Found problems: 1389
1974 IMO Longlists, 4
Let $K_a,K_b,K_c$ with centres $O_a,O_b,O_c$ be the excircles of a triangle $ABC$, touching the interiors of the sides $BC,CA,AB$ at points $T_a,T_b,T_c$ respectively.
Prove that the lines $O_aT_a,O_bT_b,O_cT_c$ are concurrent in a point $P$ for which $PO_a=PO_b=PO_c=2R$ holds, where $R$ denotes the circumradius of $ABC$. Also prove that the circumcentre $O$ of $ABC$ is the midpoint of the segment $PI$, where $I$ is the incentre of $ABC$.
India EGMO 2021 TST, 3
In acute $\triangle ABC$ with circumcircle $\Gamma$ and incentre $I$, the incircle touches side $AB$ at $F$. The external angle bisector of $\angle ACB$ meets ray $AB$ at $L$. Point $K$ lies on the arc $CB$ of $\Gamma$ not containing $A$, such that $\angle CKI=\angle IKL$. Ray $KI$ meets $\Gamma$ again at $D\ne K$. Prove that $\angle ACF =\angle DCB$.
2011 Federal Competition For Advanced Students, Part 2, 3
We are given a non-isosceles triangle $ABC$ with incenter $I$. Show that the circumcircle $k$ of the triangle $AIB$ does not touch the lines $CA$ and $CB$.
Let $P$ be the second point of intersection of $k$ with $CA$ and let $Q$ be the second point of intersection of $k$ with $CB$.
Show that the four points $A$, $B$, $P$ and $Q$ (not necessarily in this order) are the vertices of a trapezoid.
2007 Turkey MO (2nd round), 2
Let $ABC$ be a triangle with $\angle B=90$. The incircle of $ABC$ touches the side $BC$ at $D$. The incenters of triangles $ABD$ and $ADC$ are $X$ and $Z$ , respectively. The lines $XZ$ and $AD$ are intersecting at the point $K$. $XZ$ and circumcircle of $ABC$ are intersecting at $U$ and $V$. Let $M$ be the midpoint of line segment $[UV]$ . $AD$ intersects the circumcircle of $ABC$ at $Y$ other than $A$. Prove that $|CY|=2|MK|$ .
2025 EGMO, 4
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.
Kyiv City MO 1984-93 - geometry, 1991.11.5
Lines that are drawn perpendicular to the faces of a triangular pyramid through the centers of the inscribed circles intersect at one point. Prove that the sums of the opposite edges of such a pyramid are equal to each other.
2003 All-Russian Olympiad Regional Round, 9.6
Let $I$ be the intersection point of the bisectors of triangle $ABC$. Let us denote by $A', B', C'$ the points symmetrical to $I$ wrt the sides triangle $ABC$. Prove that if a circle circumscribes around triangle $A'B'C'$ passes through vertex $B$, then $\angle ABC = 60^o$.
Durer Math Competition CD 1st Round - geometry, 2012.C5
In a triangle, the line between the center of the inscribed circle and the center of gravity is parallel to one of the sides. Prove that the sidelengths form an arithmetic sequence.
2017 Bulgaria EGMO TST, 2
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 $\angle BPC$.
2024 Iran MO (2nd Round), 3
In a triangle $ABC$ the incenter, the $B$-excenter and the $C$-excenter are $I, K$ and $L$, respectively. The perpendiculars at $B$ and $C$ to $BC$ intersect the lines $AC$ and $AB$ at $E$ and $F$, respectively. Prove that the circumcircles of $AEF, FIL, EIK$ concur.
2017 All-Russian Olympiad, 8
Given a convex quadrilateral $ABCD$. We denote $I_A,I_B, I_C$ and $I_D$ centers of $\omega_A, \omega_B,\omega_C $and $\omega_D$,inscribed In the triangles $DAB, ABC, BCD$ and $CDA$, respectively.It turned out that $\angle BI_AA + \angle I_CI_AI_D = 180^\circ$. Prove that $\angle BI_BA + \angle I_CI_BI_D = 180^{\circ}$. (A. Kuznetsov)
2016 Costa Rica - Final Round, G3
Let $\vartriangle ABC$ be acute, with incircle $\Gamma$ and incenter $ I$. $\Gamma$ touches sides $AB$, $BC$ and $AC$ at $Z$, $X$ and $Y$, respectively. Let $D$ be the intersection of $XZ$ with $CI$ and $L$ the intersection of $BI$ with $XY$. Suppose $D$ and $L$ are outside of $\vartriangle ABC$. Prove that $A$, $D$, $Z$, $I$, $Y$, and $ L$ lie on a circle.
2018 Dutch IMO TST, 4
In a non-isosceles triangle $ABC$ the centre of the incircle is denoted by $I$. The other intersection point of the angle bisector of $\angle BAC$ and the circumcircle of $\vartriangle ABC$ is $D$. The line through $I$ perpendicular to $AD$ intersects $BC$ in $F$. The midpoint of the circle arc $BC$ on which $A$ lies, is denoted by $M$. The other intersection point of the line $MI$ and the circle through $B, I$ and $C$, is denoted by $N$. Prove that $FN$ is tangent to the circle through $B, I$ and $C$.
2008 Harvard-MIT Mathematics Tournament, 9
Let $ ABC$ be a triangle, and $ I$ its incenter. Let the incircle of $ ABC$ touch side $ BC$ at $ D$, and let lines $ BI$ and $ CI$ meet the circle with diameter $ AI$ at points $ P$ and $ Q$, respectively. Given $ BI \equal{} 6, CI \equal{} 5, DI \equal{} 3$, determine the value of $ \left( DP / DQ \right)^2$.
2014 Tuymaada Olympiad, 7
A parallelogram $ABCD$ is given. The excircle of triangle $\triangle{ABC}$ touches the sides $AB$ at $L$ and the extension of $BC$ at $K$. The line $DK$ meets the diagonal $AC$ at point $X$; the line $BX$ meets the median $CC_1$ of trianlge $\triangle{ABC}$ at ${Y}$. Prove that the line $YL$, median $BB_1$ of triangle $\triangle{ABC}$ and its bisector $CC^\prime$ have a common point.
[i](A. Golovanov)[/i]
2012 Sharygin Geometry Olympiad, 8
A point $M$ lies on the side $BC$ of square $ABCD$. Let $X$, $Y$ , and $Z$ be the incenters of triangles $ABM$, $CMD$, and $AMD$ respectively. Let $H_x$, $H_y$, and $H_z$ be the orthocenters of triangles $AXB$, $CY D$, and $AZD$. Prove that $H_x$, $H_y$, and $H_z$ are collinear.
1982 Brazil National Olympiad, 1
The angles of the triangle $ABC$ satisfy $\angle A / \angle C = \angle B / \angle A = 2$. The incenter is $O. K, L$ are the excenters of the excircles opposite $B$ and $A$ respectively. Show that triangles $ABC$ and $OKL$ are similar.
Indonesia MO Shortlist - geometry, g10
Given two circles with one of the centers of the circle is on the other circle. The two circles intersect at two points $C$ and $D$. The line through $D$ intersects the two circles again at $A$ and $ B$. Let $H$ be the midpoint of the arc $AC$ that does not contain $D$ and the segment $HD$ intersects circle that does not contain $H$ at point $E$. Show that $E$ is the center of the incircle of the triangle $ACD$.
2010 Romania Team Selection Test, 2
Let $ABC$ be a scalene triangle, let $I$ be its incentre, and let $A_1$, $B_1$ and $C_1$ be the points of contact of the excircles with the sides $BC$, $CA$ and $AB$, respectively. Prove that the circumcircles of the triangles $AIA_1$, $BIB_1$ and $CIC_1$ have a common point different from $I$.
[i]Cezar Lupu & Vlad Matei[/i]
2013 Online Math Open Problems, 21
Let $ABC$ be a triangle with $AB = 5$, $AC = 8$, and $BC = 7$. Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$. Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$. Suppose $DE = \frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
[i]Proposed by Ray Li[/i]
2016 Korea - Final Round, 5
An acute triangle $\triangle ABC$ has incenter $I$, and the incircle hits $BC, CA, AB$ at $D, E, F$.
Lines $BI, CI, BC, DI$ hits $EF$ at $K, L, M, Q$ and the line connecting the midpoint of segment $CL$ and $M$ hits the line segment $CK$ at $P$. Prove that $$PQ=\frac{AB \cdot KQ}{BI}$$
MathLinks Contest 7th, 2.3
Let $ ABC$ be a given triangle with the incenter $ I$, and denote by $ X$, $ Y$, $ Z$ the intersections of the lines $ AI$, $ BI$, $ CI$ with the sides $ BC$, $ CA$, and $ AB$, respectively. Consider $ \mathcal{K}_{a}$ the circle tangent simultanously to the sidelines $ AB$, $ AC$, and internally to the circumcircle $ \mathcal{C}(O)$ of $ ABC$, and let $ A^{\prime}$ be the tangency point of $ \mathcal{K}_{a}$ with $ \mathcal{C}$. Similarly, define $ B^{\prime}$, and $ C^{\prime}$.
Prove that the circumcircles of triangles $ AXA^{\prime}$, $ BYB^{\prime}$, and $ CZC^{\prime}$ all pass through two distinct points.
2021 Novosibirsk Oral Olympiad in Geometry, 7
A circle concentric with the inscribed circle of $ABC$ intersects the sides of the triangle at six points forming a convex hexagon $A_1A_2B_1B_2C_1C_2$ (points $C_1$ and $C_2$ on the $AB$ side, $A_1$ and $A_2$ on $BC$, $B_1$ and $B_2$ on $AC$). Prove that if line $A_1B_1$ is parallel to the bisector of angle $B$, then line $A_2C_2$ is parallel to the bisector of angle $C$.
1998 Iran MO (2nd round), 2
Let $ABC$ be a triangle. $I$ is the incenter of $\Delta ABC$ and $D$ is the meet point of $AI$ and the circumcircle of $\Delta ABC$. Let $E,F$ be on $BD,CD$, respectively such that $IE,IF$ are perpendicular to $BD,CD$, respectively. If $IE+IF=\frac{AD}{2}$, find the value of $\angle BAC$.
2008 Mongolia Team Selection Test, 2
The quadrilateral $ ABCD$ inscribed in a circle wich has diameter $ BD$. Let $ A',B'$ are symmetric to $ A,B$ with respect to the line $ BD$ and $ AC$ respectively. If $ A'C \cap BD \equal{} P$ and $ AC\cap B'D \equal{} Q$ then prove that $ PQ \perp AC$