Found problems: 1389
2008 Czech and Slovak Olympiad III A, 2
Two disjoint circles $W_1(S_1,r_1)$ and $W_2(S_2,r_2)$ are given in the plane. Point $A$ is on circle $W_1$ and $AB,AC$ touch the circle $W_2$ at $B,C$ respectively. Find the loci of the incenter and orthocenter of triangle $ABC$.
2016 Ukraine Team Selection Test, 8
Let $ABC$ be an acute triangle with $AB<BC$. Let $I$ be the incenter of $ABC$, and let $\omega$ be the circumcircle of $ABC$. The incircle of $ABC$ is tangent to the side $BC$ at $K$. The line $AK$ meets $\omega$ again at $T$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the arc $BAC$ of $\omega$. The segment $NT$ intersects the circumcircle of $BIC$ at $P$. Prove that $PM\parallel AK$.
1990 Canada National Olympiad, 3
The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral $q$. Show that the sum of the lengths of each pair of opposite sides of $q$ is equal.
2004 Turkey Junior National Olympiad, 1
Let $[AD]$ and $[CE]$ be internal angle bisectors of $\triangle ABC$ such that $D$ is on $[BC]$ and $E$ is on $[AB]$. Let $K$ and $M$ be the feet of perpendiculars from $B$ to the lines $AD$ and $CE$, respectively. If $|BK|=|BM|$, show that $\triangle ABC$ is isosceles.
2008 Saint Petersburg Mathematical Olympiad, 1
The graph $y=x^2+ax+b$ intersects any of the two axes at points $A$, $B$, and $C$. The incenter of triangle $ABC$ lies on the line $y=x$. Prove that $a+b+1=0$.
2012 Sharygin Geometry Olympiad, 19
Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.
1988 IMO Shortlist, 23
Let $ Q$ be the centre of the inscribed circle of a triangle $ ABC.$ Prove that for any point $ P,$
\[ a(PA)^2 \plus{} b(PB)^2 \plus{} c(PC)^2 \equal{} a(QA)^2 \plus{} b(QB)^2 \plus{} c(QC)^2 \plus{} (a \plus{} b \plus{} c)(QP)^2,
\]
where $ a \equal{} BC, b \equal{} CA$ and $ c \equal{} AB.$
2017 Turkey Team Selection Test, 3
At the $ABC$ triangle the midpoints of $BC, AC, AB$ are respectively $D, E, F$ and the triangle tangent to the incircle at $G$, $H$ and $I$ in the same order.The midpoint of $AD$ is $J$. $BJ$ and $AG$ intersect at point $K$. The $C-$centered circle passing through $A$ cuts the $[CB$ ray at point $X$. The line passing through $K$ and parallel to the $BC$ and $AX$ meet at $U$. $IU$ and $BC$ intersect at the $P$ point. There is $Y$ point chosen at incircle. $PY$ is tangent to incircle at point $Y$. Prove that $D, E, F, Y$ are cyclic.
2010 USAJMO, 6
Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.
2020 Iran Team Selection Test, 4
Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent.
[i]Proposed by Alireza Dadgarnia[/i]
2017 District Olympiad, 2
Let $ ABC $ be a triangle in which $ O,I, $ are the circumcenter, respectively, incenter. The mediators of $ IA,IB,IC, $ form a triangle $ A_1B_1C_1. $ Show that $ \overrightarrow{OI}=\overrightarrow{OA_1} +\overrightarrow{OA_2} +\overrightarrow{OA_3} . $
2011 USA Team Selection Test, 1
In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Compute $HQ/HR$.
[i]Proposed by Zuming Feng[/i]
2022 Junior Balkan Team Selection Tests - Moldova, 9
The circle inscribed in the triangle $ABC$ with center $I$ touches the side $BC$ at the point $D$. The line $DI$ intersects the side $AC$ at the point $M$. The tangent from $M$ to the inscribed circle, different from $AC$, intersects the side $AB$ at the point $N$. The line $NI$ intersects the side $BC$ at the point $P$. Prove that $AB = BP$.
2018 Sharygin Geometry Olympiad, 8
Let $I$ be the incenter of fixed triangle $ABC$, and $D$ be an arbitrary point on $BC$. The perpendicular bisector of $AD$ meets $BI,CI$ at $F$ and $E$ respectively. Find the locus of orthocenters of $\triangle IEF$ as $D$ varies.
2023 Yasinsky Geometry Olympiad, 2
Let $I$ be the incenter of triangle $ABC$. $K_1$ and $K_2$ are the points on $BC$ and $AC$ respectively, at which the inscribed circle is tangent. Using a ruler and a compass, find the center of the inscribed circle for triangle $CK_1K_2$ in the minimal possible number of steps (each step is to draw a circle or a line).
(Hryhorii Filippovskyi)
2011 Costa Rica - Final Round, 6
Let $ABC$ be a triangle. The incircle of $ABC$ touches $BC,AC,AB$ at $D,E,F$, respectively. Each pair of the incircles of triangles $AEF, BDF,CED$ has two pair of common external tangents, one of them being one of the sides of $ABC$. Show that the other three tangents divide triangle $DEF$ into three triangles and three parallelograms.
2005 Vietnam Team Selection Test, 1
Let $(I),(O)$ be the incircle, and, respectiely, circumcircle of $ABC$. $(I)$ touches $BC,CA,AB$ in $D,E,F$ respectively. We are also given three circles $\omega_a,\omega_b,\omega_c$, tangent to $(I),(O)$ in $D,K$ (for $\omega_a$), $E,M$ (for $\omega_b$), and $F,N$ (for $\omega_c$).
[b]a)[/b] Show that $DK,EM,FN$ are concurrent in a point $P$;
[b]b)[/b] Show that the orthocenter of $DEF$ lies on $OP$.
2024 Lusophon Mathematical Olympiad, 3
Let $ABC$ be a triangle with incentre $I$. A line $r$ that passes through $I$ intersects the circumcircles of triangles $AIB$ and $AIC$ at points $P$ and $Q$, respectively. Prove that the circumcentre of triangle $APQ$ is on the circumcircle of $ABC$.
2015 Bosnia Herzegovina Team Selection Test, 6
Let $D$, $E$ and $F$ be points in which incircle of triangle $ABC$ touches sides $BC$, $CA$ and $AB$, respectively, and let $I$ be a center of that circle.Furthermore, let $P$ be a foot of perpendicular from point $I$ to line $AD$, and let $M$ be midpoint of $DE$. If $\{N\}=PM\cap{AC}$, prove that $DN \parallel EF$
2020 Durer Math Competition Finals, 4
Let $ABC$ be a scalene triangle and its incentre $I$. Denote by $F_A$ the intersection of the line $BC$ and the perpendicular to the angle bisector at $A$ through $I$. Let us define points $F_B$ and $F_C$ in a similar manner. Prove that points $F_A, F_B$ and $F_C$ are collinear.
2021 Sharygin Geometry Olympiad, 13
In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.
2013 Stanford Mathematics Tournament, 1
A circle of radius $2$ is inscribed in equilateral triangle $ABC$. The altitude from $A$ to $BC$ intersects the circle at a point $D$ not on $BC$. $BD$ intersects the circle at a point $E$ distinct from $D$. Find the length of $BE$.
2012 Junior Balkan Team Selection Tests - Romania, 5
Let $ABC$ be a triangle and $A', B', C'$ the points in which its incircle touches the sides $BC, CA, AB$, respectively. We denote by $I$ the incenter and by $P$ its projection onto $AA' $. Let $M$ be the midpoint of the line segment $[A'B']$ and $N$ be the intersection point of the lines $MP$ and $AC$. Prove that $A'N $is parallel to $B'C'$
2013 India Regional Mathematical Olympiad, 5
In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.
2009 Switzerland - Final Round, 5
Let $ABC$ be a triangle with $AB \ne AC$ and incenter $I$. The incircle touches $BC$ at $D$. Let $M$ be the midpoint of $BC$ . Show that the line $IM$ bisects segment $AD$ .