Olympiad problems

2020 Estonia Team Selection Test, 2

Let $M$ be the midpoint of side BC of an acute-angled triangle $ABC$. Let $D$ and $E$ be the center of the excircle of triangle $AMB$ tangent to side $AB$ and the center of the excircle of triangle $AMC$ tangent to side $AC$, respectively. The circumscribed circle of triangle $ABD$ intersects line$ BC$ for the second time at point $F$, and the circumcircle of triangle $ACE$ is at point $G$. Prove that $| BF | = | CG|$.

2021 IMO Shortlist, G8

Tags: circumcircle , excircle , ddit , geometry

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.

Russian TST 2022, P3

Tags: geometry , circumcircle , excircle , ddit

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.

Geometry Mathley 2011-12, 13.2

Tags: euler circle , euler line , newton line , geometry , excircle

In a triangle $ABC$, the nine-point circle $(N)$ is tangent to the incircle $(I)$ and three excircles $(I_a), (I_b), (I_c)$ at the Feuerbach points $F, F_a, F_b, F_c$. Tangents of $(N)$ at $F, F_a, F_b, F_c$ bound a quadrangle $PQRS$. Show that the Euler line of $ABC$ is a Newton line of $PQRS$. Luis González

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 $\omega$.

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 $A_0BC_0$.

2020 Taiwan TST Round 3, 1

Tags: geometry , excircle

Let $\Omega$ be the $A$-excircle of triangle $ABC$, and suppose that $\Omega$ is tangent to lines $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $M$ be the midpoint of segment $EF$. Two more points $P$ and $Q$ are on $\Omega$ such that $EP$ and $FQ$ are both parallel to $DM$. Let $BP$ meet $CQ$ at point $X$. Prove that the line $AM$ is the angle bisector of $\angle XAD$. [i]Proposed by Shuang-Yen Lee[/i]

2020 Estonia Team Selection Test, 2

Tags: excircle , equal segments , circumcircle , geometry

Let $M$ be the midpoint of side BC of an acute-angled triangle $ABC$. Let $D$ and $E$ be the center of the excircle of triangle $AMB$ tangent to side $AB$ and the center of the excircle of triangle $AMC$ tangent to side $AC$, respectively. The circumscribed circle of triangle $ABD$ intersects line$ BC$ for the second time at point $F$, and the circumcircle of triangle $ACE$ is at point $G$. Prove that $| BF | = | CG|$.

1986 Bundeswettbewerb Mathematik, 2

Tags: geometry , excircle , exradius , right triangle

A triangle has sides $a, b,c$, radius of the incircle $r$ and radii of the excircles $r_a, r_b, r_c$: Prove that: a) The triangle is right-angled if and only if: $r + r_a + r_b + r_c = a + b + c$. b) The triangle is right-angled if and only if: $r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2$.

2018 Vietnam Team Selection Test, 6

Tags: geometry , incircle , excircle

Triangle $ABC$ circumscribed $(O)$ has $A$-excircle $(J)$ that touches $AB,\ BC,\ AC$ at $F,\ D,\ E$, resp. a. $L$ is the midpoint of $BC$. Circle with diameter $LJ$ cuts $DE,\ DF$ at $K,\ H$. Prove that $(BDK),\ (CDH)$ has an intersecting point on $(J)$. b. Let $EF\cap BC =\{G\}$ and $GJ$ cuts $AB,\ AC$ at $M,\ N$, resp. $P\in JB$ and $Q\in JC$ such that $$\angle PAB=\angle QAC=90{}^\circ .$$ $PM\cap QN=\{T\}$ and $S$ is the midpoint of the larger $BC$-arc of $(O)$. $(I)$ is the incircle of $ABC$. Prove that $SI\cap AT\in (O)$.

2015 IFYM, Sozopol, 6

Tags: geometry , inscribed circle , excircle , collinear , concurrency

In $\Delta ABC$ points $A_1$, $B_1$, and $C_1$ are the tangential points of the excircles of $ABC$ with its sides. a) Prove that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $N$. b) If $AC+BC=3AB$, prove that the center of the inscribed circle of $ABC$, its tangential point with $AB$, and the point $N$ are collinear.

2024 Tuymaada Olympiad, 4

Tags: tuymadaa , midpoint , excircle , geometry

A triangle $ABC$ is given. The segment connecting the points where the excircles touch $AB$ and $AC$ meets the bisector of angle $C$ at $X$. The segment connecting the points where the excircles touch $BC$ and $AC$ meets the bisector of angle $A$ at $Y$. Prove that the midpoint of $XY$ is equidistant from $A$ and $C$.

1986 Traian Lălescu, 1.4

Tags: geometry , excircle , trigonometry , 3d geometry , supremum

Let be two fixed points $ B,C. $ Find the locus of the spatial points $ A $ such that $ ABC $ is a nondegenerate triangle and the expression $$ R^2 (A)\cdot\sin \left( 2\angle ABC\right)\cdot\sin \left( 2\angle BCA\right) $$ has the greatest value possible, where $ R(A) $ denotes the radius of the excirlce of $ ABC. $

2016 USA Team Selection Test, 2

Tags: mixtilinear incircle , isogonal lines , reflection , excircle , geometry

Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and suppose the incircle of $ABC$ touches $BC$ at $D$. The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$. The circumcircle of $\triangle DEF$ intersects the $A$-excircle at $S_1$, $S_2$, and $\Omega$ at $T \neq F$. Prove that line $AT$ passes through either $S_1$ or $S_2$. [i]Proposed by Evan Chen[/i]

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 $\omega$.

2023 Bulgaria JBMO TST, 3

Tags: geometry , incircle , excircle , similarity , circumcircle , incenter

Let $ABC$ be a non-isosceles triangle with circumcircle $k$, incenter $I$ and $C$-excenter $I_C$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of arc $\widehat{ACB}$ on $k$. Prove that $\angle IMI_C + \angle INI_C = 180^{\circ}$.

1997 Tournament Of Towns, (557) 2

Tags: excircle , trisection , incircle , geometry

Let $a$ and $b$ be two sides of a triangle. How should the third side $c$ be chosen so that the points of contact of the incircle and the excircle with side $c$ divide that side into three equal segments? (The excircle corresponding to the side $c$ is the circle which is tangent to the side $c$ and to the extensions of the sides $a$ and $b$.) (Folklore)

2017 Bulgaria EGMO TST, 2

Tags: incenter , incircle , excircle , circumcircle , geometric transformation , reflection , geometry

Let $ABC$ be a triangle with incenter $I$. The line $AI$ intersects $BC$ and the circumcircle of $ABC$ at the points $T$ and $S$, respectively. Let $K$ and $L$ be the incenters of $SBT$ and $SCT$, respectively, $M$ be the midpoint of $BC$ and $P$ be the reflection of $I$ with respect to $KL$. a) Prove that $M$, $T$, $K$ and $L$ are concyclic. b) Determine the measure of $\angle BPC$.

1966 IMO Shortlist, 19

Tags: geometry , triangle , construction , excircle

Construct a triangle given the radii of the excircles.

Kharkiv City MO Seniors - geometry, 2021.11.4

Tags: geometry , excircle , bisects segment , excenter

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 $I_CL$.

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.

2020 Francophone Mathematical Olympiad, 1

Tags: excircle , congruent triangles , congruent , geometry , circumcircle

Let $ABC$ be a triangle such that $AB <AC$, $\omega$ its inscribed circle and $\Gamma$ its circumscribed circle. Let also $\omega_b$ be the excircle relative to vertex $B$, then $B'$ is the point of tangency between $\omega_b$ and $(AC)$. Similarly, let the circle $\omega_c$ be the excircle exinscribed relative to vertex $C$, then $C'$ is the point of tangency between $\omega_c$ and $(AB)$. Finally, let $I$ be the center of $\omega$ and $X$ the point of $\Gamma$ such that $\angle XAI$ is a right angle. Prove that the triangles $XBC'$ and $XCB'$ are congruent.

2006 Austria Beginners' Competition, 4

Tags: isosceles , equal circles , geometry , excircle

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 $BC$.)

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 $$\frac{m_a^2}{r_{bc}}+\frac{m_b^2}{r_{ab}}+\frac{m_c^2}{r_{ab}} \ge \frac{27\sqrt3}{8}\sqrt[3]{abc}$$

2013 Saudi Arabia BMO TST, 7

Tags: excircle , tangent , geometry

The excircle $\omega_B$ of triangle $ABC$ opposite $B$ touches side $AC$, rays $BA$ and $BC$ at $B_1, C_1$ and $A_1$, respectively. Point $D$ lies on major arc $A_1C_1$ of $\omega_B$. Rays $DA_1$ and $C_1B_1$ meet at $E$. Lines $AB_1$ and $BE$ meet at $F$. Prove that line $FD$ is tangent to $\omega_B$ (at $D$).