Found problems: 3882
2013 India IMO Training Camp, 2
Let $ABCD$ by a cyclic quadrilateral with circumcenter $O$. Let $P$ be the point of intersection of the diagonals $AC$ and $BD$, and $K, L, M, N$ the circumcenters of triangles $AOP, BOP$, $COP, DOP$, respectively. Prove that $KL = MN$.
2023 Greece National Olympiad, 3
A triangle $ABC$ with $AB>AC$ is given, $AD$ is the A-angle bisector with point $D$ on $BC$ and point $I$ is the incenter of triangle $ABC$. Point M is the midpoint of segment $AD$ and point $F$ is the second intersection of $MB$ with the circumcirle of triangle $BIC$. Prove that $AF\bot FC$.
1969 IMO Shortlist, 5
$(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$
$(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas.
$(b)$ Find the locus of the centers of these hyperbolas.
2005 IMO Shortlist, 1
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
2006 Romania Team Selection Test, 4
Let $ABC$ be an acute triangle with $AB \neq AC$. Let $D$ be the foot of the altitude from $A$ and $\omega$ the circumcircle of the triangle. Let $\omega_1$ be the circle tangent to $AD$, $BD$ and $\omega$. Let $\omega_2$ be the circle tangent to $AD$, $CD$ and $\omega$. Let $\ell$ be the interior common tangent to both $\omega_1$ and $\omega_2$, different from $AD$.
Prove that $\ell$ passes through the midpoint of $BC$ if and only if $2BC = AB + AC$.
1995 Turkey Team Selection Test, 3
Let $D$ be a point on the small arc $AC$ of the circumcircle of an equilateral triangle $ABC$, different from $A$ and $C$. Let $E$ and $F$ be the projections of $D$ onto $BC$ and $AC$ respectively. Find the locus of the intersection point of $EF$ and $OD$, where $O$ is the center of $ABC$.
2016 Brazil Team Selection Test, 1
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.
[i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]
2006 Oral Moscow Geometry Olympiad, 6
Given triangle $ABC$ and points $P$. Let $A_1,B_1,C_1$ be the second points of intersection of straight lines $AP, BP, CP$ with the circumscribed circle of $ABC$. Let points $A_2, B_2, C_2$ be symmetric to $A_1,B_1,C_1$ wrt $BC,CA,AB$, respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.
(A. Zaslavsky)
2019 Junior Balkan Team Selection Tests - Moldova, 11
Let $I$ be the center of inscribed circle of right triangle $\Delta ABC$ with $\angle A = 90$ and point $M$ is the midpoint of $(BC)$.The bisector of $\angle BAC$ intersects the circumcircle of $\Delta ABC $ in point $W$.Point $U$ is situated on the line $AB$ such that the lines $AB$ and $WU$ are perpendiculars.Point $P$ is situated on the line $WU$ such that the lines $PI$ and $WU$ are perpendiculars.Prove that the line $MP$ bisects the segment $CI$.
2012 Greece Junior Math Olympiad, 1
Let $ABC$ be an acute angled triangle (with $AB<AC<BC$) inscribed in circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ (with center $A$ and radius $AB$) intersects side $BC$ at point $D$ and the circumcircle $c(O,R)$ at point $E$. Prove that side $AC$ bisects angle $\angle DAE$.
2013 Olympic Revenge, 2
Let $ABC$ to be an acute triangle. Also, let $K$ and $L$ to be the two intersections of the perpendicular from $B$ with respect to side $AC$ with the circle of diameter $AC$, with $K$ closer to $B$ than $L$. Analogously, $X$ and $Y$ are the two intersections of the perpendicular from $C$ with respect to side $AB$ with the circle of diamter $AB$, with $X$ closer to $C$ than $Y$. Prove that the intersection of $XL$ and $KY$ lies on $BC$.
2024 Kyiv City MO Round 2, Problem 3
Let $AH_A, BH_B, CH_C$ be the altitudes of the triangle $ABC$. Points $A_1$ and $C_1$ are the projections of the point $H_B$ onto the sides $AB$ and $BC$, respectively. $B_1$ is the projection of $B$ onto $H_AH_C$. Prove that the diameter of the circumscribed circle of $\triangle A_1B_1C_1$ is equal to $BH_B$.
[i]Proposed by Anton Trygub[/i]
2022 Sharygin Geometry Olympiad, 9.6
Lateral sidelines $AB$ and $CD$ of a trapezoid $ABCD$ ($AD >BC$) meet at point $P$. Let $Q$ be a point of segment $AD$ such that $BQ = CQ$. Prove that the line passing through the circumcenters of triangles $AQC$ and $BQD$ is perpendicular to $PQ$.
2012 Korea National Olympiad, 1
Let $ ABC $ be an obtuse triangle with $ \angle A > 90^{\circ} $. Let circle $ O $ be the circumcircle of $ ABC $. $ D $ is a point lying on segment $ AB $ such that $ AD = AC $. Let $ AK $ be the diameter of circle $ O $. Two lines $ AK $ and $ CD $ meet at $ L $. A circle passing through $ D, K, L $ meets with circle $ O $ at $ P ( \ne K ) $ . Given that $ AK = 2, \angle BCD = \angle BAP = 10^{\circ} $, prove that $ DP = \sin ( \frac{ \angle A}{2} )$.
2005 Brazil National Olympiad, 5
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]
the 16th XMO, 2
In a triangle $ABC$ , let $O$ be the circumcenter , $AO$ meet $BC$ at $K$ , A circle $\Omega$ with the centre $T$ and the center $K$ and the radius $AK$ meet $AC$ again at $T$ , $D$ is a point on the plain satisfies that $BC$ is the bisector of the angle $\angle ABD$ , let the orthocenter of the triangle $ABC$ and $BCD$ be $M$ and $N$ . If $MN//AC$ than $DT$ is tangent to $\Omega$
2019 Bulgaria EGMO TST, 2
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ centered at $O$, whose diagonals intersect at $H$. Let $O_1$ and $O_2$ be the circumcenters of triangles $AHD$ and $BHC$. A line through $H$ intersects $\omega$ at $M_1$ and $M_2$ and intersects the circumcircles of triangles $O_1HO$ and $O_2HO$ at $N_1$ and $N_2$, respectively, so that $N_1$ and $N_2$ lie inside $\omega$. Prove that $M_1N_1 = M_2N_2$.
2010 N.N. Mihăileanu Individual, 3
Let $ Q $ be a point, $ H,O $ be the orthocenter and circumcenter, respectively, of a triangle $ ABC, $ and $ D,E,F, $ be the symmetric points of $ Q $ with respect to $ A,B,C, $ respectively. Also, $ M,N,P $ are the middle of the segments $ AE,BF,CD, $ and $ G,G',G'' $ are the centroids of $ ABC,MNP,DEF, $ respectively. Prove the following propositions:
[b]a)[/b] $ \frac{1}{2}\overrightarrow{OG} =\frac{1}{3}\overrightarrow{OG'}=\frac{1}{4}\overrightarrow{OG''} $
[b]b)[/b] $ Q=O\implies \overrightarrow{OG'} =\overrightarrow{G'H} $
[b]c)[/b] $ Q=H\implies G'=O $
[i]Cătălin Zîrnă[/i]
2014 APMO, 5
Circles $\omega$ and $\Omega$ meet at points $A$ and $B$. Let $M$ be the midpoint of the arc $AB$ of circle $\omega$ ($M$ lies inside $\Omega$). A chord $MP$ of circle $\omega$ intersects $\Omega$ at $Q$ ($Q$ lies inside $\omega$). Let $\ell_P$ be the tangent line to $\omega$ at $P$, and let $\ell_Q$ be the tangent line to $\Omega$ at $Q$. Prove that the circumcircle of the triangle formed by the lines $\ell_P$, $\ell_Q$ and $AB$ is tangent to $\Omega$.
[i]Ilya Bogdanov, Russia and Medeubek Kungozhin, Kazakhstan[/i]
2016 Azerbaijan Team Selection Test, 1
Tangents from the point $A$ to the circle $\Gamma$ touche this circle at $C$ and $D$.Let $B$ be a point on $\Gamma$,different from $C$ and $D$. The circle $\omega$ that passes through points $A$ and $B$ intersect with lines $AC$ and $AD$ at $F$ and $E$,respectively.Prove that the circumcircles of triangles $ABC$ and $DEB$ are tangent if and only if the points $C,D,F$ and $E$ are cyclic.
2016 China Girls Math Olympiad, 7
In acute triangle $ABC, AB<AC$, $I$ is its incenter, $D$ is the foot of perpendicular from $I$ to $BC$, altitude $AH$ meets $BI,CI$ at $P,Q$ respectively. Let $O$ be the circumcenter of $\triangle IPQ$, extend $AO$ to meet $BC$ at $L$. Circumcircle of $\triangle AIL$ meets $BC$ again at $N$. Prove that $\frac{BD}{CD}=\frac{BN}{CN}$.
2012 Online Math Open Problems, 47
Let $ABCD$ be an isosceles trapezoid with bases $AB=5$ and $CD=7$ and legs $BC=AD=2 \sqrt{10}.$ A circle $\omega$ with center $O$ passes through $A,B,C,$ and $D.$ Let $M$ be the midpoint of segment $CD,$ and ray $AM$ meet $\omega$ again at $E.$ Let $N$ be the midpoint of $BE$ and $P$ be the intersection of $BE$ with $CD.$ Let $Q$ be the intersection of ray $ON$ with ray $DC.$ There is a point $R$ on the circumcircle of $PNQ$ such that $\angle PRC = 45^\circ.$ The length of $DR$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]Author: Ray Li[/i]
2019 China Team Selection Test, 1
$ABCDE$ is a cyclic pentagon, with circumcentre $O$. $AB=AE=CD$. $I$ midpoint of $BC$. $J$ midpoint of $DE$. $F$ is the orthocentre of $\triangle ABE$, and $G$ the centroid of $\triangle AIJ$.$CE$ intersects $BD$ at $H$, $OG$ intersects $FH$ at $M$. Show that $AM\perp CD$.
2006 Germany Team Selection Test, 2
In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively.
Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ .
[i]Proposed by Hojoo Lee, Korea[/i]
2018 Slovenia Team Selection Test, 4
Let $\mathcal{K}$ be a circle centered in $A$. Let $p$ be a line tangent to $\mathcal{K}$ in $B$ and let a line parallel to $p$ intersect $\mathcal{K}$ in $C$ and $D$. Let the line $AD$ intersect $p$ in $E$ and let $F$ be the intersection of the lines $CE$ and $AB$. Prove that the line through $D$, parallel to the tangent through $A$ to the circumcircle of $AFD$ intersects the line $CF$ on $\mathcal{K}$.