Found problems: 3882
2016 Sharygin Geometry Olympiad, 6
A triangle $ABC$ is given. The point $K$ is the base of the external bisector of angle $A$. The point $M$ is the midpoint of the arc $AC$ of the circumcircle. The point $N$ on the bisector of angle $C$ is such that $AN \parallel BM$. Prove that the points $M,N,K$ are collinear.
[i](Proposed by Ilya Bogdanov)[/i]
2005 China Western Mathematical Olympiad, 2
Given three points $P$, $A$, $B$ and a circle such that the lines $PA$ and $PB$ are tangent to the circle at the points $A$ and $B$, respectively. A line through the point $P$ intersects that circle at two points $C$ and $D$. Through the point $B$, draw a line parallel to $PA$; let this line intersect the lines $AC$ and $AD$ at the points $E$ and $F$, respectively. Prove that $BE = BF$.
2021 Taiwan TST Round 2, G
Let $ABC$ be a triangle with circumcircle $\Gamma$, and points $E$ and $F$ are chosen from sides $CA$, $AB$, respectively. Let the circumcircle of triangle $AEF$ and $\Gamma$ intersect again at point $X$. Let the circumcircles of triangle $ABE$ and $ACF$ intersect again at point $K$. Line $AK$ intersect with $\Gamma$ again at point $M$ other than $A$, and $N$ be the reflection point of $M$ with respect to line $BC$. Let $XN$ intersect with $\Gamma$ again at point $S$ other that $X$.
Prove that $SM$ is parallel to $BC$.
[i] Proposed by Ming Hsiao[/i]
2015 India National Olympiad, 1
Let $ABC$ be a right-angled triangle with $\angle{B}=90^{\circ}$. Let $BD$ is the altitude from $B$ on $AC$. Let $P,Q$ and $I $be the incenters of triangles $ABD,CBD$ and $ABC$ respectively.Show that circumcenter of triangle $PIQ$ lie on the hypotenuse $AC$.
2010 Serbia National Math Olympiad, 2
In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$.
[i]Proposed by Marko Djikic[/i]
2009 China Team Selection Test, 2
In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$
2012 Middle European Mathematical Olympiad, 6
Let $ ABCD $ be a convex quadrilateral with no pair of parallel sides, such that $ \angle ABC = \angle CDA $. Assume that the intersections of the pairs of neighbouring angle bisectors of $ ABCD $ form a convex quadrilateral $ EFGH $. Let $ K $ be the intersection of the diagonals of $ EFGH$. Prove that the lines $ AB $ and $ CD $ intersect on the circumcircle of the triangle $ BKD $.
2010 Contests, 2
In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$.
[i]Proposed by Marko Djikic[/i]
2013 Sharygin Geometry Olympiad, 13
Let $A_1$ and $C_1$ be the tangency points of the incircle of triangle $ABC$ with $BC$ and $AB$ respectively, $A'$ and $C'$ be the tangency points of the excircle inscribed into the angle $B$ with the extensions of $BC$ and $AB$ respectively. Prove that the orthocenter $H$ of triangle $ABC$ lies on $A_1C_1$ if and only if the lines $A'C_1$ and $BA$ are orthogonal.
2011 International Zhautykov Olympiad, 3
Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $K.$ The midpoints of diagonals $AC$ and $BD$ are $M$ and $N,$ respectively. The circumscribed circles $ADM$ and $BCM$ intersect at points $M$ and $L.$ Prove that the points $K ,L ,M,$ and $ N$ lie on a circle. (all points are supposed to be different.)
2013 Rioplatense Mathematical Olympiad, Level 3, 6
Let $ABC$ be an acute scalene triangle, $H$ its orthocenter and $G$ its geocenter. The circumference with diameter $AH$ cuts the circumcircle of $BHC$ in $A'$ ($A' \neq H$). Points $B'$ and $C'$ are defined similarly. Show that the points $A'$, $B'$, $C'$, and $G$ lie in one circumference.
2014 ELMO Shortlist, 9
Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$.
[i]Proposed by Sammy Luo[/i]
2012 Belarus Team Selection Test, 3
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
[i]Proposed by Härmel Nestra, Estonia[/i]
2014 All-Russian Olympiad, 2
The sphere $ \omega $ passes through the vertex $S$ of the pyramid $SABC$ and intersects with the edges $SA,SB,SC$ at $A_1,B_1,C_1$ other than $S$. The sphere $ \Omega $ is the circumsphere of the pyramid $SABC$ and intersects with $ \omega $ circumferential, lies on a plane which parallel to the plane $(ABC)$.
Points $A_2,B_2,C_2$ are symmetry points of the points $A_1,B_1,C_1$ respect to midpoints of the edges $SA,SB,SC$ respectively. Prove that the points $A$, $B$, $C$, $A_2$, $B_2$, and $C_2$ lie on a sphere.
2017 Regional Olympiad of Mexico Southeast, 5
Consider an acutangle triangle $ABC$ with circumcenter $O$. A circumference that passes through $B$ and $O$ intersect sides $BC$ and $AB$ in points $P$ and $Q$. Prove that the orthocenter of triangle $OPQ$ is on $AC$.
2009 Korea - Final Round, 2
$ABC$ is an obtuse triangle. (angle $B$ is obtuse) Its circumcircle is $O$. A tangent line for $O$ passing $C$ meets with $AB$ at $B_1$. Let $O_1$ be a circumcenter of triangle $AB_1C$. $B_2$ is a point on the segment $BB_1$. Let $C_1$ be a contact point of the tangent line for $O$ passing $B_2$, which is more closer to $C$. Let $O_2$ be a circumcenter of triangle $AB_2C_1$. Prove that if $OO_2$ and $AO_1$ is perpendicular, then five points $O,O_2,O_1,C_1,C$ are concyclic.
2004 Mediterranean Mathematics Olympiad, 4
Let $z_1, z_2, z_3$ be pairwise distinct complex numbers satisfying $|z_1| = |z_2| = |z_3| = 1$ and
\[\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1.\]
If the points $A(z_1),B(z_2),C(z_3)$ are vertices of an acute-angled triangle, prove that this triangle is equilateral.
2003 Baltic Way, 13
In a rectangle $ABCD$ be a rectangle and $BC = 2AB$, let $E$ be the midpoint of $BC$ and $P$ an arbitrary inner point of $AD$. Let $F$ and $G$ be the feet of perpendiculars drawn correspondingly from $A$ to $BP$ and from $D$ to $CP$. Prove that the points $E,F,P,G$ are concyclic.
2017 Sharygin Geometry Olympiad, 8
Let $ABCD$ be a square, and let $P$ be a point on the minor arc $CD$ of its circumcircle. The lines $PA, PB$ meet the diagonals $BD, AC$ at points $K, L$ respectively. The points $M, N$ are the projections of $K, L$ respectively to $CD$, and $Q$ is the common point of lines $KN$ and $ML$. Prove that $PQ$ bisects the segment $AB$.
2002 Pan African, 2
$\triangle{AOB}$ is a right triangle with $\angle{AOB}=90^{o}$. $C$ and $D$ are moving on $AO$ and $BO$ respectively such that $AC=BD$. Show that there is a fixed point $P$ through which the perpendicular bisector of $CD$ always passes.
2011 Romanian Masters In Mathematics, 3
A triangle $ABC$ is inscribed in a circle $\omega$.
A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$).
Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$.
Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$.
[i](Russia) Vasily Mokin and Fedor Ivlev[/i]
2009 Indonesia MO, 4
Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$.
(a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$.
(b) Show that $ A_1KML$ is a cyclic quadrilateral.
2007 Italy TST, 2
Let $ABC$ a acute triangle.
(a) Find the locus of all the points $P$ such that, calling $O_{a}, O_{b}, O_{c}$ the circumcenters of $PBC$, $PAC$, $PAB$:
\[\frac{ O_{a}O_{b}}{AB}= \frac{ O_{b}O_{c}}{BC}=\frac{ O_{c}O_{a}}{CA}\]
(b) For all points $P$ of the locus in (a), show that the lines $AO_{a}$, $BO_{b}$ , $CO_{c}$ are cuncurrent (in $X$);
(c) Show that the power of $X$ wrt the circumcircle of $ABC$ is:
\[-\frac{ a^{2}+b^{2}+c^{2}-5R^{2}}4\]
Where $a=BC$ , $b=AC$ and $c=AB$.
Kyiv City MO Seniors Round2 2010+ geometry, 2018.10.3.1
The point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AC$ intersects the circumscribed circle $\Delta ABO$ for second time at the point $X$. Prove that $XO \bot BC$.
2015 Stars Of Mathematics, 3
Let $ABCD$ be cyclic quadrilateral,let $\gamma$ be it's circumscribed circle and let $M$ be the midpoint of arc $AB$ of $\gamma$,which does not contain points $C,D$.The line that passes through $M$ and the intersection point of diagonals $AC,BD$,intersects $\gamma$ in $N\neq M$.
Let $P,Q$ be two points situated on $CD$,such that $\angle{AQD}=\angle{DAP}$ and $\angle{BPC}=\angle{CBQ}$.Prove that circles $\odot(NPQ)$ and $\gamma$ are tangent.