Found problems: 3585
Given triangle $ABC$, squares $ABEF, BCGH, CAIJ$ are constructed externally on side $AB, BC, CA$, respectively. Let $AH \cap BJ = P_1$, $BJ \cap CF = Q_1$, $CF \cap AH = R_1$, $AG \cap CE = P_2$, $BI \cap AG = Q_2$, $CE \cap BI = R_2$. Prove that triangle $P_1 Q_1 R_1$ is congruent to triangle $P_2 Q_2 R_2$.
Given quadrilateral $ABCD$. $AC$ and $BD$ meets at $E$, and $M, N$ are the midpoints of $AC, BD$, respectively. Let the circumcircles of $ABE$ and $CDE$ meets again at $X\neq E$. Prove that $E, M, N, X$ are concyclic.
[i]Proposed by chengbilly[/i]
Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
Given triangle $ABC$. Point $M$ is the midpoint of side $BC$, and point $P$ is the projection of $B$ to the perpendicular bisector of segment $AC$. Line $PM$ meets $AB$ in point $Q$. Prove that triangle $QPB$ is isosceles.
A circle through vertices $A$ and $B$ of triangle $ABC$ meets side $BC$ again at $D$. A circle through $B$ and $C$ meets side $AB$ at $E$ and the first circle again at $F$. Prove that if points $A$, $E$, $D$, $C$ lie on a circle with center $O$ then $\angle BFO$ is right.
Let $ABC$ be an acute triangle such that $\angle BAC > 45^{\circ}$ with circumcenter $O$. A point $P$ is chosen inside triangle $ABC$ such that $A, P, O, B$ are concyclic and the line $BP$ is perpendicular to the line $CP$. A point $Q$ lies on the segment $BP$ such that the line $AQ$ is parallel to the line $PO$.
Prove that $\angle QCB = \angle PCO$.
Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.
Let $ABC$ be an acute angled triangle with $\angle{BAC}=60^\circ$ and $AB > AC$. Let $I$ be the incenter, and $H$ the orthocenter of the triangle $ABC$ . Prove that $2\angle{AHI}= 3\angle{ABC}$.
Let $AA_1$ and $BB_1$ be the altitudes of acute angled triangle $ABC$. Points $K$ and $M$ are midpoints of $AB$ and $A_1B_1$ respectively. Segments $AA_1$ and $KM$ intersect at point $L$. Prove that points $A$, $K$, $L$ and $B_1$ are noncyclic.
[I]Proposed by S. Berlov[/i]
A circle with center $I$ touches sides $AB,BC,CA$ of triangle $ABC$ in points $C_{1},A_{1},B_{1}$. Lines $AI, CI, B_{1}I$ meet $A_{1}C_{1}$ in points $X, Y, Z$ respectively. Prove that $\angle Y B_{1}Z = \angle XB_{1}Z$.
We consider an acute triangle $ABC$. The altitude from $A$ is $AD$, the altitude from $D$ in triangle $ABD$ is $DE,$ and the altitude from $D$ in triangle $ACD$ is $DF$.
a) Prove that the triangles $ABC$ and $AF E$ are similar.
b) Prove that the segment $EF$ and the corresponding segments constructed from the vertices $B$ and $C$ all have the same length.
The incircle of $ \triangle ABC$ touches $ BC$, $ AC$, and $ AB$ at $ A_1$, $ B_1$, and $ C_1$, respectively. The line $ AA_1$ intersects the incircle at $ Q$, again. $ A_1C_1$ and $ A_1B_1$ intersect the line, passing through $ A$ and parallel to $ BC$, at $ P$ and $ R$, respectively. If $ \angle PQC_1 \equal{} 45^\circ$ and $ \angle RQB_1 \equal{} 65^\circ$, then $ \angle PQR$ will be ?
$\textbf{(A)}\ 110^\circ \qquad\textbf{(B)}\ 115^\circ \qquad\textbf{(C)}\ 120^\circ \qquad\textbf{(D)}\ 125^\circ \qquad\textbf{(E)}\ 130^\circ$
Let $ABCD$ an isosceles trapezoid with the longer base $AB$. Denote $I$ the incenter of $\Delta ABC$ and $J$ the excenter relative to the vertex $C$ of $\Delta ACD$. Show that the lines $IJ$ and $AB$ are parallel.
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear.
[i]Proposed by David B. Rush, USA[/i]
Let $ABC$ be a triangle with incentre $I$. The circle through $B$ tangent to $AI$ at $I$ meets side $AB$ again at $P$. The circle through $C$ tangent to $AI$ at $I$ meets side $AC$ again at $Q$. Prove that $PQ$ is tangent to the incircle of $ABC.$
Let $ABC$ be a triangle with $AB < AC$ and let $D{}$ be the other intersection point of the angle bisector of $\angle A$ with the circumcircle of the triangle $ABC$. Let $E{}$ and $F{}$ be points on the sides $AB$ and $AC$ respectively, such that $AE = AF$ and let $P{}$ be the point of intersection of $AD$ and $EF$. Let $M{}$ be the midpoint of $BC{}$. Prove that $AM$ and the circumcircles of the triangles $AEF$ and $PMD$ pass through a common point.
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 $ ABC$ be a triangle, and $ D$ the midpoint of the side $ BC$. On the sides $ AB$ and $ AC$ we consider the points $ M$ and $ N$, respectively, both different from the midpoints of the sides, such that \[ AM^2\plus{}AN^2 \equal{}BM^2 \plus{} CN^2 \textrm{ and } \angle MDN \equal{} \angle BAC.\] Prove that $ \angle BAC \equal{} 90^\circ$.
Let $ABC$ be an acute triangle with $AB=AC$, let $D$ be the midpoint of the side $AC$, and let $\gamma$ be the circumcircle of the triangle $ABD$. The tangent of $\gamma$ at $A$ crosses the line $BC$ at $E$. Let $O$ be the circumcenter of the triangle $ABE$. Prove that midpoint of the segment $AO$ lies on $\gamma$.
[i]Proposed by Sam Bealing, United Kingdom[/i]
Let $M$ be the midpoint of the base $AC$ of an isosceles $\triangle ABC$. Points $E$ and $F$ on the sides $AB$ and $BC$ respectively are chosen so that $AE \neq CF$ and $\angle FMC = \angle MEF = \alpha$.
Determine $\angle AEM$. [i](6 points) [/i]
[i]Maxim Prasolov[/i]
Let $ABC$ be a triangle with circumcircle $\omega$. Let $l_B$ and $l_C$ be two lines through the points $B$ and $C$, respectively, such that $l_B \parallel l_C$. The second intersections of $l_B$ and $l_C$ with $\omega$ are $D$ and $E$, respectively. Assume that $D$ and $E$ are on the same side of $BC$ as $A$. Let $DA$ intersect $l_C$ at $F$ and let $EA$ intersect $l_B$ at $G$. If $O$, $O_1$ and $O_2$ are circumcenters of the triangles $ABC$, $ADG$ and $AEF$, respectively, and $P$ is the circumcenter of the triangle $OO_1O_2$, prove that $l_B \parallel OP \parallel l_C$.
[i]Proposed by Stefan Lozanovski, Macedonia[/i]
Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$.
Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$
A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.
In the acute-angled non-isosceles triangle $ABC$, the height $AH$ is drawn. Points $B_1$ and $C_1$ are marked on the sides $AC$ and $AB$, respectively, so that $HA$ is the angle bisector of $B_1HC_1$ and quadrangle $BC_1B_1C$ is cyclic. Prove that $B_1$ and $C_1$ are feet of the altitudes of triangle $ABC$.
In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$.
[i]Proposed by Davood Vakili, Iran[/i]