Olympiad problems

1996 IMO, 2

Tags: concurrency , incenter , angle , triangle , geometry

Let $ P$ be a point inside a triangle $ ABC$ such that \[ \angle APB \minus{} \angle ACB \equal{} \angle APC \minus{} \angle ABC. \] Let $ D$, $ E$ be the incenters of triangles $ APB$, $ APC$, respectively. Show that the lines $ AP$, $ BD$, $ CE$ meet at a point.

2012 Tournament of Towns, 4

Tags: incenter , midpoint , geometry , cyclic , obtuse

In a triangle $ABC$ two points, $C_1$ and $A_1$ are marked on the sides $AB$ and $BC$ respectively (the points do not coincide with the vertices). Let $K$ be the midpoint of $A_1C_1$ and $I$ be the incentre of the triangle $ABC$. Given that the quadrilateral $A_1BC_1I$ is cyclic, prove that the angle $AKC$ is obtuse.

2021 Korea - Final Round, P5

Tags: geometry , incenter , circumcircle , tangent circle

The incenter and $A$-excenter of $\triangle{ABC}$ is $I$ and $O$. The foot from $A,I$ to $BC$ is $D$ and $E$. The intersection of $AD$ and $EO$ is $X$. The circumcenter of $\triangle{BXC}$ is $P$. Show that the circumcircle of $\triangle{BPC}$ is tangent to the $A$-excircle if $X$ is on the incircle of $\triangle{ABC}$.

2015 Iran MO (3rd round), 4

Tags: incenter , geometry

Let $ABC$ be a triangle with incenter $I$. Let $K$ be the midpoint of $AI$ and $BI\cap \odot(\triangle ABC)=M,CI\cap \odot(\triangle ABC)=N$. points $P,Q$ lie on $AM,AN$ respectively such that $\angle ABK=\angle PBC,\angle ACK=\angle QCB$. Prove that $P,Q,I$ are collinear.

2012 Iran Team Selection Test, 3

Tags: incenter , geometric transformation , reflection , power of a point , geometry , circumcircle , radical axis

Let $O$ be the circumcenter of the acute triangle $ABC$. Suppose points $A',B'$ and $C'$ are on sides $BC,CA$ and $AB$ such that circumcircles of triangles $AB'C',BC'A'$ and $CA'B'$ pass through $O$. Let $\ell_a$ be the radical axis of the circle with center $B'$ and radius $B'C$ and circle with center $C'$ and radius $C'B$. Define $\ell_b$ and $\ell_c$ similarly. Prove that lines $\ell_a,\ell_b$ and $\ell_c$ form a triangle such that it's orthocenter coincides with orthocenter of triangle $ABC$. [i]Proposed by Mehdi E'tesami Fard[/i]

2004 All-Russian Olympiad, 2

Tags: geometry , exterior angle , complex number , incenter

Let $ABCD$ be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles $DAB$ and $ABC$ intersect each other at $K$; the exterior angle bisectors of the angles $ABC$ and $BCD$ intersect each other at $L$; the exterior angle bisectors of the angles $BCD$ and $CDA$ intersect each other at $M$; the exterior angle bisectors of the angles $CDA$ and $DAB$ intersect each other at $N$. Let $K_{1}$, $L_{1}$, $M_{1}$ and $N_{1}$ be the orthocenters of the triangles $ABK$, $BCL$, $CDM$ and $DAN$, respectively. Show that the quadrilateral $K_{1}L_{1}M_{1}N_{1}$ is a parallelogram.

2015 Germany Team Selection Test, 2

Tags: perpendicular , circumcircle , orthogonal , incenter , geometry

Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$. Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$. [i](Notation: $[\cdot]$ denotes the line segment.)[/i]

2024 Middle European Mathematical Olympiad, 6

Tags: incenter , geometry

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of the segment $BC$. Let $I, J, K$ be the incenters of triangles $ABC$, $ABM$, $ACM$, respectively. Let $P, Q$ be points on the lines $MK$, $MJ$, respectively, such that $\angle AJP=\angle ABC$ and $\angle AKQ=\angle BCA$. Let $R$ be the intersection of the lines $CP$ and $BQ$. Prove that the lines $IR$ and $BC$ are perpendicular.

2002 National Olympiad First Round, 17

Tags: symmetry , geometry , incenter , trapezoid

Let $ABCD$ be a trapezoid and a tangential quadrilateral such that $AD || BC$ and $|AB|=|CD|$. The incircle touches $[CD]$ at $N$. $[AN]$ and $[BN]$ meet the incircle again at $K$ and $L$, respectively. What is $\dfrac {|AN|}{|AK|} + \dfrac {|BN|}{|BL|}$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 16 $

1998 China Team Selection Test, 1

Tags: circumcircle , incenter , trigonometry , geometry

In acute-angled $\bigtriangleup ABC$, $H$ is the orthocenter, $O$ is the circumcenter and $I$ is the incenter. Given that $\angle C > \angle B > \angle A$, prove that $I$ lies within $\bigtriangleup BOH$.

2002 Korea Junior Math Olympiad, 7

Tags: incenter , geometry

$I$ is the incenter of $ABC$. $D$ is the intersection of $AI$ and the circumcircle of $ABC$, not $A$. And $P$ is a midpoint of $BI$. If $CI=2AI$, show that $AB=PD$.

1996 All-Russian Olympiad, 6

Tags: circumcircle , geometry , incenter

In the isosceles triangle $ABC$ ($AC = BC$) point $O$ is the circumcenter, $I$ the incenter, and $D$ lies on $BC$ so that lines $OD$ and $BI$ are perpendicular. Prove that $ID$ and $AC$ are parallel. [i]M. Sonkin[/i]

2021 Yasinsky Geometry Olympiad, 4

Tags: equal segments , geometry , incenter , concurrency

Given an acute triangle $ABC$, in which $\angle BAC = 60^o$. On the sides $AC$ and $AB$ take the points $T$ and $Q$, respectively, such that $CT = TQ = QB$. Prove that the center of the inscribed circle of triangle $ATQ$ lies on the side $BC$. (Dmitry Shvetsov)

2021-IMOC, G4

Tags: incenter , fixed point , geometry

Let $D$ be a point on the side $AC$ of a triangle $ABC$. Suppose that the incircle of triangle $BCD$ intersects $BD$ and $CD$ at $X$, $Y$, respectively. Show that $XY$ passes through a fixed point when $D$ is moving on the side $AC$.

2022 International Zhautykov Olympiad, 4

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 $IN > IM$.

2021 Czech and Slovak Olympiad III A, 2

Tags: circumcircle , incenter , geometry

Let $I$ denote the center of the circle inscribed in the right triangle $ABC$ with right angle at the vertex $A$. Next, denote by $M$ and $N$ the midpoints of the lines $AB$ and $BI$. Prove that the line $CI$ is tangent to the circumscribed circle of triangle $BMN$. (Patrik Bak, Josef Tkadlec)

2013 Online Math Open Problems, 32

Tags: geometry , trigonometry , inradius , trapezoid , incenter , circumcircle

In $\triangle ABC$ with incenter $I$, $AB = 61$, $AC = 51$, and $BC=71$. The circumcircles of triangles $AIB$ and $AIC$ meet line $BC$ at points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Determine the length of segment $DE$. [i]James Tao[/i]

2002 IMO Shortlist, 7

Tags: incenter , geometry , radical axis , polars , iran lemma

The incircle $ \Omega$ of the acute-angled triangle $ ABC$ is tangent to its side $ BC$ at a point $ K$. Let $ AD$ be an altitude of triangle $ ABC$, and let $ M$ be the midpoint of the segment $ AD$. If $ N$ is the common point of the circle $ \Omega$ and the line $ KM$ (distinct from $ K$), then prove that the incircle $ \Omega$ and the circumcircle of triangle $ BCN$ are tangent to each other at the point $ N$.

2009 Moldova Team Selection Test, 3

Tags: geometry , trigonometry , symmetry , circumcircle , geometric transformation , reflection , incenter

[color=darkred]Quadrilateral $ ABCD$ is inscribed in the circle of diameter $ BD$. Point $ A_1$ is reflection of point $ A$ wrt $ BD$ and $ B_1$ is reflection of $ B$ wrt $ AC$. Denote $ \{P\}\equal{}CA_1 \cap BD$ and $ \{Q\}\equal{}DB_1\cap AC$. Prove that $ AC\perp PQ$.[/color]

STEMS 2023 Math Cat A, 1

Tags: geometry , incenter , centroid

If in triangle $ABC$ , $AC$=$15$, $BC$=$13$ and $IG||AB$ where $I$ is the incentre and $G$ is the centroid , what is the area of triangle $ABC$ ?

2019 Moldova Team Selection Test, 10

Tags: circumcircle , incenter , geometry

The circle $\Omega$ with center $O$ is circumscribed to acute triangle $ABC$. Let $P$ be a point on the circumscribed circle of $OBC$, such that $P$ is inside $ABC$ and is different from $B$ and $C$. Bisectors of angles $BPA$ and $CPA$ intersect the sides $AB$ and $AC$ in points $E$ and $F.$ Prove that the incenters of triangles $PEF, PCA$ and $PBA$ are collinear.

2004 IMO Shortlist, 7

Tags: inradius , geometry , incenter , triangle

For a given triangle $ ABC$, let $ X$ be a variable point on the line $ BC$ such that $ C$ lies between $ B$ and $ X$ and the incircles of the triangles $ ABX$ and $ ACX$ intersect at two distinct points $ P$ and $ Q.$ Prove that the line $ PQ$ passes through a point independent of $ X$.

2009 Harvard-MIT Mathematics Tournament, 5

Tags: incenter , circumcircle , geometry

Circle $B$ has radius $6\sqrt{7}$. Circle $A$, centered at point $C$, has radius $\sqrt{7}$ and is contained in $B$. Let $L$ be the locus of centers $C$ such that there exists a point $D$ on the boundary of $B$ with the following property: if the tangents from $D$ to circle $A$ intersect circle $B$ again at $X$ and $Y$, then $XY$ is also tangent to $A$. Find the area contained by the boundary of $L$.

2007 Junior Balkan MO, 2

Tags: trigonometry , geometry , circumcircle , incenter , angle bisector , cyclic quadrilateral , function

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.

2010 Contests, 3

Tags: reflection , circumcircle , incenter , inradius , geometric transformation , euler , geometry

Let $A'\in(BC),$ $B'\in(CA),C'\in(AB)$ be the points of tangency of the excribed circles of triangle $\triangle ABC$ with the sides of $\triangle ABC.$ Let $R'$ be the circumradius of triangle $\triangle A'B'C'.$ Show that \[ R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)}\] where as usual, $R$ is the circumradius of $\triangle ABC,$ r is the inradius of $\triangle ABC,$ and $h_{a},h_{b},h_{c}$ are the lengths of altitudes of $\triangle ABC.$