Found problems: 1389
1989 Tournament Of Towns, (218) 2
The point $M$ , inside $\vartriangle ABC$, satisfies the conditions that $\angle BMC = 90^o +\frac12 \angle BAC$ and that the line $AM$ contains the centre of the circumscribed circle of $\vartriangle BMC$. Prove that $M$ is the centre of the inscribed circle of $\vartriangle ABC$.
2014 All-Russian Olympiad, 4
Given a triangle $ABC$ with $AB>BC$, $ \Omega $ is circumcircle. Let $M$, $N$ are lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. $K(.)=MN\cap AC$ and $P$ is incenter of the triangle $AMK$, $Q$ is K-excenter of the triangle $CNK$ (opposite to $K$ and tangents to $CN$). If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$.
M. Kungodjin
2008 China National Olympiad, 1
Suppose $\triangle ABC$ is scalene. $O$ is the circumcenter and $A'$ is a point on the extension of segment $AO$ such that $\angle BA'A = \angle CA'A$. Let point $A_1$ and $A_2$ be foot of perpendicular from $A'$ onto $AB$ and $AC$. $H_{A}$ is the foot of perpendicular from $A$ onto $BC$. Denote $R_{A}$ to be the radius of circumcircle of $\triangle H_{A}A_1A_2$. Similiarly we can define $R_{B}$ and $R_{C}$. Show that:
\[\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R}\]
where R is the radius of circumcircle of $\triangle ABC$.
2013 Kazakhstan National Olympiad, 2
Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.
1992 Baltic Way, 20
Let $ a\le b\le c$ be the sides of a right triangle, and let $ 2p$ be its perimeter. Show that
\[ p(p \minus{} c) \equal{} (p \minus{} a)(p \minus{} b) \equal{} S,
\]
where $ S$ is the area of the triangle.
2004 AMC 10, 22
A triangle with sides of $ 5$, $ 12$, and $ 13$ has both an inscibed and a circumscribed circle. What is the distance between the centers of those circles?
$ \textbf{(A)}\ \frac{3\sqrt{5}}{2}\qquad
\textbf{(B)}\ \frac{7}{2}\qquad
\textbf{(C)}\ \sqrt{15}\qquad
\textbf{(D)}\ \frac{\sqrt{65}}{2}\qquad
\textbf{(E)}\ \frac{9}{2}$
2011 Romania Team Selection Test, 3
The incircle of a triangle $ABC$ touches the sides $BC,CA,AB$ at points $D,E,F$, respectively. Let $X$ be a point on the incircle, different from the points $D,E,F$. The lines $XD$ and $EF,XE$ and $FD,XF$ and $DE$ meet at points $J,K,L$, respectively. Let further $M,N,P$ be points on the sides $BC,CA,AB$, respectively, such that the lines $AM,BN,CP$ are concurrent. Prove that the lines $JM,KN$ and $LP$ are concurrent.
[i]Dinu Serbanescu[/i]
2001 AIME Problems, 7
Let $\triangle{PQR}$ be a right triangle with $PQ=90$, $PR=120$, and $QR=150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{RST}$ and $C_{3}$ the inscribed circle of $\triangle{QUV}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt{10n}$. What is $n$?
2021 Taiwan Mathematics Olympiad, 4.
Let $I$ be the incenter of triangle $ABC$ and let $D$ the foot of altitude from $I$ to $BC$. Suppose the reflection point $D’$ of $D$ with respect to $I$ satisfying $\overline{AD’} = \overline{ID’}$. Let $\Gamma$ be the circle centered at $D’$ that passing through $A$ and $I$, and let $X$, $Y\neq A$ be the intersection of $\Gamma$ and $AB$, $AC$, respectively. Suppose $Z$ is a point on $\Gamma$ so that $AZ$ is perpendicular to $BC$.
Prove that $AD$, $D’Z$, $XY$ concurrent at a point.
1996 Romania Team Selection Test, 4
Let $ ABCD $ be a cyclic quadrilateral and let $ M $ be the set of incenters and excenters of the triangles $ BCD $, $ CDA $, $ DAB $, $ ABC $ (so 16 points in total). Prove that there exist two sets $ \mathcal{K} $ and $ \mathcal{L} $ of four parallel lines each, such that every line in $ \mathcal{K} \cup \mathcal{L} $ contains exactly four points of $ M $.
1979 Austrian-Polish Competition, 5
The circumcenter and incenter of a given tetrahedron coincide. Prove that all its faces are congruent.
2015 Czech-Polish-Slovak Junior Match, 1
Let $I$ be the center of the circle of the inscribed triangle $ABC$ and $M$ be the center of its side $BC$.
If $|AI| = |MI|$, prove that there are two of the sides of triangle $ABC$, of which one is twice of the other.
2009 IberoAmerican, 4
Given a triangle $ ABC$ of incenter $ I$, let $ P$ be the intersection of the external bisector of angle $ A$ and the circumcircle of $ ABC$, and $ J$ the second intersection of $ PI$ and the circumcircle of $ ABC$. Show that the circumcircles of triangles $ JIB$ and $ JIC$ are respectively tangent to $ IC$ and $ IB$.
2014 District Olympiad, 3
Let $ABC$ be a triangle in which $\measuredangle{A}=135^{\circ}$. The perpendicular to the line $AB$ erected at $A$ intersects the side $BC$ at $D$, and the angle bisector of $\angle B$ intersects the side $AC$ at $E$.
Find the measure of $\measuredangle{BED}$.
Kyiv City MO Seniors 2003+ geometry, 2009.10.4
In the triangle $ABC$ the angle bisectors $AL$ and $BT$ are drawn, which intersect at the point $I$, and their extensions intersect the circle circumscribed around the triangle $ABC$ at the points $E$ and $D$ respectively. The segment $DE$ intersects the sides $AC$ and $BC$ at the points $F$ and $K$, respectively. Prove that:
a) quadrilateral $IKCF$ is rhombus;
b) the side of this rhombus is $\sqrt {DF \cdot EK}$.
(Rozhkova Maria)
2014 PUMaC Individual Finals A, 1
Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) for which $AB+AC=3BC$. Let the point where $AC$ is tangent to $\gamma$ be $D$. Let the incenter of $I$. Let the intersection of the circumcircle of $\triangle BCI$ with $\gamma$ that is closer to $B$ be $P$. Show that $PID$ is collinear.
2024 Sharygin Geometry Olympiad, 1
Bisectors $AI$ and $CI$ meet the circumcircle of triangle $ABC$ at points $A_1, C_1$ respectively.
The circumcircle of triangle $AIC_1$ meets $AB$ at point $C_0$; point $A_0$ is defined similarly.
Prove that $A_0, A_1, C_0, C_1$ are collinear.
2015 India Regional MathematicaI Olympiad, 5
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I.$ Let the internal angle bisectors of $\angle A,\angle B,\angle C$ meet $\Gamma$ in $A',B',C'$ respectively. Let $B'C'$ intersect $AA'$ at $P,$ and $AC$ in $Q.$ Let $BB'$ intersect $AC$ in $R.$ Suppose the quadrilateral $PIRQ$ is a kite; that is, $IP=IR$ and $QP=QR.$ Prove that $ABC$ is an equilateral triangle.
2011 Northern Summer Camp Of Mathematics, 3
Given an acute triangle $ABC$ such that $\angle C< \angle B< \angle A$. Let $I$ be the incenter of $ABC$. Let $M$ be the midpoint of the smaller arc $BC$, $N$ be the midpoint of the segment $BC$ and let $E$ be a point such that $NE=NI$. The line $ME$ intersects circumcircle of $ABC$ at $Q$ (different from $A, B$, and $C$). Prove that
[b](i)[/b] The point $Q$ is on the smaller arc $AC$ of circumcircle of $ABC$.
[b](ii)[/b] $BQ=AQ+CQ$
2010 Regional Olympiad of Mexico Northeast, 3
In triangle $ABC$, $\angle BAC= 60^o$. Angle bisector of $\angle ABC$ meets side $AC$ at $X$ and angle bisector of $\angle BCA$ meets side $AB$ at $Y$. Prove that if $I$ is the incenter of triangle $ABC$, then $IX=IY$.
2001 Korea - Final Round, 2
In a triangle $ABC$ with $\angle B < 45^{\circ}$, $D$ is a point on $BC$ such that the incenter of $\triangle ABD$ coincides with the circumcenter $O$ of $\triangle ABC$. Let $P$ be the intersection point of the tangent lines to the circumcircle $\omega$ of $\triangle AOC$ at points $A$ and $C$. The lines $AD$ and $CO$ meet at $Q$. The tangent to $\omega$ at $O$ meets $PQ$ at $X$. Prove that $XO=XD$.
2009 Sharygin Geometry Olympiad, 4
Three parallel lines $d_a, d_b, d_c$ pass through the vertex of triangle $ABC$. The reflections of $d_a, d_b, d_c$ in $BC, CA, AB$ respectively form triangle $XYZ$. Find the locus of incenters of such triangles.
(C.Pohoata)
2017 Brazil National Olympiad, 3.
[b]3.[/b] A quadrilateral $ABCD$ has the incircle $\omega$ and is such that the semi-lines $AB$ and $DC$ intersect at point $P$ and the semi-lines $AD$ and $BC$ intersect at point $Q$. The lines $AC$ and $PQ$ intersect at point $R$. Let $T$ be the point of $\omega$ closest from line $PQ$. Prove that the line $RT$ passes through the incenter of triangle $PQC$.
2012 All-Russian Olympiad, 2
The points $A_1,B_1,C_1$ lie on the sides sides $BC,AC$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $I_A, I_B, I_C$ be the incentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the circumcentre of triangle $I_AI_BI_C$ is the incentre of triangle $ABC$.
2014 Taiwan TST Round 2, 1
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. A straight line $L$ is parallel to $BC$ and tangent to the incircle. Suppose $L$ intersects $IO$ at $X$, and select $Y$ on $L$ such that $YI$ is perpendicular to $IO$. Prove that $A$, $X$, $O$, $Y$ are cyclic.
[i]Proposed by Telv Cohl[/i]