Found problems: 1389
2012 Sharygin Geometry Olympiad, 20
Point $D$ lies on side $AB$ of triangle $ABC$. Let $\omega_1$ and $\Omega_1,\omega_2$ and $\Omega_2$ be the incircles and the excircles (touching segment $AB$) of triangles $ACD$ and $BCD.$ Prove that the common external tangents to $\omega_1$ and $\omega_2,$ $\Omega_1$ and $\Omega_2$ meet on $AB$.
2007 China Team Selection Test, 1
Let $ ABC$ be a triangle. Circle $ \omega$Â passes through points $ B$ and $ C.$ Circle $ \omega_{1}$ is tangent internally to $ \omega$Â and also to sides $ AB$ and $ AC$ at $ T,\, P,$ and $ Q,$ respectively. Let $ M$ be midpoint of arc $ BC\, ($containing $ T)$ of Â$ \omega.$ Prove that lines $ PQ,\,BC,$ and $ MT$ are concurrent.
2013 Sharygin Geometry Olympiad, 8
Two fixed circles are given on the plane, one of them lies inside the other one. From a point $C$ moving arbitrarily on the external circle, draw two chords $CA, CB$ of the larger circle such that they tangent to the smalaler one. Find the locus of the incenter of triangle $ABC$.
2010 Tournament Of Towns, 6
Quadrilateral $ABCD$ is circumscribed around the circle with centre $I$. Let points $M$ and $N$ be the midpoints of sides $AB$ and $CD$ respectively and let $\frac{IM}{AB} = \frac{IN}{CD}$. Prove that $ABCD$ is either a trapezoid or a parallelogram.
2005 Harvard-MIT Mathematics Tournament, 4
Let $XYZ$ be a triangle with $ \angle X = 60^\circ $ and $ \angle Y = 45^\circ $. A circle with center $P$ passes through points $A$ and $B$ on side $XY$, $C$ and $D$ on side $YZ$, and $E$ and $F$ on side $ZX$. Suppose $AB=CD=EF$. Find $ \angle XPY $ in degrees.
1983 Tournament Of Towns, (051) 3
The centre $O$ of the circumcircle of $\vartriangle ABC$ lies inside the triangle. Perpendiculars are drawn rom $O$ on the sides. When produced beyond the sides they meet the circumcircle at points $K, M$ and $P$. Prove that $\overrightarrow{OK} + \overrightarrow{OM} + \overrightarrow{OP} = \overrightarrow{OI}$, where $I$ is the centre of the inscribed circle of $\vartriangle ABC$.
(V Galperin, Moscow)
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.
2006 National Olympiad First Round, 25
Let $E$ be the midpoint of the side $[BC]$ of $\triangle ABC$ with $|AB|=7$, $|BC|=6$, and $|AC|=5$. The line, which passes through $E$ and is perpendicular to the angle bisector of $\angle A$, intersects $AB$ at $D$. What is $|AD|$?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ \frac 92
\qquad\textbf{(D)}\ 3\sqrt 2
\qquad\textbf{(E)}\ \text{None of above}
$
2005 IMAR Test, 1
The incircle of triangle $ABC$ touches the sides $BC,CA,AB$ at the points $D,E,F$, respectively. Let $K$ be a point on the side $BC$ and let $M$ be the point on the line segment $AK$ such that $AM=AE=AF$. Denote by $L,N$ the incenters of triangles $ABK,ACK$, respectively.
Prove that $K$ is the foot of the altitude from $A$ if and only if $DLMN$ is a square.
[hide="Remark"]This problem is slightly connected to [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=344774#p344774]GMB-IMAR 2005, Juniors, Problem 2[/url]
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[i]Bogdan Enescu[/i]
2010 Korea Junior Math Olympiad, 3
In an acute triangle $\triangle ABC$, let there be point $D$ on segment $AC, E$ on segment $AB$ such that $\angle ADE = \angle ABC$. Let the bisector of $\angle A$ hit $BC$ at $K$. Let the foot of the perpendicular from $K$ to $DE$ be $P$, and the foot of the perpendicular from $A$ to $DE$ be $L$. Let $Q$ be the midpoint of $AL$. If the incenter of $\triangle ABC$ lies on the circumcircle of $\triangle ADE$, prove that $P,Q$ and the incenter of $\triangle ADE$ are collinear.
2022 India National Olympiad, 1
Let $D$ be an interior point on the side $BC$ of an acute-angled triangle $ABC$. Let the circumcircle of triangle $ADB$ intersect $AC$ again at $E(\ne A)$ and the circumcircle of triangle $ADC$ intersect $AB$ again at $F(\ne A)$. Let $AD$, $BE$, and $CF$ intersect the circumcircle of triangle $ABC$ again at $D_1(\ne A)$, $E_1(\ne B)$ and $F_1(\ne C)$, respectively. Let $I$ and $I_1$ be the incentres of triangles $DEF$ and $D_1E_1F_1$, respectively. Prove that $E,F, I, I_1$ are concyclic.
2006 Iran MO (3rd Round), 5
Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$, $ b$, $ c$, we have
\[ a^{3}\plus{}b^{3}\plus{}c^{3}\geq k\left(a\plus{}b\plus{}c\right)^{3}.\]
2010 USAMO, 4
Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.
2012 All-Russian Olympiad, 4
The point $E$ is the midpoint of the segment connecting the orthocentre of the scalene triangle $ABC$ and the point $A$. The incircle of triangle $ABC$ incircle is tangent to $AB$ and $AC$ at points $C'$ and $B'$ respectively. Prove that point $F$, the point symmetric to point $E$ with respect to line $B'C'$, lies on the line that passes through both the circumcentre and the incentre of triangle $ABC$.
2025 Nepal National Olympiad, 3
Let the incircle of $\triangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $D'$ be the diametrically opposite point of $D$ with respect to the incircle. Let lines $AD'$ and $AD$ intersect the incircle again at $X$ and $Y$, respectively. Prove that the lines $DX$, $D'Y$, and $EF$ are concurrent, i.e., the lines intersect at the same point.
[i](Kritesh Dhakal, Nepal)[/i]
2010 ELMO Shortlist, 1
Let $ABC$ be a triangle. Let $A_1$, $A_2$ be points on $AB$ and $AC$ respectively such that $A_1A_2 \parallel BC$ and the circumcircle of $\triangle AA_1A_2$ is tangent to $BC$ at $A_3$. Define $B_3$, $C_3$ similarly. Prove that $AA_3$, $BB_3$, and $CC_3$ are concurrent.
[i]Carl Lian.[/i]
2006 MOP Homework, 5
Let $ABC$ be an acute triangle with $AC \neq BC$. Points $H$ and $I$ are the orthocenter and incenter of the triangle, respectively. Line $CH$ and $CI$ meet the circumcircle of triangle $ABC$ again at $D$ and $L$ (other than $C$), respectively. Prove that $\angle CIH=90^{\circ}$ if and only if $\angle IDL=90^{\circ}$.
2015 India IMO Training Camp, 1
In a triangle $ABC$, a point $D$ is on the segment $BC$, Let $X$ and $Y$ be the incentres of triangles $ACD$ and $ABD$ respectively. The lines $BY$ and $CX$ intersect the circumcircle of triangle $AXY$ at $P\ne Y$ and $Q\ne X$, respectively. Let $K$ be the point of intersection of lines $PX$ and $QY$. Suppose $K$ is also the reflection of $I$ in $BC$ where $I$ is the incentre of triangle $ABC$. Prove that $\angle BAC=\angle ADC=90^{\circ}$.
2007 Peru IMO TST, 1
Let $P$ be an interior point of the semicircle whose diameter is $AB$ ($\angle APB$ is obtuse). The incircle of $\triangle ABP$ touches $AP$ and $BP$ at $M$ and $N$ respectively. The line $MN$ intersects the semicircle in $X$ and $Y$. Prove that $\widehat{XY}= \angle APB$.
2007 Sharygin Geometry Olympiad, 5
Reconstruct a triangle, given the incenter, the midpoint of some side and the foot of the altitude drawn on this side.
2008 Stars Of Mathematics, 3
Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals.
[i]Dan Schwarz[/i]
1994 China National Olympiad, 6
Let $M$ be a point which has coordinates $(p\times 1994,7p\times 1994)$ in the Cartesian plane ($p$ is a prime). Find the number of right-triangles satisfying the following conditions:
(1) all vertexes of the triangle are lattice points, moreover $M$ is on the right-angled corner of the triangle;
(2) the origin ($0,0$) is the incenter of the triangle.
2024 CAPS Match, 3
Let $ABC$ be a triangle and $D$ a point on its side $BC.$ Points $E, F$ lie on the lines $AB, AC$ beyond vertices $B, C,$ respectively, such that $BE = BD$ and $CF = CD.$ Let $P$ be a point such that $D$ is the incenter of triangle $P EF.$ Prove that $P$ lies inside the circumcircle $\Omega$ of triangle $ABC$ or on it.
Kyiv City MO Seniors 2003+ geometry, 2020.10.5
Given an acute isosceles triangle $ABC, AK$ and $CN$ are its angle bisectors, $I$ is their intersection point . Let point $X$ be the other intersection point of the circles circumscribed around $\vartriangle ABC$ and $\vartriangle KBN$. Let $M$ be the midpoint of $AC$. Prove that the Euler line of $\vartriangle ABC$ is perpendicular to the line $BI$ if and only if the points $X, I$ and $M$ lie on the same line.
(Kivva Bogdan)
2016 Latvia National Olympiad, 2
Triangle $ABC$ has incircle $\omega$ and incenter $I$. On its sides $AB$ and $BC$ we pick points $P$ and $Q$ respectively, so that $PI = QI$ and $PB > QB$. Line segment $QI$ intersects $\omega$ in $T$. Draw a tangent line to $\omega$ passing through $T$; it intersects the sides $AB$ and $BC$ in $U$ and $V$ respectively. Prove that $PU = UV + VQ$!