Found problems: 316
2005 Bosnia and Herzegovina Team Selection Test, 1
Let $H$ be an orthocenter of an acute triangle $ABC$. Prove that midpoints of $AB$ and $CH$ and intersection point of angle bisectors of $\angle CAH$ and $\angle CBH$ lie on the same line.
2023 Yasinsky Geometry Olympiad, 4
Let $BE$ and $CF$ be the altitudes of acute triangle $ABC$. Let $H$ be the orthocenter of $ABC$ and $M$ be the midpoint of side $BC$. The points of intersection of the midperpendicular line to $BC$ with segments $BE$ and $CF$ are denoted by $K$ and $L$ respectively. The point $Q$ is the orthocenter of triangle $KLH$. Prove that $Q$ belongs to the median $AM$.
(Bohdan Zheliabovskyi)
2019 Costa Rica - Final Round, G2
Let $H$ be the orthocenter and $O$ the circumcenter of the acute triangle $\vartriangle ABC$. The circle with center $H$ and radius $HA$ intersects the lines $AC$ and $AB$ at points $P$ and $Q$, respectively. Let point $O$ be the orthocenter of triangle $\vartriangle APQ$, determine the measure of $\angle BAC$.
2014 Thailand TSTST, 3
Let $O$ be the incenter of a tangential quadrilateral $ABCD$. Prove that the orthocenters of $\vartriangle AOB$, $\vartriangle BOC$, $\vartriangle COD$, $\vartriangle DOA$ lie on a line.
2006 Belarusian National Olympiad, 4
Given a quadrilateral $ABCD$ with $\angle ABC = \angle ADC$. Let $BM$ be the altitude of the triangle $ABC$, and $M$ belongs to $AC$. Point $M'$ is marked on the diagonal $AC$ so that $$\frac{AM \cdot CM'}{ AM' \cdot CM}= \frac{AB \cdot CD }{ BC \cdot AD}$$ Prove that the intersection point of $DM'$ and $BM$ coincides with the orthocenter of the triangle $ABC$.
(M. Zhikhovich)
1986 Tournament Of Towns, (123) 5
Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle .
(A. Andjans, Riga)
1997 Austrian-Polish Competition, 4
In a trapezoid $ABCD$ with $AB // CD$, the diagonals $AC$ and $BD$ intersect at point $E$. Let $F$ and $G$ be the orthocenters of the triangles $EBC$ and $EAD$. Prove that the midpoint of $GF$ lies on the perpendicular from $E$ to $AB$.
2024 Canadian Mathematical Olympiad Qualification, 7a
In triangle $ABC$, let $I$ be the incentre. Let $H$ be the orthocentre of triangle $BIC$. Show that $AH$ is parallel to $BC$ if and only if $H$ lies on the circle with diameter $AI$.
2019 Centers of Excellency of Suceava, 3
The circumcenter, circumradius and orthocenter of a triangle $ ABC $ satisfying $ AB<AC $ are notated with $ O,R,H, $ respectively. Prove that the middle of the segment $ OH $ belongs to the line $ BC $ if
$$ AC^2-AB^2=2R\cdot BC. $$
[i]Marius Marchitan[/i]
2022 Kyiv City MO Round 1, Problem 3
Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$.
[i](Proposed by Oleksii Masalitin)[/i]
2011 Saudi Arabia Pre-TST, 4.4
In a triangle $ABC$, let $O$ be the circumcenter, $H$ the orthocenter, and $M$ the midpoint of the segment $AH$. The perpendicular at $M$ onto $OM$ intersects lines $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that $MP = MQ$.
2015 Balkan MO Shortlist, G7
Let scalene triangle $ABC$ have orthocentre $H$ and circumcircle $\Gamma$. $AH$ meets $\Gamma$ at $D$ distinct from $A$. $BH$ and $CH$ meet $CA$ and $AB$ at $E$ and $F$ respectively, and $EF$ meets $BC$ at $P$. The tangents to $\Gamma$ at $B$ and $C$ meet at $T$. Show that $AP$ and $DT$ are concurrent on the circumcircle of $AFE$.
2021 Dutch IMO TST, 3
Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.
2006 Petru Moroșan-Trident, 3
In an acute-angled triangle $ ABC $ consider $ A_1,B_1,C_1 $ to be the symmetric points of the orthocenter of $ ABC $ to the sides $ BC,AC,AB, $ respectively. Show that if the centroids of the triangles $ ABC,A_1B_1C_1 $ are the same, then $ ABC $ is equilateral.
[i]Carmen Botea[/i]
2017 China Team Selection Test, 2
Let $ABCD$ be a non-cyclic convex quadrilateral. The feet of perpendiculars from $A$ to $BC,BD,CD$ are $P,Q,R$ respectively, where $P,Q$ lie on segments $BC,BD$ and $R$ lies on $CD$ extended. The feet of perpendiculars from $D$ to $AC,BC,AB$ are $X,Y,Z$ respectively, where $X,Y$ lie on segments $AC,BC$ and $Z$ lies on $BA$ extended. Let the orthocenter of $\triangle ABD$ be $H$. Prove that the common chord of circumcircles of $\triangle PQR$ and $\triangle XYZ$ bisects $BH$.
2018 India PRMO, 21
Let $\Delta ABC$ be an acute-angled triangle and let $H$ be its orthocentre. Let $G_1, G_2$ and $G_3$ be the centroids of the triangles $\Delta HBC , \Delta HCA$ and $\Delta HAB$ respectively. If the area of $\Delta G_1G_2G_3$ is $7$ units, what is the area of $\Delta ABC $?
2017 Abels Math Contest (Norwegian MO) Final, 4
Let $a > 0$ and $0 < \alpha <\pi$ be given. Let $ABC$ be a triangle with $BC = a$ and $\angle BAC = \alpha$ , and call the cicumcentre $O$, and the orthocentre $H$. The point $P$ lies on the ray from $A$ through $O$. Let $S$ be the mirror image of $P$ through $AC$, and $T$ the mirror image of $P$ through $AB$. Assume that $SATH$ is cyclic. Show that the length $AP$ depends only on $a$ and $\alpha$.
2006 Sharygin Geometry Olympiad, 18
Two perpendicular lines are drawn through the orthocenter $H$ of triangle $ABC$, one of which intersects $BC$ at point $X$, and the other intersects $AC$ at point $Y$. Lines $AZ, BZ$ are parallel, respectively with $HX$ and $HY$. Prove that the points $X, Y, Z$ lie on the same line.
2019 Grand Duchy of Lithuania, 3
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. The perpendicular bisector of segment $CH$ intersects the sides $AC$ and $BC$ in points $X$ and $Y$ , respectively. The lines $XO$ and $YO$ intersect the side $AB$ in points $P$ and $Q$, respectively. Prove that if $XP + Y Q = AB + XY$ then $\angle OHC = 90^o$.
2015 EGMO, 6
Let $H$ be the orthocentre and $G$ be the centroid of acute-angled triangle $ABC$ with $AB\ne AC$. The line $AG$ intersects the circumcircle of $ABC$ at $A$ and $P$. Let $P'$ be the reflection of $P$ in the line $BC$. Prove that $\angle CAB = 60$ if and only if $HG = GP'$
1996 Iran MO (3rd Round), 5
Let $O$ be the circumcenter and $H$ the orthocenter of an acute-angled triangle $ABC$ such that $BC>CA$. Let $F$ be the foot of the altitude $CH$ of triangle $ABC$. The perpendicular to the line $OF$ at the point $F$ intersects the line $AC$ at $P$. Prove that $\measuredangle FHP=\measuredangle BAC$.
2020-21 KVS IOQM India, 27
Let $ABC$ be an acute-angled triangle and $P$ be a point in its interior. Let $P_A,P_B$ and $P_c$ be the images of $P$ under reflection in the sides $BC,CA$, and $AB$, respectively. If $P$ is the orthocentre of the triangle $P_AP_BP_C$ and if the largest angle of the triangle that can be formed by the line segments$ PA, PB$. and $PC$ is $x^o$, determine the value of $x$.
2025 6th Memorial "Aleksandar Blazhevski-Cane", P2
Let $\triangle ABC$ be a scalene and acute triangle in which the angle at $A$ is second largest, $H$ is the orthocenter, and $k$ is the circumcircle with center $O$. Let the circumcircle of $\triangle AHO$ intersect the sides $AB$ and $AC$ again at $M$ and $N$, respectively, whereas the altitudes $CH$ and $BH$ intersect $k$ again at $K$ and $L$, respectively. Prove that the intersection of $KL$ and the perpendicular bisector of $AH$ is the orthocenter of $\triangle AMN$.
Proposed by [i]Ilija Jovcevski[/i]
2002 Junior Balkan Team Selection Tests - Moldova, 3
Let $ABC$ be a an acute triangle. Points $A_1, B_1$ and $C_1$ are respectively the projections of the vertices $A, B$ and $C$ on the opposite sides of the triangle, the point $H$ is the orthocenter of the triangle, and the point $P$ is the middle of the segment $[AH]$. The lines $BH$ and $A_1C_1$, $P B_1$ and $AB$ intersect respectively at the points $M$ and $N$. Prove that the lines $MN$ and $BC$ are perpendicular.
2014 Bulgaria National Olympiad, 1
Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies.
[i]Proposed by T. Vitanov, E. Kolev[/i]