Found problems: 1389
1998 Mediterranean Mathematics Olympiad, 3
In a triangle $ABC$, $I$ is the incenter and $D,E, F$ are the points of tangency of the incircle with $BC,CA,AB$, respectively. The bisector of angle $BIC$ meets $BC$ at $M$, and the line $AM$ intersects $EF$ at $P$. Prove that $DP$ bisects the angle $FDE$.
2023 Turkey EGMO TST, 1
Let $O_1O_2O_3$ be an acute angled triangle.Let $\omega_1$, $\omega_2$, $\omega_3$ be the circles with centres $O_1$, $O_2$, $O_3$ respectively such that any of two are tangent to each other. Circumcircle of $O_1O_2O_3$ intersects $\omega_1$ at $A_1$ and $B_1$, $\omega_2$ at $A_2$ and $B_2$, $\omega_3$ at $A_3$ and $B_3$ respectively. Prove that the incenter of triangle which can be constructed by lines $A_1B_1$, $A_2B_2$, $A_3B_3$ and the incenter of $O_1O_2O_3$ are coincide.
IV Soros Olympiad 1997 - 98 (Russia), 9.10
A circle is drawn through vertices $A$ and $B$ of triangle $ABC$, intersecting sides $AC$ and $BC$ at points $M$ and $P$. It is known that the segment $MP$ contains the center of the circle inscribed in $ABC$. Find $MP$ if $AB = c$, $BC = a$, $CA=b$.
2014 Contests, 3
Let $ABC$ be a triangle with $AB < AC$ and incentre $I$. Let $E$ be the point on the side $AC$ such that $AE = AB$. Let $G$ be the point on the line $EI$ such that $\angle IBG = \angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$.
Prove that the line $AI$, the line perpendicular to $AE$ at $E$, and the bisector of the angle $\angle BGI$ are concurrent.
2007 Moldova Team Selection Test, 3
Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid.
Prove that these three Euler lines pass through one common point.
[i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$.
[i]Floor van Lamoen[/i]
1990 IMO Longlists, 15
Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.
2005 IMAR Test, 2
Let $P$ be an arbitrary point on the side $BC$ of triangle $ABC$ and let $D$ be the tangency point between the incircle of the triangle $ABC$ and the side $BC$. If $Q$ and $R$ are respectively the incenters in the triangles $ABP$ and $ACP$, prove that $\angle QDR$ is a right angle.
Prove that the triangle $QDR$ is isosceles if and only if $P$ is the foot of the altitude from $A$ in the triangle $ABC$.
2004 China Team Selection Test, 2
Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$.
Prove that $ M$, $ N$, $ O$ are collinear.
2010 Contests, 3
A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively.
Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic.
Babis
1967 IMO Shortlist, 3
Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$
2019 Saudi Arabia IMO TST, 3
Let $ABC$ be an acute nonisosceles triangle with incenter $I$ and $(d)$ is an arbitrary line tangent to $(I)$ at $K$. The lines passes through $I$, perpendicular to $IA, IB, IC$ cut $(d)$ at $A_1, B_1,C_1$ respectively. Suppose that $(d)$ cuts $BC, CA, AB$ at $M,N, P$ respectively. The lines through $M,N,P$ and respectively parallel to the internal bisectors of $A, B, C$ in triangle $ABC$ meet each other to define a triange $XYZ$. Prove that three lines $AA_1, BB_1, CC_1$ are concurrent and $IK$ is tangent to the circle $(XY Z)$
2015 Postal Coaching, Problem 1
$O$ is the centre of the circumcircle of triangle $ABC$, and $M$ is its orthocentre. Point $A$ is reflected in the perpendicular bisector of the side $BC$,$ B$ is reflected in the perpendicular bisector of the side $CA$, and finally $C$ is reflected in the perpendicular bisector of the side $AB$. The images are denoted by $A_1, B_1, C_1$ respectively. Let $K$ be the centre of the inscribed circle of triangle $A_1B_1C_1$. Prove that $O$ bisects the line segment $MK$.
2001 Saint Petersburg Mathematical Olympiad, 10.3
Let $I$ be the incenter of triangle $ABC$ and let $D$ be the midpoint of side $AB$. Prove that if the angle $\angle AOD$ is right, then $AB+BC=3AC$.
[I]Proposed by S. Ivanov[/i]
2008 APMO, 1
Let $ ABC$ be a triangle with $ \angle A < 60^\circ$. Let $ X$ and $ Y$ be the points on the sides $ AB$ and $ AC$, respectively, such that $ CA \plus{} AX \equal{} CB \plus{} BX$ and $ BA \plus{} AY \equal{} BC \plus{} CY$ . Let $ P$ be the point in the plane such that the lines $ PX$ and $ PY$ are perpendicular to $ AB$ and $ AC$, respectively. Prove that $ \angle BPC < 120^\circ$.
1992 Romania Team Selection Test, 10
In a tetrahedron $VABC$, let $I$ be the incenter and $A',B',C'$ be arbitrary points on the edges $AV,BV,CV$, and let $S_a,S_b,S_c,S_v$ be the areas of triangles $VBC,VAC,VAB,ABC$, respectively. Show that points $A',B',C',I$ are coplanar if and only if $\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v$
2002 China Girls Math Olympiad, 7
An acute triangle $ ABC$ has three heights $ AD, BE$ and $ CF$ respectively. Prove that the perimeter of triangle $ DEF$ is not over half of the perimeter of triangle $ ABC.$
2012 Federal Competition For Advanced Students, Part 1, 4
Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the second point of intersection different from $C$ of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$.
Show that $\gamma$ is the second-largest angle in the triangle $ABC$.
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 Saudi Arabia GMO TST, 3
$ABC$ is a triangle, $H$ its orthocenter, $I$ its incenter, $O$ its circumcenter and $\omega$ its circumcircle. Line $CI$ intersects circle $\omega$ at point $D$ different from $C$. Assume that $AB = ID$ and $AH = OH$. Find the angles of triangle $ABC$.
2007 District Olympiad, 3
Let $ABC$ be a triangle with $BC=a$ $AC=b$ $AB=c$. For each line $\Delta$ we denote $d_{A}, d_{B}, d_{C}$ the distances from $A,B, C$ to $\Delta$ and we consider the expresion $E(\Delta)=ad_{A}^{2}+bd_{B}^{2}+cd_{C}^{2}$. Prove that if $E(\Delta)$ is minimum, then $\Delta$ passes through the incenter of $\Delta ABC$.
Durer Math Competition CD 1st Round - geometry, 2021.D4
In the triangle $ABC$ we have $30^o$ at the vertex $A$, and $50^o$ at the vertex $B$. Let $O$ be the center of inscribed circle. Show that $AC + OC = AB$.
2016 JBMO Shortlist, 3
A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.
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 $AB = BP$.
2012 Kazakhstan National Olympiad, 3
Line $PQ$ is tangent to the incircle of triangle $ABC$ in such a way that the points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively. On the sides $AB$ and $AC$ are selected points $M$ and $N$, respectively, so that $AM = BP$ and $AN = CQ$. Prove that all lines constructed in this manner $MN$ pass through one point
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.