Found problems: 63
2016 Saudi Arabia BMO TST, 2
Let $I_a$ be the excenter of triangle $ABC$ with respect to $A$. The line $AI_a$ intersects the circumcircle of triangle ABC at $T$. Let $X$ be a point on segment $TI_a$ such that $X I_a^2 = XA \cdot X T$ The perpendicular line from $X$ to $BC$ intersects $BC$ at $A'$. Define $B'$ and $C'$ in the same way. Prove that $AA',BB'$ and $CC'$ are concurrent.
Ukrainian TYM Qualifying - geometry, 2020.10
In triangle $ABC$, point $I$ is the center, point $I_a$ is the center of the excircle tangent to the side $BC$. From the vertex $A$ inside the angle $BAC$ draw rays $AX$ and $AY$. The ray $AX$ intersects the lines $BI$, $CI$, $BI_a$, $CI_a$ at points $X_1$, $...$, $X_4$, respectively, and the ray $AY$ intersects the same lines at points $Y_1$, $...$, $Y_4$ respectively. It turned out that the points $X_1,X_2,Y_1,Y_2$ lie on the same circle. Prove the equality $$\frac{X_1X_2}{X_3X_4}= \frac{Y_1Y_2}{Y_3Y_4}.$$
2021 SAFEST Olympiad, 3
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
2021 Saudi Arabia Training Tests, 12
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$, ex-center in angle $A$ is $J$. Denote $D$ as the tangent point of $(I)$ on $BC$ and the angle bisector of angle $A$ cuts $BC$, $(O)$ respectively at $E, F$. The circle $(DEF )$ meets $(O)$ again at $T$. Prove that $AT$ passes through an intersection of $(J)$ and $(DEF )$.
2021 Taiwan TST Round 3, G
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
Kharkiv City MO Seniors - geometry, 2021.11.4
In the triangle $ABC$, the segment $CL$ is the angle bisector. The $C$-exscribed circle with center at the point $ I_c$ touches the side of the $AB$ at the point $D$ and the extension of sides $CA$ and $CB$ at points $P$ and $Q$, respectively. It turned out that the length of the segment $CD$ is equal to the radius of this exscribed circle. Prove that the line $PQ$ bisects the segment $I_CL$.
2024 Alborz Mathematical Olympiad, P4
In triangle \( ABC \), let \( I \) be the \( A \)-excenter. Points \( X \) and \( Y \) are placed on line \( BC \) such that \( B \) is between \( X \) and \( C \), and \( C \) is between \( Y \) and \( B \). Moreover, \( B \) and \( C \) are the contact points of \( BC \) with the \( A \)-excircle of triangles \( BAY \) and \( AXC \), respectively. Let \( J \) be the \( A \)-excenter of triangle \( AXY \), and let \( H' \) be the reflection of the orthocenter of triangle \( ABC \) with respect to its circumcenter. Prove that \( I \), \( J \), and \( H' \) are collinear.
Proposed by Ali Nazarboland
2019 Saudi Arabia BMO TST, 2
Let $I $be the incenter of triangle $ABC$and $J$ the excenter of the side $BC$: Let $M$ be the midpoint of $CB$ and $N$ the midpoint of arc $BAC$ of circle $(ABC)$. If $T$ is the symmetric of the point $N$ by the point $A$, prove that the quadrilateral $JMIT$ is cyclic
2014 Oral Moscow Geometry Olympiad, 1
In triangle $ABC, \angle A= 45^o, BH$ is the altitude, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.
2006 MOP Homework, 4
Let $ABC$ be a right triangle with$ \angle A = 90^o$. Point $D$ lies on side $BC$ such that $\angle BAD = \angle CAD$. Point $I_a$ is the excenter of the triangle opposite $A$. Prove that $\frac{AD}{DI_a } \le \sqrt{2} -1$
2011 Saudi Arabia IMO TST, 2
In triangle $ABC$, let $I_a$ $,I_b$, $I_c$ be the centers of the excircles tangent to sides $BC$, $CA$, $AB$, respectively. Let $P$ and $Q$ be the tangency points of the excircle of center $I_a$ with lines $AB$ and $AC$. Line $PQ$ intersects $I_aB$ and $I_aC$ at $D$ and $E$. Let $A_1$ be the intersection of $DC$ and $BE$. In an analogous way we define points $B_1$ and $C_1$. Prove that $AA_1$, $BB_1$ , $CC_1$ are concurrent.
2016 Germany National Olympiad (4th Round), 3
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 $I_aCB$.
2004 India IMO Training Camp, 1
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]