Found problems: 1389
2012 IMO Shortlist, G7
Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.
1991 India National Olympiad, 9
Triangle $ABC$ has an incenter $I$ l its incircle touches the side $BC$ at $T$. The line through $T$ parallel to $IA$ meets the incircle again at $S$ and the tangent to the incircle at $S$ meets $AB , AC$ at points $C' , B'$ respectively. Prove that triangle $AB'C'$ is similar to triangle $ABC$.
2018 Thailand TST, 3
A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.
2017 India IMO Training Camp, 2
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2021 Harvard-MIT Mathematics Tournament., 9
Let scalene triangle $ABC$ have circumcenter $O$ and incenter $I$. Its incircle $\omega$ is tangent to sides $BC,CA,$ and $AB$ at $D,E,$ and $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $EF$, and let line $DP$ intersect $\omega$ again at $Q \ne D$. The line $OI$ intersects the altitude from $A$ to$ BC$ at $T$. Given that $OI \|BC,$ show that $PQ=PT$.
2017 Bosnia and Herzegovina Team Selection Test, Problem 1
Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.
2021 Yasinsky Geometry Olympiad, 6
In an acute-angled triangle $ABC$, point $I$ is the center of the inscribed circle, point $T$ is the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$. It turned out that $\angle AIT = 90^o$ . Prove that $AB + AC = 3BC$.
(Matthew of Kursk)
2025 Korea - Final Round, P4
Triangle $ABC$ satisfies $\overline{CA} > \overline{AB}$. Let the incenter of triangle $ABC$ be $\omega$, which touches $BC, CA, AB$ at $D, E, F$, respectively. Let $M$ be the midpoint of $BC$. Let the circle centered at $M$ passing through $D$ intersect $DE, DF$ at $P(\neq D), Q(\neq D)$, respecively. Let line $AP$ meet $BC$ at $N$, line $BP$ meet $CA$ at $L$. Prove that the three lines $EQ, FP, NL$ are concurrent.
2020 GQMO, 8
Let $ABC$ be an acute scalene triangle, with the feet of $A,B,C$ onto $BC,CA,AB$ being $D,E,F$ respectively. Let $W$ be a point inside $ABC$ whose reflections over $BC,CA,AB$ are $W_a,W_b,W_c$ respectively. Finally, let $N$ and $I$ be the circumcenter and the incenter of $W_aW_bW_c$ respectively. Prove that, if $N$ coincides with the nine-point center of $DEF$, the line $WI$ is parallel to the Euler line of $ABC$.
[i]Proposed by Navneel Singhal, India and Massimiliano Foschi, Italy[/i]
2012 Iran MO (2nd Round), 3
The incircle of triangle $ABC$, is tangent to sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. The reflection of $F$ with respect to $B$ and the reflection of $E$ with respect to $C$ are $T$ and $S$ respectively. Prove that the incenter of triangle $AST$ is inside or on the incircle of triangle $ABC$.
[i]Proposed by Mehdi E'tesami Fard[/i]
2012 Bosnia And Herzegovina - Regional Olympiad, 4
Let $S$ be an incenter of triangle $ABC$ and let incircle touch sides $AC$ and $AB$ in points $P$ and $Q$, respectively. Lines $BS$ and $CS$ intersect line $PQ$ in points $M$ and $N$, respectively. Prove that points $M$, $N$, $B$ and $C$ are concyclic
2014 India PRMO, 16
In a triangle $ABC$, let $I$ denote the incenter. Let the lines $AI,BI$ and $CI$ intersect the incircle at $P,Q$ and $R$, respectively. If $\angle BAC = 40^o$, what is the value of $\angle QPR$ in degrees ?
2012 Tournament of Towns, 7
Let $AH$ be an altitude of an equilateral triangle $ABC$. Let $I$ be the incentre of triangle $ABH$, and let $L, K$ and $J$ be the incentres of triangles $ABI,BCI$ and $CAI$ respectively. Determine $\angle KJL$.
2021 Taiwan Mathematics Olympiad, 4.
Let $I$ be the incenter of triangle $ABC$ and let $D$ the foot of altitude from $I$ to $BC$. Suppose the reflection point $D’$ of $D$ with respect to $I$ satisfying $\overline{AD’} = \overline{ID’}$. Let $\Gamma$ be the circle centered at $D’$ that passing through $A$ and $I$, and let $X$, $Y\neq A$ be the intersection of $\Gamma$ and $AB$, $AC$, respectively. Suppose $Z$ is a point on $\Gamma$ so that $AZ$ is perpendicular to $BC$.
Prove that $AD$, $D’Z$, $XY$ concurrent at a point.
1991 Iran MO (2nd round), 2
Triangle $ABC$ is inscribed in circle $C.$ The bisectors of the angles $A,B$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC,$ prove that
\[\frac{IA'}{IA} + \frac{IB'}{IB}+\frac{IC'}{IC} \geq 3\]\[, IA'+IB'+IC' \geq IA+IB+IC\]
2018 China Second Round Olympiad, 2
In triangle $\triangle ABC$, $AB<AC$, $M,D,E$ are the midpoints of $BC$, the arcs $BAC$ and $BC$ of the circumcircle of $\triangle ABC$ respectively. The incircle of $\triangle ABC$ touches $AB$ at $F$, $AE$ meets $BC$ at $G$, and the perpendicular to $AB$ at $B$ meets segment $EF$ at $N$. If $BN=EM$, prove that $DF$ is perpendicular to $FG$.
2007 CentroAmerican, 3
Consider a circle $S$, and a point $P$ outside it. The tangent lines from $P$ meet $S$ at $A$ and $B$, respectively. Let $M$ be the midpoint of $AB$. The perpendicular bisector of $AM$ meets $S$ in a point $C$ lying inside the triangle $ABP$. $AC$ intersects $PM$ at $G$, and $PM$ meets $S$ in a point $D$ lying outside the triangle $ABP$. If $BD$ is parallel to $AC$, show that $G$ is the centroid of the triangle $ABP$.
[i]Arnoldo Aguilar (El Salvador)[/i]
2020 Middle European Mathematical Olympiad, 3#
Let $ABC$ be an acute scalene triangle with circumcircle $\omega$ and incenter $I$. Suppose the orthocenter $H$ of $BIC$ lies inside $\omega$. Let $M$ be the midpoint of the longer arc $BC$ of $\omega$. Let $N$ be the midpoint of the shorter arc $AM$ of $\omega$.
Prove that there exists a circle tangent to $\omega$ at $N$ and tangent to the circumcircles of $BHI$ and $CHI$.
2019 India PRMO, 28
Let $ABC$ be a triangle with sides $51, 52, 53$. Let $\Omega$ denote the incircle of $\bigtriangleup ABC$. Draw tangents to $\Omega$ which are parallel to the sides of $ABC$. Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed $r_1 + r_2 + r_3$.
2010 Contests, 3
Let $A'\in(BC),$ $B'\in(CA),C'\in(AB)$ be the points of tangency of the excribed circles of triangle $\triangle ABC$ with the sides of $\triangle ABC.$ Let $R'$ be the circumradius of triangle $\triangle A'B'C'.$ Show that \[
R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)}\]
where as usual, $R$ is the circumradius of $\triangle ABC,$ r is the inradius of $\triangle ABC,$ and $h_{a},h_{b},h_{c}$ are the lengths of altitudes of $\triangle ABC.$
Kyiv City MO Juniors Round2 2010+ geometry, 2013.9.5
Given a triangle $ ABC $, $ AD $ is its angle bisector. Let $ E, F $ be the centers of the circles inscribed in the triangles $ ADC $ and $ ADB $, respectively. Denote by $ \omega $, the circle circumscribed around the triangle $ DEF $, and by $ Q $, the intersection point of $ BE $ and $ CF $, and $ H, J, K, M $ , respectively the second intersection point of the lines $ CE, CF, BE, BF $ with circle $ \omega $. Let $\omega_1, \omega_2 $ the circles be circumscribed around the triangles $ HQJ $ and $ KQM $ Prove that the intersection point of the circles $\omega_1, \omega_2 $ different from $ Q $ lies on the line $ AD $.
(Kivva Bogdan)
2022 Macedonian Mathematical Olympiad, Problem 5
An acute $\triangle ABC$ with circumcircle $\Gamma$ is given. $I$ and $I_a$ are the incenter and $A-$excenter of $\triangle ABC$ respectively. The line $AI$ intersects $\Gamma$ again at $D$ and $A'$ is the antipode of $A$ with respect to $\Gamma$.
$X$ and $Y$ are point on $\Gamma$ such that $\angle IXD = \angle I_aYD = 90^\circ$. The tangents to $\Gamma$ at $X$ and $Y$ intersect in point $Z$. Prove that $A', D$ and $Z$ are collinear.
[i]Proposed by Nikola Velov[/i]
2011 Romania Team Selection Test, 2
In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.
2019 Turkey Junior National Olympiad, 3
In $ABC$ triangle $I$ is incenter and incircle of $ABC$ tangents to $BC,AC,AB$ at $D,E,F$, respectively. If $AI$ intersects $DE$ and $DF$ at $P$ and $Q$, prove that the circumcenter of $DPQ$ triangle is the midpoint of $BC$.
2020 South East Mathematical Olympiad, 2
In a scalene triangle $\Delta ABC$, $AB<AC$, $PB$ and $PC$ are tangents of the circumcircle $(O)$ of $\Delta ABC$. A point $R$ lies on the arc $\widehat{AC}$(not containing $B$), $PR$ intersects $(O)$ again at $Q$. Suppose $I$ is the incenter of $\Delta ABC$, $ID \perp BC$ at $D$, $QD$ intersects $(O)$ again at $G$. A line passing through $I$ and perpendicular to $AI$ intersects $AG,AC$ at $M,N$, respectively. $S$ is the midpoint of arc $\widehat{AR}$, and$SN$ intersects $(O)$ again at $T$.
Prove that, if $AR \parallel BC$, then $M,B,T$ are collinear.