Found problems: 3585
Let $ABC$ be a scalene triangle with $\angle BCA = 90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK = BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL = AC$. The circumcircle of triangle $DKL$ intersects segment $AB$ at a second point $T$ (other than $D$). Prove that $\angle ACT = \angle BCT$.
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. The points $D,E$ and $F$ on the sides $BC,CA$ and $AB$ respectively are such that $BD+BF=CA$ and $CD+CE=AB$. The circumcircles of the triangles $BFD$ and $CDE$ intersect at $P \neq D$. Prove that $OP=OI$.
A point $C$ lies on the bisector of an acute angle with vertex $S$. Let $P$, $Q$ be the projections of $C$ to the sidelines of the angle. The circle centered at $C$ with radius $PQ$ meets the sidelines at points $A$ and $B$ such that $SA\ne SB$. Prove that the circle with center $A$ touching $SB$ and the circle with center $B$ touching $SA$ are tangent.
Proposed by: A.Zaslavsky
Let $a$ be real number with $\sqrt{2}<a<2$, and let $ABCD$ be a convex cyclic quadrilateral whose circumcentre $O$ lies in its interior. The quadrilateral's circumcircle $\omega$ has radius $1$, and the longest and shortest sides of the quadrilateral have length $a$ and $\sqrt{4-a^2}$, respectively. Lines $L_A,L_B,L_C,L_D$ are tangent to $\omega$ at $A,B,C,D$, respectively.
Let lines $L_A$ and $L_B$, $L_B$ and $L_C$,$L_C$ and $L_D$,$L_D$ and $L_A$ intersect at $A',B',C',D'$ respectively. Determine the minimum value of $\frac{S_{A'B'C'D'}}{S_{ABCD}}$.
Let $A,B,P$ be three points on a circle. Prove that if $a,b$ are the distances from $P$ to the tangents at $A,B$ respectively, and $c$ is the distance from $P$ to the chord $AB$, then $c^2 =ab$.
In $\triangle ABC$ with $AB\neq{AC}$ let $M$ be the midpoint of $AB$, let $K$ be the midpoint of the arc $BAC$ in the circumcircle of $\triangle ABC$, and let the perpendicular bisector of $AC$ meet the bisector of $\angle BAC$ at $P$ . Prove that $A, M, K, P$ are concyclic.
Let $ABC$ be a acute angled triangle and let $AD,BE,CF$ be its altitudes. $X \not=A,B$ and $Y \not=A,C$ lie on sides $AB$ and $AC$, respectively, so that $ADXY$ is a cyclic quadrilateral. Let $H$ be the orthocenter of triangle $AXY$.
Prove that $H$ lies on line $EF$.
Let $ABC$ be a triangle such that $BC>AB$, $L$ be the internal angle bisector of $\angle ABC$. Let $P,Q$ be the feet from $A,C$ to $L$, respectively. Suppose $M,N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively. Let $O$ be the circumcenter of triangle $PQM$, and the circumcircle intersects $AC$ at point $H$. Prove that $O,M,N,H$ are concyclic.
Let the set $S = \{P_1, P_2, \cdots, P_{12}\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called communal if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)
In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .
A circle $\omega$ with diameter $AK$ is given. The point $M$ lies in the interior of the circle, but not on $AK$. The line $AM$ intersects $\omega$ in $A$ and $Q$. The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to $AK$, at $P$. The point $L$ lies on $\omega$, and is such that $PL$ is tangent to $\omega$ and $L\neq Q$.
Show that $K, L$, and $M$ are collinear.
Let $ABCD$ be a cyclic quadrilateral. Prove that \[ |AB - CD| + |AD - BC| \geq 2|AC - BD|. \]
An acute triangle $ABC$ is given and let $H$ be its orthocenter. Let $\omega$ be the circle through $B$, $C$ and $H$, and let $\Gamma$ be the circle with diameter $AH$. Let $X\neq H$ be the other intersection point of $\omega$ and $\Gamma$, and let $\gamma$ be the reflection of $\Gamma$ over $AX$.
Suppose $\gamma$ and $\omega$ intersect again at $Y\neq X$, and line $AH$ and $\omega$ intersect again at $Z \neq H$. Show that the circle through $A,Y,Z$ passes through the midpoint of segment $BC$.
There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that
\[
\angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ}
\]
if and only if the diagonals $ AC$ and $ BD$ are perpendicular.
[i]Proposed by Dusan Djukic, Serbia[/i]
In the convex pentagon $ABCDE$ we know that $AB=DE, BC=EA$ but $AB \neq EA$.
$B,C,D,E$ are concyclic .
Prove that $A,B,C,D$ are concyclic if and only if $AC=AD.$
Let $ABCD$ be a convex quadrilateral and let the diagonals $AC$ and $BD$ intersect at $O$. Let $I_1, I_2, I_3, I_4$ be respectively the incentres of triangles $AOB, BOC, COD, DOA$. Let $J_1, J_2, J_3, J_4$ be respectively the excentres of triangles $AOB, BOC, COD, DOA$ opposite $O$. Show that $I_1, I_2, I_3, I_4$ lie on a circle if and only if $J_1, J_2, J_3, J_4$ lie on a circle.
Points $E, F$ are selected on sides $AC, AB$ respectively of triangle $ABC$ with $AC=AB$ so that $AE = BF$. Point $D$ is chosen so that $D, A$ are in the same halfplane with respect to line $EF$, and $\triangle DFE \sim \triangle ABC$. Lines $EF, BC$ intersect at point $K$. Prove that the line $DK$ is tangent to the circumscribed circle of $\triangle ABC$.
[i]Proposed by Fedir Yudin[/i]
Let $H$ be the orthocentre of an acute triangle $ABC$ with $BC > AC$, inscribed in a circle $\Gamma$. The circle with centre $C$ and radius $CB$ intersects $\Gamma$ at the point $D$, which is on the arc $AB$ not containing $C$. The circle with centre $C$ and radius $CA$ intersects the segment $CD$ at the point $K$. The line parallel to $BD$ through $K$, intersects $AB$ at point $L$. If $M$ is the midpoint of $AB$ and $N$ is the foot of the perpendicular from $H$ to $CL$, prove that the line $MN$ bisects the segment $CH$.
Let $ABC$ be a triangle with incenter $I$. Let $U$, $V$ and $W$ be the intersections of the angle bisectors of angles $A$, $B$, and $C$ with the incircle, so that $V$ lies between $B$ and $I$, and similarly with $U$ and $W$. Let $X$, $Y$, and $Z$ be the points of tangency of the incircle of triangle $ABC$ with $BC$, $AC$, and $AB$, respectively. Let triangle $UVW$ be the [i]David Yang triangle[/i] of $ABC$ and let $XYZ$ be the [i]Scott Wu triangle[/i] of $ABC$. Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if $ABC$ is equilateral.
[i]Proposed by Owen Goff[/i]
Let $ABC$ be an acute, scalene triangle with circumcenter $O$, circumcircle $(O)$, orthocenter $H$. Line $AH$ meets $(O)$ again at $D \neq A$. Let $E, F$ be the midpoint of segments $AB, AC$ respectively. The line through $H$ and perpendicular to $HF$ meets line $BC$ at $K$.
a) Line $DK$ meets $(O)$ again at $Y \neq D$. Prove that the intersection of line $BY$ and the perpendicular bisector of $BK$ lies on the circumcircle of triangle $OFY$.
b) The line through $H$ and perpendicular to $HE$ meets line $BC$ at $L$. Line $DL$ meets $(O)$ again at $Z \neq D$. Let $M$ be the intersection of lines $BZ, OE$; $N$ be the intersection of lines $CY, OF$; $P$ be the intersection of lines $BY, CZ$. Let $T$ be the intersection of lines $YZ, MN$ and $d$ be the line through $T$ and perpendicular to $OA$. Prove that $d$ bisects $AP$.
Given a $ \triangle ABC $ and a point $ P. $ Let $ O$, $D$, $E$, $F $ be the circumcenter of $ \triangle ABC$, $\triangle BPC$, $\triangle CPA$, $\triangle APB, $ respectively and let $ T $ be the intersection of $ BC $ with $ EF. $ Prove that the reflection of $ O $ in $ EF $ lies on the perpendicular from $ D $ to $ PT. $
[i]Proposed by Telv Cohl[/i]
Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other.
$\emph{Slovakia}$
A tangent line to circumcircle of acute triangle $ABC$ ($AC>AB$) at $A$ intersects with the extension of $BC$ at $P$. $O$ is the circumcenter of triangle $ABC$.Point $X$ lying on $OP$ such that $\measuredangle AXP=90^\circ$.Points $E$ and $F$ lying on $AB$ and $AC$,respectively,and they are in one side of line $OP$ such that $ \measuredangle EXP=\measuredangle ACX $ and $\measuredangle FXO=\measuredangle ABX $.
$K$,$L$ are points of intersection $EF$ with circumcircle of triangle $ABC$.prove that $OP$ is tangent to circumcircle of triangle $KLX$.
Author:Mehdi E'tesami Fard , Iran
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
Let $ABC$ be an acute scalene triangle and let $\omega$ be its Euler circle. The tangent $t_A$ of $\omega$ at the foot of the height $A$ of the triangle ABC, intersects the circle of diameter $AB$ at the point $K_A$ for the second time. The line determined by the feet of the heights $A$ and $C$ of the triangle $ABC$ intersects the lines $AK_A$ and $BK_A$ at the points $L_A$ and $M_A$, respectively, and the lines $t_A$ and $CM_A$ intersect at the point $N_A$.
Points $K_B, L_B, M_B, N_B$ and $K_C, L_C, M_C, N_C$ are defined similarly for $(B, C, A)$ and $(C, A, B)$ respectively. Show that the lines $L_AN_A, L_BN_B,$ and $L_CN_C$ are concurrent.