Found problems: 241
2008 Postal Coaching, 5
Consider the triangle $ABC$ and the points $D \in (BC),E \in (CA), F \in (AB)$, such that $\frac{BD}{DC}=\frac{CE}{EA}=\frac{AF}{FB}$. Prove that if the circumcenters of triangles $DEF$ and $ABC$ coincide, then the triangle $ABC$ is equilateral.
2019 Romanian Master of Mathematics Shortlist, G1
Let $BM$ be a median in an acute-angled triangle $ABC$. A point $K$ is chosen on the line through $C$ tangent to the circumcircle of $\vartriangle BMC$ so that $\angle KBC = 90^\circ$. The segments $AK$ and $BM$ meet at $J$. Prove that the circumcenter of $\triangle BJK$ lies on the line $AC$.
Aleksandr Kuznetsov, Russia
2022-IMOC, G1
The circumcenter and orthocenter of $ABC$ are $O$ and $H$, respectively. Let $XACH$ be a parallelogram. Show that if $OH$ is parallel to $BC$, then $OX$ and $AB$ intersect at some point on the perpendicular bisector of $AH$.
[i]proposed by USJL[/i]
Mathley 2014-15, 4
Points $E, F$ are in the plane of triangle $ABC$ so that triangles $ABE$ and $ACF$ are the opposite directed, and the two triangles are isosceles in that $BE = AE, AF = CF$. Let $H, K$ be the orthocenter of triangle $ABE, ACF$ respectively. Points $M, N$ are the intersections of $BE$ and $CF, CK$ and $CH$. Prove that $MN$ passes through the center of the circumcircle of triangle $ABC$.
Nguyen Minh Ha, High School for Education, Hanoi Pedagogical University
Brazil L2 Finals (OBM) - geometry, 2010.2
Let $ABCD$ be a parallelogram and $\omega$ be the circumcircle of the triangle $ABD$. Let $E ,F$ be the intersections of $\omega$ with lines $BC ,CD$ respectively . Prove that the circumcenter of the triangle $CEF$ lies on $\omega$.
2022 Saudi Arabia BMO + EGMO TST, 2.1
Let $ABC$ be an acute-angled triangle. Point $P$ is such that $AP = AB$ and $PB \parallel AC$. Point $Q$ is such that $AQ = AC$ and $CQ \parallel AB$. Segments $CP$ and $BQ$ meet at point $X$. Prove that the circumcenter of triangle $ABC$ lies on the circumcircle of triangle $PXQ$.
2022 Korea National Olympiad, 2
In a scalene triangle $ABC$, let the angle bisector of $A$ meets side $BC$ at $D$. Let $E, F$ be the circumcenter of the triangles $ABD$ and $ADC$, respectively. Suppose that the circumcircles of the triangles $BDE$ and $DCF$ intersect at $P(\neq D)$, and denote by $O, X, Y$ the circumcenters of the triangles $ABC, BDE, DCF$, respectively. Prove that $OP$ and $XY$ are parallel.
1965 All Russian Mathematical Olympiad, 058
A circle is circumscribed around the triangle $ABC$. Chords, from the midpoint of the arc $AC$ to the midpoints of the arcs $AB$ and $BC$, intersect sides $[AB]$ and $[BC]$ in the points $D$ and $E$. Prove that $(DE)$ is parallel to $(AC)$ and passes through the centre of the inscribed circle.
2001 IMO Shortlist, 2
Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.
2015 Sharygin Geometry Olympiad, P2
Let $O$ and $H$ be the circumcenter and the orthocenter of a triangle $ABC$. The line passing through the midpoint of $OH$ and parallel to $BC$ meets $AB$ and $AC$ at points $D$ and $E$. It is known that $O$ is the incenter of triangle $ADE$. Find the angles of $ABC$.
2013 BAMO, 3
Let $H$ be the orthocenter of an acute triangle $ABC$. (The orthocenter is the point at the intersection of the three altitudes. An acute triangle has all angles less than $90^o$.) Draw three circles: one passing through $A, B$, and $H$, another passing through $B, C$, and $H$, and finally, one passing through $C, A$, and $H$. Prove that the triangle whose vertices are the centers of those three circles is congruent to triangle $ABC$.
2005 Sharygin Geometry Olympiad, 5
There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles:
a) points of intersection of heights,
b) the intersection points of the medians,
c) the centers of the circumscribed circles.
2019 Canada National Olympiad, 1
Points $A,B,C$ are on a plane such that $AB=BC=CA=6$. At any step, you may choose any three existing points and draw that triangle's circumcentre. Prove that you can draw a point such that its distance from an previously drawn point is:
$(a)$ greater than 7
$(b)$ greater than 2019
2021 Baltic Way, 14
Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. Denote by $M$ the midpoint of $BC$. The point $D$ is the reflection of $A$ over $BC$, and the point $E$ is the intersection of $\Gamma$ and the ray $MD$. Let $S$ be the circumcentre of the triangle $ADE$. Prove that the points $A$, $E$, $M$, $O$, and $S$ lie on the same circle.
2018 Brazil Team Selection Test, 4
Consider an isosceles triangle $ABC$ with $AB = AC$. Let $\omega(XYZ)$ be the circumcircle of the triangle $XY Z$. The tangents to $\omega(ABC)$ through $B$ and $C$ meet at the point $D$. The point $F$ is marked on the arc $AB$ of $\omega(ABC)$ that does not contain $C$. Let $K$ be the point of intersection of lines $AF$ and $BD$ and $L$ the point of intersection of the lines $AB$ and $CF$. Let $T$ and $S$ be the centers of the circles $\omega(BLC)$ and $\omega(BLK)$, respectively. Suppose that the circles $\omega(BTS)$ and $\omega(CFK)$ are tangent to each other at the point $P$. Prove that $P$ belongs to the line $AB$.
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 $\vartriangle A_1B_1C_1$.
2016 Federal Competition For Advanced Students, P1, 2
We are given an acute triangle $ABC$ with $AB > AC$ and orthocenter $H$. The point $E$ lies symmetric to $C$ with respect to the altitude $AH$. Let $F$ be the intersection of the lines $EH$ and $AC$. Prove that the circumcenter of the triangle $AEF$ lies on the line $AB$.
(Karl Czakler)
2020 Kosovo National Mathematical Olympiad, 3
Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of triangle $\triangle ABC$ and $P$ a variable point in line segment $BC$. The circle with center $P$ and radius $PA$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $R$ and $RP$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $Q$. Show that points $A$, $O$, $P$ and $Q$ are concyclic.
Brazil L2 Finals (OBM) - geometry, 2008.5
Let $ABC$ be an acutangle triangle and $O, H$ its circumcenter, orthocenter, respectively. If $\frac{AB}{\sqrt2}=BH=OB$, calculate the angles of the triangle $ABC$ .
2003 Estonia Team Selection Test, 6
Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ .
(J. Willemson)
2019 Final Mathematical Cup, 1
Let $ABC$ be an acute triangle with $AB<AC<BC$ and let $D$ be a point on it's extension of $BC$ towards $C$. Circle $c_1$, with center $A$ and radius $AD$, intersects lines $AC,AB$ and $CB$ at points $E,F$, and $G$ respectively. Circumscribed circle $c_2$ of triangle $AFG$ intersects again lines $FE,BC,GE$ and $DF$ at points $J,H,H' $ and $J'$ respectively. Circumscribed circle $c_3$ of triangle $ADE$ intersects again lines $FE,BC,GE$ and $DF$ at points $I,K,K' $ and $I' $ respectively. Prove that the quadrilaterals $HIJK$ and $H'I'J'K '$ are cyclic and the centers of their circumscribed circles coincide.
by Evangelos Psychas, Greece
Brazil L2 Finals (OBM) - geometry, 2023.2
Consider a triangle $ABC$ with $AB < AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. A line starting from $B$ cuts the lines $AO$ and $AH$ at $M$ and $M'$ so that $M'$ is the midpoint of $BM$. Another line starting from $C$ cuts the lines $AH$ and $AO$ at $N$ and $N'$ so that $N'$ is the midpoint of $CN$. Prove that $M, M', N, N'$ are on the same circle.
2019 Bulgaria EGMO TST, 1
Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$, $OG = 1$ and $OG \parallel BC$. (As usual $O$ is the circumcenter and $G$ is the centroid.)
2007 Sharygin Geometry Olympiad, 18
Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.
2003 Croatia Team Selection Test, 2
Let $B$ be a point on a circle $k_1, A \ne B$ be a point on the tangent to the circle at $B$, and $C$ a point not lying on $k_1$ for which the segment $AC$ meets $k_1$ at two distinct points. Circle $k_2$ is tangent to line $AC$ at $C$ and to $k_1$ at point $D$, and does not lie in the same half-plane as $B$. Prove that the circumcenter of triangle $BCD$ lies on the circumcircle of $\vartriangle ABC$