Found problems: 316
2016 Switzerland - Final Round, 8
Let $ABC$ be an acute-angled triangle with height intersection $H$. Let $G$ be the intersection of parallel of $AB$ through $H$ with the parallel of $AH$ through $B$. Let $I$ be the point on the line $GH$, so that $AC$ bisects segment $HI$. Let $J$ be the second intersection of $AC$ and the circumcircle of the triangle $CGI$. Show that $IJ = AH$
2023 Czech-Polish-Slovak Junior Match, 3
Given is an acute triangle $ABC$. Point $P$ lies inside this triangle and lies on the bisector of angle $\angle BAC$. Suppose that the point of intersection of the altitudes $H$ of triangle $ABP$ lies inside triangle $ABC$. Let $Q$ be the intersection of the line $AP$ and the line perpendicular to $AC$ passing through $H$. Prove that $Q$ is the point symmetrical to $P$ wrt the line $BH$.
2015 Sharygin Geometry Olympiad, P7
The altitudes $AA_1$ and $CC_1$ of a triangle $ABC$ meet at point $H$. Point $H_A$ is symmetric to $H$ about $A$. Line $H_AC_1$ meets $BC$ at point $C' $, point $A' $ is defined similarly. Prove that $A' C' // AC$.
Estonia Open Senior - geometry, 2010.1.4
Circle $c$ passes through vertices $A$ and $B$ of an isosceles triangle $ABC$, whereby line $AC$ is tangent to it. Prove that circle $c$ passes through the circumcenter or the incenter or the orthocenter of triangle $ABC$.
1968 Poland - Second Round, 2
Given a circle $ k $ and a point inside it $ H $. Inscribe a triangle in the circle such that this point $ H $ is the point of intersection of the triangle's altitudes.
2013 Junior Balkan Team Selection Tests - Romania, 4
Let $H$ be the orthocenter of an acute-angled triangle $ABC$ and $P$ a point on the circumcenter of triangle $ABC$. Prove that the Simson line of $P$ bisects the segment $[P H]$.
2023 Brazil National Olympiad, 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.
2020 Ukrainian Geometry Olympiad - April, 3
Let $H$ be the orthocenter of the acute-angled triangle $ABC$. Inside the segment $BC$ arbitrary point $D$ is selected. Let $P$ be such that $ADPH$ is a parallelogram. Prove that $\angle BCP< \angle BHP$.
2013 Saudi Arabia GMO TST, 3
$ABC$ is a triangle, $H$ its orthocenter, $I$ its incenter, $O$ its circumcenter and $\omega$ its circumcircle. Line $CI$ intersects circle $\omega$ at point $D$ different from $C$. Assume that $AB = ID$ and $AH = OH$. Find the angles of triangle $ABC$.
2022 JBMO Shortlist, G5
Given is an acute angled triangle $ABC$ with orthocenter $H$ and circumcircle $k$. Let $\omega$ be the circle with diameter $AH$ and $P$ be the point of intersection of $\omega$ and $k$ other than $A$. Assume that $BP$ and $CP$ intersect $\omega$ for the second time at points $Q$ and $R$, respectively. If $D$ is the foot of the altitude from $A$ to $BC$ and $S$ is the point of the intersection of $\omega$ and $QD$, prove that $HR = HS$.
2015 Estonia Team Selection Test, 4
Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $C_1 (H,HE)$ and $C_2(B,BE)$ be two circles tangent at $AC$ at point $E$. Let $P\ne E$ be the second point of tangency of the circle $C_1 (H,HE)$ with its tangent line going through point $C$, and $Q\ne E$ be the second point of tangency of the circle $C_2(B,BE)$ with its tangent line going through point $C$. Prove that points $D, P$, and $Q$ are collinear.
2002 Estonia National Olympiad, 2
Let $ABC$ be a non-right triangle with its altitudes intersecting in point $H$. Prove that $ABH$ is an acute triangle if and only if $\angle ACB$ is obtuse.
2017 Mexico National Olympiad, 3
Let $ABC$ be an acute triangle with orthocenter $H$. The circle through $B, H$, and $C$ intersects lines $AB$ and $AC$ at $D$ and $E$ respectively, and segment $DE$ intersects $HB$ and $HC$ at $P$ and $Q$ respectively. Two points $X$ and $Y$, both different from $A$, are located on lines $AP$ and $AQ$ respectively such that $X, H, A, B$ are concyclic and $Y, H, A, C$ are concyclic. Show that lines $XY$ and $BC$ are parallel.
2006 Sharygin Geometry Olympiad, 19
Through the midpoints of the sides of the triangle $T$, straight lines are drawn perpendicular to the bisectors of the opposite angles of the triangle. These lines formed a triangle $T_1$. Prove that the center of the circle circumscribed about $T_1$ is in the midpoint of the segment formed by the center of the inscribed circle and the intersection point of the heights of triangle $T$.
2015 Junior Balkan Team Selection Tests - Romania, 5
Let $ABCD$ be a convex quadrilateral with non perpendicular diagonals and with the sides $AB$ and $CD$ non parallel . Denote by $O$ the intersection of the diagonals , $H_1$ the orthocenter of the triangle $AOB$ and $H_2$ the orthocenter of the triangle $COD$ . Also denote with $M$ the midpoint of the side $AB$ and with $N$ the midpoint of the side $CD$ . Prove that $H_1H_2$ and $MN$ are parallel if and only if $AC=BD$
2023 Bangladesh Mathematical Olympiad, P4
Let $ABCD$ be an isosceles trapezium inscribed in circle $\omega$, such that $AB||CD$. Let $P$ be a point on the circle $\omega$. Let $H_1$ and $H_2$ be the orthocenters of triangles $PAD$ and $PBC$ respectively. Prove that the length of $H_1H_2$ remains constant, when $P$ varies on the circle.
2023 Sharygin Geometry Olympiad, 9
It is known that the reflection of the orthocenter of a triangle $ABC$ about its circumcenter lies on $BC$. Let $A_1$ be the foot of the altitude from $A$. Prove that $A_1$ lies on the circle passing through the midpoints of the altitudes of $ABC$.
2021 Iran RMM TST, 1
Suppose that two circles $\alpha, \beta$ with centers $P,Q$, respectively , intersect orthogonally at $A$,$B$. Let $CD$ be a diameter of $\beta$ that is exterior to $\alpha$. Let $E,F$ be points on $\alpha$ such that $CE,DF$ are tangent to $\alpha$ , with $C,E$ on one side of $PQ$ and $D,F$ on the other side of $PQ$. Let $S$ be the intersection of $CF,AQ$ and $T$ be the intersection of $DE,QB$. Prove that $ST$ is parallel to $CD$ and is tangent to $\alpha$
2020 ELMO Problems, P3
Janabel has a device that, when given two distinct points $U$ and $V$ in the plane, draws the perpendicular bisector of $UV$. Show that if three lines forming a triangle are drawn, Janabel can mark the orthocenter of the triangle using this device, a pencil, and no other tools.
[i]Proposed by Fedir Yudin.[/i]
2018 Regional Olympiad of Mexico West, 5
Let $ABC$ be a scalene triangle, $\Gamma$ its circumscribed circle and $H$ the point where the altitudes of triangle $ABC$ meet. The circumference with center at $H$ passing through $A$ cuts $\Gamma$ at a second point $D$. In the same way, the circles with center at $H$ and passing through $B$ and $C$ cut $\Gamma$ again at points $E$ and $F$, respectively. Prove that $H$ is also the point in which the altitudes of the triangle $DEF$ meet.
Kyiv City MO Seniors 2003+ geometry, 2004.10.5
Let the points $M$ and $N$ in the triangle $ABC$ be the midpoints of the sides $BC$ and $AC$, respectively. It is known that the point of intersection of the altitudes of the triangle $ABC$ coincides with the point of intersection of the medians of the triangle $AMN$. Find the value of the angle $ABC$.
2017 Saudi Arabia JBMO TST, 7
Let $ABC$ be a triangle inscribed in the circle $(O)$, with orthocenter $H$. Let d be an arbitrary line which passes through $H$ and intersects $(O)$ at $P$ and $Q$. Draw diameter $AA'$ of circle $(O)$. Lines $A'P$ and $A'Q$ meet $BC$ at $K$ and $L$, respectively. Prove that $O, K, L$ and $A'$ are concyclic.
2024 Indonesia MO, 3
The triangle $ABC$ has $O$ as its circumcenter, and $H$ as its orthocenter. The line $AH$ and $BH$ intersect the circumcircle of $ABC$ for the second time at points $D$ and $E$, respectively. Let $A'$ and $B'$ be the circumcenters of triangle $AHE$ and $BHD$ respectively. If $A', B', O, H$ are [b]not[/b] collinear, prove that $OH$ intersects the midpoint of segment $A'B'$.
1999 Estonia National Olympiad, 3
Prove that the line segment, joining the orthocenter and the intersection point of the medians of the acute-angled triangle $ABC$ is parallel to the side $AB$ iff $\tan \angle A \cdot \tan \angle B = 3$.
Kvant 2019, M2553
A circle centred at $I$ is tangent to the sides $BC, CA$, and $AB$ of an acute-angled triangle $ABC$ at $A_1, B_1$, and $C_1$, respectively. Let $K$ and $L$ be the incenters of the quadrilaterals $AB_1IC_1$ and $BA_1IC_1$, respectively. Let $CH$ be an altitude of triangle $ABC$. Let the internal angle bisectors of angles $AHC$ and $BHC$ meet the lines $A_1C_1$ and $B_1C_1$ at $P$ and $Q$, respectively. Prove that $Q$ is the orthocenter of the triangle $KLP$.
Kolmogorov Cup 2018, Major League, Day 3, Problem 1; A. Zaslavsky