Found problems: 320
2009 Bulgaria National Olympiad, 2
In the triangle $ABC$ its incircle with center $I$ touches its sides $BC, CA$ and $AB$ in the points $A_1, B_1, C_1$ respectively. Through $I$ is drawn a line $\ell$. The points $A', B'$ and $C'$ are reflections of $A_1, B_1, C_1$ with respect to the line $\ell$. Prove that the lines $AA', BB'$ and $CC'$ intersects at a common point.
2012 Belarus Team Selection Test, 2
Let $\Gamma$ be the incircle of an non-isosceles triangle $ABC$, $I$ be it’s center. Let $A_1, B_1, C_1$ be the tangency points of $\Gamma$ with the sides $BC, AC, AB$, respectively. Let $A_2 = \Gamma \cap AA_1, M = C_1B_1 \cup AI$, $P$ and $Q$ be the other (different from $A_1, A_2$) intersection points of $A_1M, A_2M$ and $\Gamma$, respectively. Prove that $A, P, Q$ are collinear.
(A. Voidelevich)
2011 Sharygin Geometry Olympiad, 8
The incircle of right-angled triangle $ABC$ ($\angle B = 90^o$) touches $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. Points $A_2, C_2$ are the reflections of $B_1$ in lines $BC, AB$ respectively. Prove that lines $A_1A_2$ and $C_1C_2$ meet on the median of triangle $ABC$.
Indonesia MO Shortlist - geometry, g3
Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.
1998 Belarus Team Selection Test, 2
The incircle of the triangle $ABC$ touches its sides $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. If $r$ is the inradius of $\vartriangle ABC, P,P_1$ are the perimeters of $\vartriangle ABC, \vartriangle A_1B_1C_1$ respectively, prove that $P+P_1 \ge 9 \sqrt3 r$.
I. Voronovich
2024 Israel TST, P2
Triangle $ABC$ is inscribed in circle $\Omega$ with center $O$. The incircle of $ABC$ is tangent to $BC$, $AC$, $AB$ at $D$, $E$, $F$ respectively, and its center is $I$. The reflection of the tangent line to $\Omega$ at $A$ with respect to $EF$ will be denoted $\ell_A$. We similarly define $\ell_B$, $\ell_C$.
Show that the orthocenter of the triangle with sides $\ell_A$, $\ell_B$, $\ell_C$ lies on $OI$.
2018 Federal Competition For Advanced Students, P1, 2
Let $ABC$ be a triangle with incenter $I$. The incircle of the triangle is tangent to the sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let $P$ denote the common point of lines $AI$ and $DE$, and let $M$ and $N$ denote the midpoints of sides $BC$ and $AB$, respectively. Prove that points $M, N$ and $P$ are collinear.
[i](Proposed by Karl Czakler)[/i]
1991 Spain Mathematical Olympiad, 4
The incircle of $ABC$ touches the sides $BC,CA,AB$ at $A' ,B' ,C'$ respectively. The line $A' C'$ meets the angle bisector of $\angle A$ at $D$. Find $\angle ADC$.
2021 Saudi Arabia Training Tests, 4
Let $ABC$ be a triangle with incircle $(I)$, tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. On the line $DF$, take points $M, P$ such that $CM \parallel AB$, $AP \parallel BC$. On the line $DE$, take points $N$, $Q$ such that $BN \parallel AC$, $AQ \parallel BC$. Denote $X$ as intersection of $PE$, $QF$ and $K$ as the midpoint of $BC$. Prove that if $AX = IK$ then $\angle BAC \le 60^o$.
2010 Sharygin Geometry Olympiad, 6
The incircle of triangle $ABC$ touches its sides in points $A', B',C'$ . It is known that the orthocenters of triangles $ABC$ and $A' B'C'$ coincide. Is triangle $ABC$ regular?
2020 Hong Kong TST, 5
In $\Delta ABC$, let $D$ be a point on side $BC$. Suppose the incircle $\omega_1$ of $\Delta ABD$ touches sides $AB$ and $AD$ at $E,F$ respectively, and the incircle $\omega_2$ of $\Delta ACD$ touches sides $AD$ and $AC$ at $F,G$ respectively. Suppose the segment $EG$ intersects $\omega_1$ and $\omega_2$ again at $P$ and $Q$ respectively. Show that line $AD$, tangent of $\omega_1$ at $P$ and tangent of $\omega_2$ at $Q$ are concurrent.
Ukrainian TYM Qualifying - geometry, 2014.9
Construct a point $Q$ in triangle $ABC$ such that at least two of the segments $CQ, BQ, AQ$, divide the inscribed circle in half. For which triangles is this possible?
2022 Junior Balkan Team Selection Tests - Moldova, 9
The circle inscribed in the triangle $ABC$ with center $I$ touches the side $BC$ at the point $D$. The line $DI$ intersects the side $AC$ at the point $M$. The tangent from $M$ to the inscribed circle, different from $AC$, intersects the side $AB$ at the point $N$. The line $NI$ intersects the side $BC$ at the point $P$. Prove that $AB = BP$.
2019 Oral Moscow Geometry Olympiad, 1
In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$
2014 Saudi Arabia Pre-TST, 4.4
Let $\vartriangle ABC$ be an acute triangle, with $\angle A> \angle B \ge \angle C$. Let $D, E$ and $F$ be the tangency points between the incircle of triangle and sides $BC, CA, AB$, respectively. Let $J$ be a point on $(BD)$, $K$ a point on $(DC)$, $L$ a point on $(EC)$ and $M$ a point on $(FB)$, such that $$AF = FM = JD = DK = LE = EA.$$Let $P$ be the intersection point between $AJ$ and $KM$ and let $Q$ be the intersection point between $AK$ and $JL$. Prove that $PJKQ$ is cyclic.
2016 Sharygin Geometry Olympiad, P20
The incircle $\omega$ of a triangle $ABC$ touches $BC, AC$ and $AB$ at points $A_0, B_0$ and $C_0$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to segment $AA_0$ at points $Q$ and $P$ respectively. Prove that $PC_0$ and $QB_0$ meet on $\omega$ .
1968 All Soviet Union Mathematical Olympiad, 112
The circle inscribed in the triangle $ABC$ touches the side $[AC]$ in the point $K$. Prove that the line connecting the midpoint of the side $[AC]$ with the centre of the circle halves the segment $[BK]$ .
2016 Saudi Arabia BMO TST, 2
Let $ABC$ be a triangle with $AB \ne AC$. The incirle of triangle $ABC$ is tangent to $BC, CA, AB$ at $D, E, F$, respectively. The perpendicular line from $D$ to $EF$ intersects $AB$ at $X$. The second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX \perp T F$
Russian TST 2022, P2
The quadrilateral $ABCD$ is inscribed in the circle $\Gamma$. Let $I_B$ and $I_D$ be the centers of the circles $\omega_B$ and $\omega_D$ inscribed in the triangles $ABC$ and $ADC$, respectively. A common external tangent to $\omega_B$ and $\omega_D$ intersects $\Gamma$ at $K$ and $L{}$. Prove that $I_B,I_D,K$ and $L{}$ lie on the same circle.
2019 Bundeswettbewerb Mathematik, 3
Let $ABC$ be atriangle with $\overline{AC}> \overline{BC}$ and incircle $k$. Let $M,W,L$ be the intersections of the median, angle bisector and altitude from point $C$ respectively. The tangent to $k$ passing through $M$, that is different from $AB$, touch $k$ in $T$. Prove that the angles $\angle MTW$ and $\angle TLM$ are equal.
1992 IMO Shortlist, 8
Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions:
[i](i)[/i] its side lengths are $ 1, 2, 3, \ldots, 1992$ in some order;
[i](ii)[/i] the polygon is circumscribable about a circle.
[i]Alternative formulation:[/i] Does there exist a 1992-gon with side lengths $ 1, 2, 3, \ldots, 1992$ circumscribed about a circle? Answer the same question for a 1990-gon.
2013 North Korea Team Selection Test, 5
The incircle $ \omega $ of a quadrilateral $ ABCD $ touches $ AB, BC, CD, DA $ at $ E, F, G, H $, respectively. Choose an arbitrary point $ X$ on the segment $ AC $ inside $ \omega $. The segments $ XB, XD $ meet $ \omega $ at $ I, J $ respectively. Prove that $ FJ, IG, AC $ are concurrent.
2012 Argentina National Olympiad, 3
In the triangle $ABC$ the incircle is tangent to the sides $AB$ and $AC$ at $D$ and $E$ respectively. The line $DE$ intersects the circumcircle at $P$ and $Q$, with $P$ in the small arc $AB$ and $Q$ in the small arc $AC$. If $P$ is the midpoint of the arc $AB$, find the angle A and the ratio $\frac{PQ}{BC}$.
Kharkiv City MO Seniors - geometry, 2012.11.4
The incircle $\omega$ of triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $D, E$ and $E$, respectively. Point $G$ lies on circle $\omega$ in such a way that $FG$ is a diameter. Lines $EG$ and $FD$ intersect at point $H$. Prove that $AB \parallel CH$.
1999 Kazakhstan National Olympiad, 3
The circle inscribed in the triangle $ ABC $ , with center $O$, touches the sides $ AB $ and $ BC $ at the points $ C_1 $ and $ A_1 $, respectively. The lines $ CO $ and $ AO $ intersect the line $ C_1A_1 $ at the points $ K $ and $ L $. $ M $ is the midpoint of $ AC $ and $ \angle ABC = 60^\circ $. Prove that $ KLM $ is a regular triangle.