Found problems: 333
1914 Eotvos Mathematical Competition, 3
The circle k intersects the sides $BC$, $CA$, $AB$ of triangle $ABC$ in points $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$. The perpendiculars to $BC$, $CA$, $AB$ through $A_1$, $B_1$, $C_1$, respectively, meet at a point $M$. Prove that the three perpendiculars to $BC$, $CA$, $AB$ through $A_2$, $B_2$, and $C_2$, respectively, also meet in one point.
2020-IMOC, G5
Let $O, H$ be the circumcentor and the orthocenter of a scalene triangle $ABC$. Let $P$ be the reflection of $A$ w.r.t. $OH$, and $Q$ is a point on $\odot (ABC)$ such that $AQ, OH, BC$ are concurrent. Let $A'$ be a points such that $ABA'C$ is a parallelogram. Show that $A', H, P, Q$ are concylic.
(ltf0501).
2014 Oral Moscow Geometry Olympiad, 3
The bisectors $AA_1$ and $CC_1$ of triangle $ABC$ intersect at point $I$. The circumscribed circles of triangles $AIC_1$ and $CIA_1$ intersect the arcs $AC$ and $BC$ (not containing points $B$ and $A$ respectively) of the circumscribed circle of triangle $ABC$ at points $C_2$ and $A_2$, respectively. Prove that lines $A_1A_2$ and $C_1C_2$ intersect on the circumscribed circle of triangle $ABC$.
2005 China Team Selection Test, 1
Point $P$ lies inside triangle $ABC$. Let the projections of $P$ onto sides $BC$,$CA$,$AB$ be $D$, $E$, $F$ respectively. Let the projections from $A$ to the lines $BP$ and $CP$ be $M$ and $N$ respectively. Prove that $ME$, $NF$ and $BC$ are concurrent.
2000 Junior Balkan Team Selection Tests - Moldova, 7
Let a triangle $ABC, A_1$ be the midpoint of the segment $[BC], B_1 \in (AC)$ ¸and $C_1 \in (AB)$ such that $[A_1B_1$ is the bisector of the angle $AA_1C$ and $A_1C_1$ is perpendicular to $AB$. Show that the lines $AA_1, BB_1$ and $CC_1$ are concurrent if and only if $ \angle BAC = 90^o$
2006 Sharygin Geometry Olympiad, 9
$L(a)$ is the line connecting the points of the unit circle corresponding to the angles $a$ and $\pi - 2a$.
Prove that if $a + b + c = 2\pi$, then the lines $L (a), L (b)$ and $L (c)$ intersect at one point.
2004 Bosnia and Herzegovina Team Selection Test, 6
It is given triangle $ABC$ and parallelogram $ASCR$ with diagonal $AC$. Let line constructed through point $B$ parallel with $CS$ intersects line $AS$ and $CR$ in $M$ and $P$, respectively. Let line constructed through point $B$ parallel with $AS$ intersects line $AR$ and $CS$ in $N$ and $Q$, respectively. Prove that lines $RS$, $MN$ and $PQ$ are concurrent
1962 Polish MO Finals, 3
What condition should the angles of triangle $ ABC $ satisfy so that the bisector of angle $ A $, the median drawn from vertex $ B $ and the altitude drawn from vertex $ C $ intersect at one point?
2023 Yasinsky Geometry Olympiad, 6
Let $ABC$ be a scalene triangle. Let $\ell$ be a line passing through point $B$ that lies outside of the triangle $ABC$ and creates different angles with sides $AB$ and $BC$ . The point $M$ is the midpoint of side $AC$ and the ponts $H_a$ and $H_c$ are the bases of the perpendicular lines on the line $\ell$ drawn from points $A$ and $C$ respectively. The circle circumscribing triangle $MBH_a$ intersects AB at the point $A_1$ and the circumscribed circle of triangle $MBH_c$ intersects $BC$ at point $C_1$. The point $A_2$ is symmetric to the point $A$ relative to the point $A_1$ and the point $C_2$ is symmetric to the point $C_1$ relative to the point $C_1$. Prove that the lines $\ell$, $AC_2$ and $CA_2$ intersect at one point.
(Yana Kolodach)
2020 European Mathematical Cup, 1
Let $ABCD$ be a parallelogram such that $|AB| > |BC|$. Let $O$ be a point on the line $CD$ such that $|OB| = |OD|$. Let $\omega$ be a circle with center $O$ and radius $|OC|$. If $T$ is the second intersection of $\omega$ and $CD$, prove that $AT, BO$ and $\omega$ are concurrent.
[i]Proposed by Ivan Novak[/i]
2019 Romania Team Selection Test, 3
Let $AD, BE$, and $CF$ denote the altitudes of triangle $\vartriangle ABC$. Points $E'$ and $F'$ are the reflections of $E$ and $F$ over $AD$, respectively. The lines $BF'$ and $CE'$ intersect at $X$, while the lines $BE'$ and $CF'$ intersect at the point $Y$. Prove that if $H$ is the orthocenter of $\vartriangle ABC$, then the lines $AX, YH$, and $BC$ are concurrent.
1956 Moscow Mathematical Olympiad, 333
Let $O$ be the center of the circle circumscribed around $\vartriangle ABC$, let $A_1, B_1, C_1$ be symmetric to $O$ through respective sides of $\vartriangle ABC$. Prove that all altitudes of $\vartriangle A_1B_1C_1$ pass through $O$, and all altitudes of $\vartriangle ABC$ pass through the center of the circle circumscribed around $\vartriangle A_1B_1C_1$.
2009 Bulgaria National Olympiad, 2
In the triangle $ABC$ its incircle with center $I$ touches its sides $BC, CA$ and $AB$ in the points $A_1, B_1, C_1$ respectively. Through $I$ is drawn a line $\ell$. The points $A', B'$ and $C'$ are reflections of $A_1, B_1, C_1$ with respect to the line $\ell$. Prove that the lines $AA', BB'$ and $CC'$ intersects at a common point.
2013 Saudi Arabia Pre-TST, 3.4
$\vartriangle ABC$ is a triangle with $AB < BC, \Gamma$ its circumcircle, $K$ the midpoint of the minor arc $CA$ of the circle $C$ and $T$ a point on $\Gamma$ such that $KT$ is perpendicular to $BC$. If $A',B'$ are the intouch points of the incircle of $\vartriangle ABC$ with the sides $BC,AC$, prove that the lines $AT,BK,A'B'$ are concurrent.
2008 Korea Junior Math Olympiad, 5
Let there be a pentagon $ABCDE$ inscribed in a circle $O$. The tangent to $O$ at $E$ is parallel to $AD$. A point $F$ lies on $O$ and it is in the opposite side of $A$ with respect to $CD$, and satisfi es $AB \cdot BC \cdot DF = AE \cdot ED \cdot CF$ and $\angle CFD = 2\angle BFE$. Prove that the tangent to $O$ at $B,E$ and line $AF$ concur at one point.
Geometry Mathley 2011-12, 16.1
Let $ABCD$ be a cyclic quadrilateral with two diagonals intersect at $E$. Let $ M$, $N$, $P$, $Q$ be the reflections of $ E $ in midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Prove that the Euler lines of $ \triangle MAB$, $\triangle NBC$, $\triangle PCD,$ $\triangle QDA$ are concurrent.
Trần Quang Hùng
2004 Switzerland - Final Round, 9
Let $ABCD$ be a cyclic quadrilateral, so that $|AB| + |CD| = |BC|$. Show that the intersection of the bisector of $\angle DAB$ and $\angle CDA$ lies on the side $BC$.
2007 Sharygin Geometry Olympiad, 2
Points $A', B', C'$ are the feet of the altitudes $AA', BB'$ and $CC'$ of an acute triangle $ABC$. A circle with center $B$ and radius $BB'$ meets line $A'C'$ at points $K$ and $L$ (points $K$ and $A$ are on the same side of line $BB'$). Prove that the intersection point of lines $AK$ and $CL$ belongs to line $BO$ ($O$ is the circumcenter of triangle $ABC$).
2003 Cuba MO, 9
Let $D$ be the midpoint of the base $AB$ of the isosceles and acute angle triangle $ABC$, $E$ is a point on $AB$ and $O$ circumcenter of the triangle $ACE$. Prove that the line that passes through $D$ perpendicular to $DO$, the line that passes through $E$ perpendicular to $BC$ and the line that passes through$ B$ parallel to $AC$, they intersect at a point.
Swiss NMO - geometry, 2016.5
Let $ABC$ be a right triangle with $\angle ACB = 90^o$ and M the center of $AB$. Let $G$ br any point on the line $MC$ and $P$ a point on the line $AG$, such that $\angle CPA = \angle BAC$ . Further let $Q$ be a point on the straight line $BG$, such that $\angle BQC = \angle CBA$ . Show that the circles of the triangles $AQG$ and $BPG$ intersect on the segment $AB$.
2021 Iranian Geometry Olympiad, 2
Two circles $\Gamma_1$ and $\Gamma_2$ meet at two distinct points $A$ and $B$. A line passing through $A$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ respectively, such that $A$ lies between $C$ and $D$. The tangent at $A$ to $\Gamma_2$ meets $\Gamma_1$ again at $E$. Let $F$ be a point on $\Gamma_2$ such that $F$ and $A$ lie on different sides of $BD$, and $2\angle AFC=\angle ABC$. Prove that the tangent at $F$ to $\Gamma_2$, and lines $BD$ and $CE$ are concurrent.
Kyiv City MO Seniors 2003+ geometry, 2015.11.4
In the acute-angled triangle $ ABC $, the sides $ AB $ and $BC$ have different lengths, and the extension of the median $ BM $ intersects the circumscribed circle at the point $ N $. On this circle we note such a point $ D $ that $ \angle BDH = 90 {} ^ \circ $, where $ H $ is the point of intersection of the altitudes of the triangle $ ABC $. The point $K$ is chosen so that $ ANCK $ is a parallelogram. Prove that the lines $ AC $, $ KH $ and $ BD $ intersect at one point.
(Igor Nagel)
1993 Poland - Second Round, 2
Let be given a circle with center $O$ and a point $P$ outside the circle. A line $l$ passes through $P$ and cuts the circle at $A$ and $B$. Let $C$ be the point symmetric to $A$ with respect to $OP$, and let $m$ be the line $BC$. Prove that all lines $m$ have a common point as $l$ varies.
2014 Czech-Polish-Slovak Match, 4
Let $ABC$ be a triangle, and let $P$ be the midpoint of $AC$. A circle intersects $AP, CP, BC, AB$ sequentially at their inner points $K, L, M, N$. Let $S$ be the midpoint of $KL$. Let also $2 \cdot | AN |\cdot |AB |\cdot |CL | = 2 \cdot | CM | \cdot| BC | \cdot| AK| = | AC | \cdot| AK |\cdot |CL |.$ Prove that if $P\ne S$, then the intersection of $KN$ and $ML$ lies on the perpendicular bisector of the $PS$.
(Jan Mazák)
2016 Saint Petersburg Mathematical Olympiad, 3
In a tetrahedron, the midpoints of all the edges lie on the same sphere. Prove that it's altitudes intersect at one point.