Found problems: 3585
Let $k$ be incircle of acute triangle $ABC$, $AC\neq BC$, and $l$ be excircle that touches $AB$. Line $p$ through the $C$ is orthogonal to $AB$, $p\cap k = \{X, Y\}$ , $p\cap l = \{Z, T\}$ and the point arrangement is $X-Y-Z-T$. Circle $m$ through $X$ and $Z$ intersects $AB$ at $D$ and $E$. Prove that points $D,Y,E,T$ are concyclic.
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $O$. The circumcircles of triangles $AOB$ and $COD$ intersect again at $K$. Point $L$ is such that the triangles $BLC$ and $AKD$ are similar and equally oriented. Prove that if the quadrilateral $BLCK$ is convex, then it is tangent [has an incircle].
Let $ABC$ be a triangle. A point $P$ is chosen inside $\triangle ABC$ such that $\angle BPC+\angle BAC=180^{\circ}$. The lines $AP,BP,CP$ intersect $BC,CA,AB$ at $P_A,P_B,P_C$ respectively. Let $X_A$ be the second intersection of the circumcircles of $\triangle ABC$ and $\triangle AP_BP_C$ . Similarly define $X_B,X_C$. Let $B'$ be the intersection of lines $AX_A,CX_C$, and let $C'$ be the intersection of lines $AX_A,BX_B$. Prove that lines $BB'$ and $CC'$ intersect on the circumcircle of $\triangle AP_BP_C$.
Let $ ABC$ be an equilateral triangle of center $ O$, and $ M\in BC$. Let $ K,L$ be projections of $ M$ onto the sides $ AB$ and $ AC$ respectively. Prove that line $ OM$ passes through the midpoint of the segment $ KL$.
$ABCDE$ is a regular pentagon. Two circles $C_1$ and $C_2$ are drawn through $B$ with centers $A$ and $C$ respectively. Let the other intersection of $C_1$ and $C_2$ be $P$. The circle with center $P$ which passes through $E$ and $D$ intersects $C_2$ at $X$ and $AE$ at $Y$. Prove that $AX = AY$.
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.
Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear.
[i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]
Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. The internal angle bisectors of \(\angle DAB\), \(\angle ABC\), \(\angle BCD\), \(\angle CDA\) create a convex quadrilateral $Q_1$. The external bisectors of the same angles create another convex quadrilateral $Q_2$.
Prove $Q_1$, $Q_2$ are cyclic, and that $O$ is the midpoint of their circumcenters.
Given a pointed triangle $ABC$ with orthocenter $H$. Let $D$ be the point such that the quadrilateral $AHCD$ is parallelogram. Let $p$ be the perpendicular to the direction $AB$ through the midpoint $A_1$ of the side $BC$. Denote the intersection of the lines $p$ and $AB$ with $E$, and the midpoint of the length $A_1E$ with $F$. The point where the parallel to the line $BD$ through point $A$ intersects $p$ denote by $G$. Prove that the quadrilateral $AFA_1C$ is cyclic if and only if the lines $BF$ passes through the midpoint of the length $CG$.
Let $ ABC$ be a triangle, and $ H$ its orthocenter. Let $ P$ be a point on the circumcircle of triangle $ ABC$ (distinct from the vertices $ A$, $ B$, $ C$), and let $ E$ be the foot of the altitude of triangle $ ABC$ from the vertex $ B$. Let the parallel to the line $ BP$ through the point $ A$ meet the parallel to the line $ AP$ through the point $ B$ at a point $ Q$. Let the parallel to the line $ CP$ through the point $ A$ meet the parallel to the line $ AP$ through the point $ C$ at a point $ R$. The lines $ HR$ and $ AQ$ intersect at some point $ X$. Prove that the lines $ EX$ and $ AP$ are parallel.
Let there be cyclic quadrilateral $ABCD$ with $L$ as the midpoint of $AB$ and $M$ as the midpoint of $CD$. Let $AC \cap BD = E$, and let rays $AB$ and $DC$ meet again at $F$. Let $LM \cap DE = P$. Let $Q$ be the foot of the perpendicular from $P$ to $EM$. If the orthocenter of $\triangle FLM$ is $E$, prove the following equality.
$$\frac{EP^2}{EQ} = \frac{1}{2} \left( \frac{BD^2}{DF} - \frac{BC^2}{CF} \right)$$
Let $T$ be a point outside circle $\omega$ centered at $O$. Tangents from $T$ to $\omega$ touch $\omega$ at $A;B$. Line $TO$ intersects bigger $AB$ arc at $C$.The line drawn from $T$ parallel to $AC$ intersects $CB$ at $E$. Ray $TE$ intersects small $BC$ arc at $F$. Prove that the circumcircle of $OEF$ is tangent to $\omega$.
Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic.
[i]Proposed by John Cuya, Peru[/i]
Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic.
[i]Proposed by John Cuya, Peru[/i]
Let $P$ be a point in parallelogram $ABCD$ such that $$PA \cdot PC + PB \cdot PD = AB \cdot BC.$$ Prove that the reflections of $P$ over lines $AB$, $BC$, $CD$, and $DA$ are concyclic.
Let $ABC$ be a triangle with $AB<AC$ and incenter $I.$ A point $D$ lies on segment $AC$ such that $AB=AD,$ and the line $BI$ intersects $AC$ at $E.$ Suppose the line $CI$ intersects $BD$ at $F,$ and $G$ lies on segment $DI$ such that $FD=FG.$ Prove that the lines $AG$ and $EF$ intersect on the circumcircle of triangle $CEI.$ \\
Proposed by Avan Lim Zenn Ee, Malaysia
Point $P$ is the intersection of diagonals $AC,BD$ of the trapezoid $ABCD$ with $AB \parallel CD$. Reflections of the lines $AD$ and $BC$ into the internal angle bisectors of $\angle PDC$ and $\angle PCD$ intersects the circumcircles of $\bigtriangleup APD$ and $\bigtriangleup BPC$ at $D'$ and $C'$. Line $C'A$ intersects the circumcircle of $\bigtriangleup BPC$ again at $Y$ and $D'C$ intersects the circumcricle of $\bigtriangleup APD$ again at $X$. Prove that $P,X,Y$ are collinear.
[i]Proposed by Iman Maghsoudi - Iran[/i]
Let $ABC$ be an acute triangle and $H$ its orthocenter. Let $D$ be the intersection of the altitude from $A$ to $BC$. Let $M$ and $N$ be the midpoints of $BH$ and $CH$, respectively. Let the lines $DM$ and $DN$ intersect $AB$ and $AC$ at points $X$ and $Y$ respectively. If $P$ is the intersection of $XY$ with $BH$ and $Q$ the intersection of $XY$ with $CH$, show that $H, P, D, Q$ lie on a circumference.
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let the lines $AB$ and $CD$ intersect at $P$, and the lines $AD$ and $BC$ intersect at $Q$. Let then the circumcircle of the triangle $\triangle APQ$ intersect $\Omega$ at $R \neq A$. Prove that the line $CR$ goes through the midpoint of the segment $PQ$.
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.
Let $ABCD$ be a convex cyclic quadrilateral with circumcircle $\omega$. Let $BA$ produced beyond $A$ meet $CD$ produced beyond $D$, at $L$. Let $\ell$ be a line through $L$ meeting $AD$ and $BC$ at $M$ and $N$ respectively, so that $M,D$ (respectively $N,C$) are on opposite sides of $A$ (resp. $B$). Suppose $K$ and $J$ are points on the arc $AB$ of $\omega$ not containing $C,D$ so that $MK, NJ$ are tangent to $\omega$. Prove that $K,J,L$ are collinear.
[i]Proposed by Rijul Saini[/i]
$In$ $triangle$ $ABC$ $bisectors$ $AA_1$, $BB_1$ $and$ $CC_1$ $are$ $drawn$. $Bisectors$ $AA_1$ $and$ $CC_1$ $intersect$ $segments$ $C_1B_1$ $and$ $B_1A_1$ $at$ $points$ $M$ $and$ $N$, $respectively$. $Prove$ $that$ $\angle$$MBB_1$ = $\angle$$NBB_1$.
Let $ABC$ be an acute, nonisosceles triangle with inscribe in a circle $(O)$ and has orthocenter $H$. Denote $M,N,P$ as the midpoints of sides $BC,CA,AB$ and $D,E,F$ as the feet of the altitudes from vertices $A,B,C$ of triangle $ABC$. Let $K$ as the reflection of $H$ through $BC$. Two lines $DE,MP$ meet at $X$; two lines $DF,MN$ meet at $Y$.
a) The line $XY$ cut the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic.
b) Two lines $KE,KF$ cuts $(O)$ second time at $S,T$. Prove that $BS,CT,XY$ are concurrent.
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.
Let $ABC$ be an acute-angled triangle, and let $D, E$ and $K$ be the midpoints of its sides $AB, AC$ and $BC$ respectively. Let $O$ be the circumcentre of triangle $ABC$, and let $M$ be the foot of the perpendicular from $A$ on the line $BC$. From the midpoint $P$ of $OM$ we draw a line parallel to $AM$, which meets the lines $DE$ and $OA$ at the points $T$ and $Z$ respectively. Prove that:
(a) the triangle $DZE$ is isosceles
(b) the area of the triangle $DZE$ is given by the formula
\[E_{DZE}=\frac{BC\cdot OK}{8}\]