Found problems: 3585
Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold:
\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]
Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $AB$.
[i]Proposed by Dominik Burek, Poland[/i]
The incircle of triangle $ABC$, is tangent to sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. The reflection of $F$ with respect to $B$ and the reflection of $E$ with respect to $C$ are $T$ and $S$ respectively. Prove that the incenter of triangle $AST$ is inside or on the incircle of triangle $ABC$.
[i]Proposed by Mehdi E'tesami Fard[/i]
Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$.
(Slovakia)
A convex quadrilateral $ABCD$ is circumscribed about a circle $\omega$. Let $PQ$ be the diameter of $\omega$ perpendicular to $AC$. Suppose lines $BP$ and $DQ$ intersect at point $X$, and lines $BQ$ and $DP$ intersect at point $Y$. Show that the points $X$ and $Y$ lie on the line $AC$.
[i]Géza Kós[/i]
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.
Let $ABC$ be an acute triangle and $D$ an interior point of segment $BC$. Points $E$ and $F$ lie in the half-plane determined by the line $BC$ containing $A$ such that $DE$ is perpendicular to $BE$ and $DE$ is tangent to the circumcircle of $ACD$, while $DF$ is perpendicular to $CF$ and $DF$ is tangent to the circumcircle of $ABD$. Prove that the points $A, D, E$ and $F$ are concyclic.
Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$.
[i]Proposed by Sammy Luo[/i]
Let $ABC$ be a triangle in which $AB=AC$ and let $I$ be its incenter. It is known that $BC=AB+AI$. Let $D$ be a point on line $BA$ extended beyond $A$ such that $AD=AI$. Prove that $DAIC$ is a cyclic quadrilateral.
Let $ABC$ be an acute, non-isosceles triangle with circumcircle $(O)$. $BE, CF$ are the heights of $\triangle ABC$, and $BE, CF$ intersect at $H$. Let $M$ be the midpoint of $AH$, and $K$ be the point on $EF$ such that $HK \perp EF$. A line not going through $A$ and parallel to $BC$ intersects the minor arc $AB$ and $AC$ of $(O)$ at $P$, $Q$, respectively.
Show that the tangent line of $(CQE)$ at $E$, the tangent line of $(BPF)$ at $F$, and $MK$ concur.
$ABCD$ is a cyclic quadrilateral with circumcenter $O$ and circumradius $7$. $AB$ intersects $CD$ at $E$, $DA$ intersects $CB$ at $F$. $OE=13$, $OF=14$. Let $\cos\angle FOE=\dfrac pq$, with $p$, $q$ coprime. Find $p+q$.
Acute-angled triangle $ABC$ is inscribed into circle $\Omega$. Lines tangent to $\Omega$ at $B$ and $C$ intersect at $P$. Points $D$ and $E$ are on $AB$ and $AC$ such that $PD$ and $PE$ are perpendicular to $AB$ and $AC$ respectively. Prove that the orthocentre of triangle $ADE$ is the midpoint of $BC$.
Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$.
Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.
Consider $\triangle MAB$ with a right angle at $A$ and circumcircle $\omega$. Take any chord $CD$ perpendicular to $AB$ such that $A, C, B, D, M$ lie on $\omega$ in this order. Let $AC$ and $MD$ intersect at point $E$, and let $O$ be the circumcenter of $\triangle EMC$. Show that if $J$ is the intersection of $BC$ and $OM$, then $JB = JM$.
[i](Proposed by Matthew Kung Wei Sheng and Ivan Chan Kai Chin)[/i]
The length of the largest side of the cyclic quadrilateral $ABCD$ is $a$, while the radius of the circumcircle of $\vartriangle ACD$ is $1$. Find the smallest possible value for $a$. Which cyclic quadrilateral $ABCD$ gives the value $a$ equal to the smallest value?
In the cyclic quadrilateral $ABCD$, the sides $AB, DC$ meet at $Q$, the sides $AD,BC$ meet at $P, M$ is the midpoint of $BD$, If $\angle APQ=90^o$, prove that $PM$ is perpendicular to $AB$.
Let $ABC$ be a triangle and let $D$ be a point on the segment $BC, D\neq B$ and $D\neq C$. The circle $ABD$ meets the segment $AC$ again at an interior point $E$. The circle $ACD$ meets the segment $AB$ again at an interior point $F$. Let $A'$ be the reflection of $A$ in the line $BC$. The lines $A'C$ and $DE$ meet at $P$, and the lines $A'B$ and $DF$ meet at $Q$. Prove that the lines $AD, BP$ and $CQ$ are concurrent (or all parallel).
Let O be the circumcenter of triangle ABC.
Let H be the orthocenter of triangle ABC.
the perpendicular bisector of AB meet AC,BC at D,E.
the circumcircle of triangle DEH meet AC,BC,OH again at F,G,L.
CH meet FG at T,and ABCT is concyclic.
Prove that LHBC is concyclic.
graph: https://cdn.luogu.com.cn/upload/image_hosting/w6z6mvm4.png
Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.
Two circles intersect at points $A\neq B$. A line passing through $A{}$ intersects the circles again at $C$ and $D$. Let $E$ and $F$ be the midpoints of the arcs $\overarc{BC}$ and $\overarc{BD}$ which do not contain $A{}$ and let $M$ be the midpoint of the segment $CD$. Prove that $ME$ and $MF$ are perpendicular.
Let $A, B,C$ be colinear points in this order, $\omega$ an arbitrary circle passing through $B$ and $C$, and $l$ an arbitrary line different from $BC$, passing through A and intersecting $\omega$ at $M$ and $N$. The bisectors of the angles $\angle CMB$ and $\angle CNB$ intersect $BC$ at $P$ and $Q$. Prove that $AP\cdot AQ = AB \cdot AC$.
Let acute scalene $\Delta ABC$ have circumcircle $\omega$. Let $M$ be the midpoint of $BC$, define $X$ as the other intersection of $AM$ with $\omega$. Let $E,F$ be the feet of altitudes from $B,C$ to $AC, AB$ respectively. Let $Q$ be the second intersection of the circumcircle of $\Delta AEF$ and $\omega$. Let $Y\neq X$ be a point such that $MX=MY$ and $QMXY$ is cyclic. Finally, let $S$ be a point on $BC$ such that $\angle BAS=\angle MAC.$ Prove that the quadrilaterals $BFYS$ and $CEYS$ are cyclic.
Proposed by Kanav Talwar and Malay Mahajan
Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$.
[i]Proposed by Evan Chen[/i]
Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$.
[i]Proposed by Carlos Yuzo Shine, Brazil[/i]
The diagonals of the cyclic quadrilateral $ ABCD$ are intersecting at the point $ E$.
$ K$ and $ M$ are the midpoints of $ AB$ and $ CD$, respectively. Let the points $ L$ on $ BC$ and $ N$ on $ AD$ s.t.
$ EL\perp BC$ and $ EN\perp AD$.Prove that $ KM\perp LN$.
Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.