Found problems: 100
2010 Ukraine Team Selection Test, 11
Let $ABC$ be the triangle in which $AB> AC$. Circle $\omega_a$ touches the segment of the $BC$ at point $D$, the extension of the segment $AB$ towards point $B$ at the point $F$, and the extension of the segment $AC$ towards point $C$ at the point $E$. The ray $AD$ intersects circle $\omega_a$ for second time at point $M$. Denote the circle circumscribed around the triangle $CDM$ by $\omega$. Circle $\omega$ intersects the segment $DF$ at N. Prove that $FN > ND$.
2025 Sharygin Geometry Olympiad, 3
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
2000 Saint Petersburg Mathematical Olympiad, 10.6
One of the excircles of triangle $ABC$ is tangent to the side $AB$ and to the extensions of sides $CA$ and $CB$ at points $C_1$, $B_1$ and $A_1$ respectively. Another excircle is tangent to side $AB$ and to the extensions of sides $BA$ and $BC$ at points $B_2$, $C_2$ and $A_2$ respectively. Line $A_1B_1$ and $A_2B_2$ intersect at point $P$,. lines $A_1C_1$ and $A_2C_2$ intersect at point $Q$. Prove that the points $A$, $P$, $Q$ are collinear
[I]Proposed by S. Berlov[/i]
Ukrainian TYM Qualifying - geometry, 2019.11
Let $\omega_a, \omega_b, \omega_c$ be the exscribed circles tangent to the sides $a, b, c$ of a triangle $ABC$, respectively, $ I_a, I_b, I_c$ be the centers of these circles, respectively, $T_a, T_b, T_c$ be the points of contact of these circles to the line $BC$, respectively. The lines $T_bI_c$ and $T_cI_b$ intersect at the point $Q$. Prove that the center of the circle inscribed in triangle $ABC$ lies on the line $T_aQ$.
2011 Sharygin Geometry Olympiad, 11
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 $AC||A'C'$.
2014 Bosnia And Herzegovina - Regional Olympiad, 3
Excircle of triangle $ABC$ to side $AB$ of triangle $ABC$ touches side $AB$ in point $D$. Determine ratio $AD : BD$ if $\angle CAB = 2 \angle ADC$
2021 Adygea Teachers' Geometry Olympiad, 3
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.
2022 Taiwan TST Round 2, 3
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 $\overline{AR} \perp \overline{BC}$.
2019 Czech and Slovak Olympiad III A, 4
Let be $ABC$ an acute-angled triangle. Consider point $P$ lying on the opposite ray to the ray $BC$ such that $|AB|=|BP|$. Similarly, consider point $Q$ on the opposite ray to the ray $CB$ such that $|AC|=|CQ|$. Denote $J$ the excenter of $ABC$ with respect to $A$ and $D,E$ tangent points of this excircle with the lines $AB$ and $AC$, respectively. Suppose that the opposite rays to $DP$ and $EQ$ intersect in $F\neq J$. Prove that $AF\perp FJ$.
2016 Saudi Arabia BMO TST, 2
Let $I$ be the incenter of an acute triangle $ABC$. Assume that $K_1$ is the point such that $AK_1 \perp BC$ and the circle with center $K_1$ of radius $K_1A$ is internally tangent to the incircle of triangle $ABC$ at $A_1$. The points $B_1, C_1$ are defined similarly.
a) Prove that $AA_1, BB_1, CC_1$ are concurrent at a point $P$.
b) Let $\omega_1,\omega_2,\omega_3$ be the excircles of triangle $ABC$ with respect to $A, B, C$, respectively. The circles $\gamma_1,\gamma_2\gamma_3$ are the reflections of $\omega_1,\omega_2,\omega_3$ with respect to the midpoints of $BC, CA, AB$, respectively. Prove that P is the radical center of $\gamma_1,\gamma_2,\gamma_3$.
Indonesia MO Shortlist - geometry, g10
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 $E_AE_BE_C$.
2020 Brazil National Olympiad, 4
Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$, respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$, $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$ the straight line that passes through W and is perpendicular to $AB$. Prove that the lines $r_u$, $r_v$ and $r_w$ pass through the same point.
2022 3rd Memorial "Aleksandar Blazhevski-Cane", P4
Let $ABC$ be an acute triangle with incircle $\omega$, incenter $I$, and $A$-excircle $\omega_{a}$. Let $\omega$ and $\omega_{a}$ meet $BC$ at $X$ and $Y$, respectively. Let $Z$ be the intersection point of $AY$ and $\omega$ which is closer to $A$. The point $H$ is the foot of the altitude from $A$. Show that $HZ$, $IY$ and $AX$ are concurrent.
[i]Proposed by Nikola Velov[/i]
1992 All Soviet Union Mathematical Olympiad, 571
$ABCD$ is a parallelogram. The excircle of $ABC$ opposite $A$ has center $E$ and touches the line $AB$ at $X$. The excircle of $ADC$ opposite $A$ has center $F$ and touches the line $AD$ at $Y$. The line $FC$ meets the line$ AB$ at $W$, and the line $EC$ meets the line $AD$ at $Z$. Show that $WX = YZ$.
2020 Iran MO (3rd Round), 3
The circle $\Omega$ with center $I_A$, is the $A$-excircle of triangle $ABC$. Which is tangent to $AB,AC$ at $F,E$ respectivly. Point $D$ is the reflection of $A$ through $I_AB$. Lines $DI_A$ and $EF$ meet at $K$. Prove that ,circumcenter of $DKE$ , midpoint of $BC$ and $I_A$ are collinear.
2007 Silk Road, 2
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 $B C$ .
2008 Bulgarian Autumn Math Competition, Problem 12.2
Let $ABC$ be a triangle, such that the midpoint of $AB$, the incenter and the touchpoint of the excircle opposite $A$ with $\overline{AC}$ are collinear. Find $AB$ and $BC$ if $AC=3$ and $\angle ABC=60^{\circ}$.
2024 Israel TST, P2
In triangle $ABC$ the incenter is $I$. The center of the excircle opposite $A$ is $I_A$, and it is tangent to $BC$ at $D$. The midpoint of arc $BAC$ is $N$, and $NI$ intersects $(ABC)$ again at $T$. The center of $(AID)$ is $K$. Prove that $TI_A\perp KI$.
2013 JBMO Shortlist, 4
Let $I$ be the incenter and $AB$ the shortest side of the triangle $ABC$. The circle centered at $I$ passing through $C$ intersects the ray $AB$ in $P$ and the ray $BA$ in $Q$. Let $D$ be the point of tangency of the $A$-excircle of the triangle $ABC$ with the side $BC$. Let $E$ be the reflection of $C$ with respect to the point $D$. Prove that $PE\perp CQ$.
2022 AMC 12/AHSME, 25
A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$. What is the ratio $\frac{c_{14}}{c_1}$ for the least possible value of $r$?
$\textbf{(A)} ~\frac{21}{5} \qquad\textbf{(B)} ~\frac{85}{13} \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~\frac{39}{5} \qquad\textbf{(E)} ~17 $
2021 Sharygin Geometry Olympiad, 14
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 $\gamma_A$.
2021 Balkan MO Shortlist, G7
Let $ABC$ be an acute scalene triangle. Its $C$-excircle tangent to the segment $AB$ meets
$AB$ at point $M$ and the extension of $BC$ beyond $B$ at point $N$. Analogously, its $B$-excircle
tangent to the segment $AC$ meets $AC$ at point $P$ and the extension of $BC$ beyond $C$ at point
$Q$. Denote by $A_1$ the intersection point of the lines $MN$ and $PQ$, and let $A_2$ be defined as the
point, symmetric to $A$ with respect to $A_1$. Define the points $B_2$ and $C_2$, analogously. Prove
that $\triangle ABC$ is similar to $\triangle A_2B_2C_2$.
2024 Bangladesh Mathematical Olympiad, P5
Let $I$ be the incenter of $\triangle ABC$ and $P$ be a point such that $PI$ is perpendicular to $BC$ and $PA$ is parallel to $BC$. Let the line parallel to $BC$, which is tangent to the incircle of $\triangle ABC$, intersect $AB$ and $AC$ at points $Q$ and $R$ respectively. Prove that $\angle BPQ = \angle CPR$.
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 $\Omega$.
2011 Sharygin Geometry Olympiad, 5
The touching point of the excircle with the side of a triangle and the base of the altitude to this side are symmetric wrt the base of the corresponding bisector. Prove that this side is equal to one third of the perimeter.