Olympiad problems

India EGMO 2022 TST, 5

Tags: incenter , angle chasing , excenter , inversion , geometry , tangent circle

Let $I$ and $I_A$ denote the incentre and excentre opposite to $A$ of scalene $\triangle ABC$ respectively. Let $A'$ be the antipode of $A$ in $\odot (ABC)$ and $L$ be the midpoint of arc $(BAC)$. Let $LB$ and $LC$ intersect $AI$ at points $Y$ and $Z$ respectively. Prove that $\odot (LYZ)$ is tangent to $\odot (A'II_A)$. [i]~Mahavir Gandhi[/i]

2005 Croatia National Olympiad, 2

Tags: geometry , circumcircle , homothety , incenter , geometric transformation

Let $U$ be the incenter of a triangle $ABC$ and $O_{1}, O_{2}, O_{3}$ be the circumcenters of the triangles $BCU, CAU, ABU$ , respectively. Prove that the circumcircles of the triangles $ABC$ and $O_{1}O_{2}O_{3}$ have the same center.

1990 Kurschak Competition, 2

Tags: trig identities , geometry , trigonometry , incenter , law of sines , ratio , angle bisector

The incenter of $\triangle A_1A_2A_3$ is $I$, and the center of the $A_i$-excircle is $J_i$ ($i=1,2,3$). Let $B_i$ be the intersection point of side $A_{i+1}A_{i+2}$ and the bisector of $\angle A_{i+1}IA_{i+2}$ ($A_{i+3}:=A_i$ $\forall i$). Prove that the three lines $B_iJ_i$ are concurrent.

2018 PUMaC Geometry B, 4

Tags: geometry , incenter

Let $\triangle ABC$ satisfy $AB = 17, AC = \frac{70}{3}$ and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with diameter $IE$ intersects $AB$ beyond $B$ at $D$. If $BD = \frac{a}{b}$ where $a, b$ are coprime positive integers, find $a + b$.

2007 Moldova Team Selection Test, 2

Tags: trig identities , circumcircle , trigonometry , law of sines , geometry , inequalities , incenter

If $I$ is the incenter of a triangle $ABC$ and $R$ is the radius of its circumcircle then \[AI+BI+CI\leq 3R\]

2010 Contests, 1

Tags: geometry , incenter

Two circles $\Gamma_1$ and $\Gamma_2$ meet at $A$ and $B$. A line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ repsectively. Another line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $E$ and $F$ repsectively. Line $CF$ meets $\Gamma_1$ and $\Gamma_2$ again at $P$ and $Q$ respectively. $M$ and $N$ are midpoints of arc $PB$ and arc $QB$ repsectively. Show that if $CD = EF$, then $C,F,M,N$ are concyclic.

2017 Germany Team Selection Test, 3

Tags: geometry , cyclic quadrilateral , inversion , reflection , incenter

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 $BDE$.

2023 Thailand Mathematical Olympiad, 2

Tags: perpendicular bisector , incenter , geometry

Let $\triangle ABC$ which $\angle ABC$ are right angle, Let $D$ be point on $AB$ ( $D \neq A , B$ ), Let $E$ be point on line $AB$ which $B$ is the midpoint of $DE$, Let $I$ be incenter of $\triangle ACE$ and $J$ be $A$-excenter of $\triangle ACD$. Prove that perpendicular bisector of $BC$ bisects $IJ$

2018 Iranian Geometry Olympiad, 5

Tags: geometry , concyclic , incenter , cyclic quadrilateral

$ABCD$ is a cyclic quadrilateral. A circle passing through $A,B$ is tangent to segment $CD$ at point $E$. Another circle passing through $C,D$ is tangent to $AB$ at point $F$. Point $G$ is the intersection point of $AE,DF$, and point $H$ is the intersection point of $BE$, $CF$. Prove that the incenters of triangles $AGF$, $BHF$, $CHE$, $DGE$ lie on a circle. Proposed by Le Viet An (Vietnam)

2016 Korea National Olympiad, 2

Tags: geometry , incenter

A non-isosceles triangle $\triangle ABC$ has its incircle tangent to $BC, CA, AB$ at points $D, E, F$. Let the incenter be $I$. Say $AD$ hits the incircle again at $G$, at let the tangent to the incircle at $G$ hit $AC$ at $H$. Let $IH \cap AD = K$, and let the foot of the perpendicular from $I$ to $AD$ be $L$. Prove that $IE \cdot IK= IC \cdot IL$.

2007 All-Russian Olympiad Regional Round, 10.4

Tags: geometric transformation , geometry , circumcircle , incenter , homothety

Given a triangle $ ABC$. A circle passes through vertices $ B$ and $ C$ and intersects sides $ AB$ and $ AC$ at points $ D$ and $ E$, respectively. Segments $ CD$ and $ BE$ intersect at point $ O$. Denote the incenters of triangles $ ADE$ and $ ODE$ by $ M$ and $ N$, respectiely. Prove that the midpoint of the smaller arc $ DE$ lies on line $ MN$.

2020 Dutch IMO TST, 1

Tags: equal angles , incenter , geometry

In acute-angled triangle $ABC, I$ is the center of the inscribed circle and holds $| AC | + | AI | = | BC |$. Prove that $\angle BAC = 2 \angle ABC$.

2007 Italy TST, 1

Tags: incenter , rectangle , geometry , homothety , geometric transformation , reflection

Let $ABC$ an acute triangle. (a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$; (b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.

2020 Yasinsky Geometry Olympiad, 4

Tags: geometry , circumcircle , trapezoid , incenter , isosceles

In an isosceles trapezoid $ABCD$, the base $AB$ is twice as large as the base $CD$. Point $M$ is the midpoint of $AB$. It is known that the center of the circle inscribed in the triangle $MCB$ lies on the circle circumscribed around the triangle $MDC$. Find the angle $\angle MBC$. [img]https://cdn.artofproblemsolving.com/attachments/8/a/7af6a1d32c4e2affa49cb3eed9c10ba1e7ab71.png[/img]

2020 Brazil Cono Sur TST, 1

Tags: circumcircle , incenter , geometry

Let $D$ and $E$ be points on sides $AB$ and $AC$ of a triangle $ABC$ such that $DB = BC = CE$. The segments $BE$ and $CD$ intersect at point $P$. Prove that the incenter of triangle $ABC$ lies on the circles circumscribed around the triangles $BDP$ and $CEP$.

2006 IMO Shortlist, 1

Tags: geometry , incenter , circumcircle

Let $ABC$ be triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies \[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\] Show that $AP \geq AI$, and that equality holds if and only if $P=I$.

2011 Postal Coaching, 1

Tags: geometry , circumcircle , incenter

Let $ABC$ be a triangle in which $\angle BAC = 60^{\circ}$ . Let $P$ (similarly $Q$) be the point of intersection of the bisector of $\angle ABC$(similarly of $\angle ACB$) and the side $AC$(similarly $AB$). Let $r_1$ and $r_2$ be the in-radii of the triangles $ABC$ and $AP Q$, respectively. Determine the circum-radius of $APQ$ in terms of $r_1$ and $r_2$.

1997 Belarusian National Olympiad, 4

Tags: geometry , incenter , construction , projective geometry , orthocenter

A triangle $A_1B_1C_1$ is a parallel projection of a triangle $ABC$ in space. The parallel projections $A_1H_1$ and $C_1L_1$ of the altitude $AH$ and the bisector $CL$ of $\vartriangle ABC$ respectively are drawn. Using a ruler and compass, construct a parallel projection of : (a) the orthocenter, (b) the incenter of $\vartriangle ABC$.

2011 Federal Competition For Advanced Students, Part 2, 3

Tags: circumcircle , geometry , trapezoid , incenter

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.

2012 IMO Shortlist, G7

Tags: geometry , incenter , homothety

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 $CD$.

2024 European Mathematical Cup, 3

Tags: parallelogram , incenter , geometry

Let $\omega$ be a semicircle with diamater $AB$. Let $M$ be the midpoint of $AB$. Let $X,Y$ be points on the same semiplane with $\omega$ with respect to the line $AB$ such that $AMXY$ is a parallelogram. Let $XM\cap \omega = C$ and $YM \cap \omega = D$. Let $I$ be the incenter of $\triangle XYM$. Let $AC \cap BD= E$ and $ME$ intersects $XY$ at $T$. Let the intersection point of $TI$ and $AB$ be $Q$ and let the perpendicular projection of $T$ onto $AB$ be $P$. Prove that $M$ is midpoint of $PQ$

Russian TST 2022, P1

Tags: incenter , geometry

In triangle $ABC$, a point $M$ is the midpoint of $AB$, and a point $I$ is the incentre. Point $A_1$ is the reflection of $A$ in $BI$, and $B_1$ is the reflection of $B$ in $AI$. Let $N$ be the midpoint of $A_1B_1$. Prove that $IN > IM$.

2007 Germany Team Selection Test, 1

Tags: geometry , incenter , triangle , midpoint , congruent triangles

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

2005 Bulgaria National Olympiad, 2

Tags: circumcircle , incenter , radical axis , geometry , power of a point , ratio , trigonometry

Consider two circles $k_{1},k_{2}$ touching externally at point $T$. a line touches $k_{2}$ at point $X$ and intersects $k_{1}$ at points $A$ and $B$. Let $S$ be the second intersection point of $k_{1}$ with the line $XT$ . On the arc $\widehat{TS}$ not containing $A$ and $B$ is chosen a point $C$ . Let $\ CY$ be the tangent line to $k_{2}$ with $Y\in k_{2}$ , such that the segment $CY$ does not intersect the segment $ST$ . If $I=XY\cap SC$ . Prove that : (a) the points $C,T,Y,I$ are concyclic. (b) $I$ is the excenter of triangle $ABC$ with respect to the side $BC$.

1992 Taiwan National Olympiad, 5

Tags: geometry , circumcircle , incenter

A line through the incenter $I$ of triangle $ABC$, perpendicular to $AI$, intersects $AB$ at $P$ and $AC$ at $Q$. Prove that the circle tangent to $AB$ at $P$ and to $AC$ at $Q$ is also tangent to the circumcircle of triangle $ABC$.