Olympiad problems

2021 Sharygin Geometry Olympiad, 14

Tags: geometry , excircle

Let $\gamma_A, \gamma_B, \gamma_C$ be excircles of triangle $ABC$, touching the sides $BC$, $CA$, $AB$ respectively. Let $l_A$ denote the common external tangent to $\gamma_B$ and $\gamma_C$ distinct from $BC$. Define $l_B, l_C$ similarly. The tangent from a point $P$ of $l_A$ to $\gamma_B$ distinct from $l_A$ meets $l_C$ at point $X$. Similarly the tangent from $P$ to $\gamma_C$ meets $l_B$ at $Y$. Prove that $XY$ touches $\gamma_A$.

2014 Ukraine Team Selection Test, 4

Tags: geometry , circumcircle , excircle , circumcenter

The $A$-excircle of the triangle $ABC$ touches the side $BC$ at point $K$. The circumcircles of triangles $AKB$ and $AKC$ intersect for the second time with the bisector of angle $A$ at points $X$ and $Y$ respectively. Let $M$ be the midpoint of $BC$. Prove that the circumcenter of triangle $XYM$ lies on $BC$.

2018 India IMO Training Camp, 2

Tags: homothety , power of a point , excircle , geometry , tangent circle

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

2018 Brazil Team Selection Test, 3

Tags: homothety , power of a point , excircle , geometry , tangent circle

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

1966 IMO Shortlist, 19

Tags: geometry , triangle , construction , excircle

Construct a triangle given the radii of the excircles.

2021 Adygea Teachers' Geometry Olympiad, 3

Tags: geometry , excircle

In a triangle, one excircle touches side $AB$ at point $C_1$ and the other touches side $BC$ at point $A_1$. Prove that on the straight line $A_1C_1$ the constructed excircles cut out equal segments.

2010 Ukraine Team Selection Test, 2

Tags: geometry , cyclic quadrilateral , collinear , incircle , excircle

Let $ABCD$ be a quadrilateral inscribled in a circle with the center $O, P$ be the point of intersection of the diagonals $AC$ and $BD$, $BC\nparallel AD$. Rays $AB$ and $DC$ intersect at the point $E$. The circle with center $I$ inscribed in the triangle $EBC$ touches $BC$ at point $T_1$. The $E$-excircle with center $J$ in the triangle $EAD$ touches the side $AD$ at the point T$_2$. Line $IT_1$ and $JT_2$ intersect at $Q$. Prove that the points $O, P$, and $Q$ lie on a straight line.

2010 IFYM, Sozopol, 5

Tags: geometry , excircle

We are given $\Delta ABC$, for which the excircle to side $BC$ is tangent to the continuations of $AB$ and $AC$ in points $E$ and $F$ respectively. Let $D$ be the reflection of $A$ in line $EF$. If it is known that $\angle BAC=2\angle BDC$, then determine $\angle BAC$.

2018 Oral Moscow Geometry Olympiad, 3

Tags: geometry , circumcircle , excircle , parallel

On the extensions of sides $CA$ and $AB$ of triangle $ABC$ beyond points $A$ and $B$, respectively, the segments $AE = BC$ and $BF = AC$ are drawn. A circle is tangent to segment $BF$ at point $N$, side $BC$ and the extension of side $AC$ beyond point $C$. Point $M$ is the midpoint of segment $EF$. Prove that the line $MN$ is parallel to the bisector of angle $A$.

2021 Adygea Teachers' Geometry Olympiad, 2

Tags: geometry , incircle , isosceles , excircle

In triangle $ABC$, the incircle touches the side $AC$ at point $B_1$ and one excircle is touching the same side at point $B_2$. It is known that the segments $BB_1$ and $BB_2$ are equal. Is it true that $\vartriangle ABC$ is isosceles?

2010 Sharygin Geometry Olympiad, 5

Tags: geometry , incircle , angle chasing , excircle

The incircle of a right-angled triangle $ABC$ ($\angle ABC =90^o$) touches $AB, BC, AC$ in points $C_1, A_1, B_1$, respectively. One of the excircles touches the side $BC$ in point $A_2$. Point $A_0$ is the circumcenter or triangle $A_1A_2B_1$, point $C_0$ is defined similarly. Find angle $A_0BC_0$.

2018 Azerbaijan IMO TST, 2

Tags: homothety , power of a point , excircle , geometry , tangent circle

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

Indonesia MO Shortlist - geometry, g10

Tags: incenter , excenter , excircle , geometry

Given a triangle $ABC$ with incenter $I$ . It is known that $E_A$ is center of the ex-circle tangent to $BC$. Likewise, $E_B$ and $E_C$ are the centers of the ex-circles tangent to $AC$ and $AB$, respectively. Prove that $I$ is the orthocenter of the triangle $E_AE_BE_C$.

2007 Silk Road, 2

Tags: incircle , excircle , tangent circle , geometry

Let $\omega$ be the incircle of triangle $ABC$ touches $BC$ at point $K$ . Draw a circle passing through points $B$ and $C$ , and touching $\omega$ at the point $S$ . Prove that $S K$ passes through the center of the exscribed circle of triangle $A B C$ , tangent to side $B C$ .

2024 Myanmar IMO Training, 8

Tags: geometry , config geo , symmedian , excircle

Let $ABC$ be a triangle and let $X$ and $Y$ be points on the $A$-symmedian such that $AX = XB$ and $AY = YC$. Let $BX$ and $CY$ meet at $Z$. Let the $Z$-excircle of triangle $XYZ$ touch $ZX$ and $ZY$ at $E$ and $F$. Show that $A$, $E$, $F$ are collinear.

2015 IFYM, Sozopol, 1

Tags: geometry , excircle

Let $AA_1$ be an altitude in $\Delta ABC$. Let $H_a$ be the orthocenter of the triangle with vertices the tangential points of the excircle to $\Delta ABC$, opposite to $A$. The points $B_1$, $C_1$, $H_b$, and $H_c$ are defined analogously. Prove that $A_1 H_a$, $B_1 H_b$, and $C_1 H_c$ are concurrent.

2016 APMC, 4

Tags: excircle , apmc , tangent circle , geometry

Let $ABC$ be a triangle with $AB\neq AC$. Let the excircle $\omega$ opposite $A$ touch $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Suppose $X$ and $Y$ are points on the segments $AC$ and $AB$, respectively, such that $XY$ and $BC$ are parallel, and let $\Gamma$ be a circle through $X$ and $Y$ which is externally tangent to $\omega$ at $Z$. Prove that the lines $EF$, $DZ$, and $XY$ are concurrent.

2008 Balkan MO Shortlist, G1

Tags: geometric inequality , geometry , median , exradius , excircle

In acute angled triangle $ABC$ we denote by $a,b,c$ the side lengths, by $m_a,m_b,m_c$ the median lengths and by $r_{b}c,r_{ca},r_{ab}$ the radii of the circles tangents to two sides and to circumscribed circle of the triangle, respectively. Prove that $$\frac{m_a^2}{r_{bc}}+\frac{m_b^2}{r_{ab}}+\frac{m_c^2}{r_{ab}} \ge \frac{27\sqrt3}{8}\sqrt[3]{abc}$$

Mathematical Minds 2024, P8

Tags: olympiad geometry , excircle , incircle , collinearity , geometry

Let $ABC$ be a triangle with circumcircle $\Omega$, incircle $\omega$, and $A$-excircle $\omega_A$. Let $X$ and $Y$ be the tangency points of $\omega_A$ with $AB$ and $AC$. Lines $XY$ and $BC$ intersect in $T$. The tangent from $T$ to $\omega$ different from $BC$ intersects $\omega$ at $K$. The radical axis of $\omega_A$ and $\Omega$ intersects $BC$ in $S$. The tangent from $S$ to $\omega_A$ different from $BC$ intersects $\omega_A$ at $L$. Prove that $A$, $K$ and $L$ are collinear. [i]Proposed by Ana Boiangiu[/i]

2014 Regional Olympiad of Mexico Center Zone, 3

Tags: geometry , inradius , excircle

Let $AB$ be a triangle and $\Gamma$ the excircle, relative to the vertex $A$, with center $D$. The circle $\Gamma$ is tangent to the lines $AB$ and $AC$ at $E$ and $F$, respectively. Let $P$ and $Q$ be the intersections of $EF$ with $BD$ and $CD$, respectively. If $O$ is the point of intersection of $BQ$ and $CP$, show that the distance from $O$ to the line $BC$ is equal to the radius of the inscribed circle in the triangle $ABC$.

Croatia MO (HMO) - geometry, 2012.7

Tags: geometry , cyclic quadrilateral , concyclic , incircle , right triangle , excircle

Let the points $M$ and $N$ be the intersections of the inscribed circle of the right-angled triangle $ABC$, with sides $AB$ and $CA$ respectively , and points $P$ and $Q$ respectively be the intersections of the ex-scribed circles opposite to vertices $B$ and $C$ with direction $BC$. Prove that the quadrilateral $MNPQ$ is a cyclic if and only if the triangle $ABC$ is right-angled with a right angle at the vertex $A$.

2025 Sharygin Geometry Olympiad, 3

Tags: geometry , law of sines , excircle , pedal circles

An excircle centered at $I_{A}$ touches the side $BC$ of a triangle $ABC$ at point $D$. Prove that the pedal circles of $D$ with respect to the triangles $ABI_{A}$ and $ACI_{A}$ are congruent. Proposed by:K.Belsky

2011 Sharygin Geometry Olympiad, 11

Tags: geometry , parallel , incircle , excircle

The excircle of right-angled triangle $ABC$ ($\angle B =90^o$) touches side $BC$ at point $A_1$ and touches line $AC$ in point $A_2$. Line $A_1A_2$ meets the incircle of $ABC$ for the first time at point $A'$, point $C'$ is defined similarly. Prove that $AC||A'C'$.

2006 Austria Beginners' Competition, 4

Tags: isosceles , equal circles , excircle , geometry

Show that if a triangle has two excircles of the same size, then the triangle is isosceles. (Note: The excircle $ABC$ to the side $ a$ touches the extensions of the sides $AB$ and $AC$ and the side $BC$.)

2018 Germany Team Selection Test, 3

Tags: homothety , power of a point , excircle , geometry , tangent circle

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.