$ meets the circumcircle of triangle $APE$ at a point $F$ (apart from $E$). Show that the ratio $\frac{\left|APEF\right|}{\left|ABP\right|}$ does not depend on the point $P$, where the notation $\left|P_1P_2...P_n\right|$ stands for the area of an arbitrary polygon $P_1P_2...P_n$.

From a point $P$ exterior to a circle $K$, two rays are drawn intersecting $K$ in the respective pairs of points $A, A'$ and $B,B' $. For any other pair of points $C, C'$ on $K$, let $D$ be the point of intersection of the circumcircles of triangles $PAC$ and $PB'C'$ other than point $P$. Similarly, let $D'$ be the point of intersection of the circumcircles of triangles $PA'C'$ and $PBC$ other than point $P$. Prove that the points $P, D$, and $D'$ are collinear.

Let $ ABC $ be a isosceles triangle with $ AC=BC$. Let $ D $ a point on a line $ BA $ such that $ A $ lies between $ B, D $. Let $O_1 $ be the circumcircle of triangle $ DAC $. $ O_1 $ meets $ BC $ at point $ E $. Let $ F $ be the point on $ BC $ such that $ FD $ is tangent to circle $O_1 $, and let $O_2 $ be the circumcircle of $ DBF$. Two circles $O_1 , O_2 $ meet at point $ G ( \ne D) $. Let $ O $ be the circumcenter of triangle $ BEG$. Prove that the line $FG$ is tangent to circle $O$ if and only if $ DG \bot FO$.

Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$. Prove \[ AN \cdot NC = CD \cdot BN. \]

The vertices of two acute-angled triangles lie on the same circle. The Euler circle (nine-point circle) of one of the triangles passes through the midpoints of two sides of the other triangle. Prove that the triangles have the same Euler circle. EDIT by pohoatza (in concordance with Luis' PS): [hide=Alternate/initial version ]Let $ABC$ be a triangle with circumcenter $\Gamma$ and nine-point center $\gamma$. Let $X$ be a point on $\Gamma$ and let $Y$, $Z$ be on $\Gamma$ so that the midpoints of segments $XY$ and $XZ$ are on $\gamma$. Prove that the midpoint of $YZ$ is on $\gamma$.[/hide]

Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$ are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is equal to area of $O_1CO_2$, then find all possible values for $\angle CAA'$

Two points $X, Y$ lie on the arc $BC$ of the circumcircle of $\triangle ABC$ (this arc does not contain $A$) such that $\angle BAX = \angle CAY$ . Let $M$ denotes the midpoint of the chord $AX$ . Show that $BM +CM > AY$ . by Mahan Tajrobekar

Triangle $ ABC$ is scalene with angle $ A$ having a measure greater than 90 degrees. Determine the set of points $ D$ that lie on the extended line $ BC$, for which \[ |AD|\equal{}\sqrt{|BD| \cdot |CD|}\] where $ |BD|$ refers to the (positive) distance between $ B$ and $ D$.

Given is an acute angled triangle $ABC$. A circle going through $B$ and the triangle's circumcenter, $O$, intersects $BC$ and $BA$ at points $P$ and $Q$ respectively. Prove that the intersection of the heights of the triangle $POQ$ lies on line $AC$.