This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 2

2023 Sharygin Geometry Olympiad, 11
Tags: othorcenter , antipode , geometry
Let $H$ be the orthocenter of an acute-angled triangle $ABC$; $E$, $F$ be points on $AB, AC$ respectively, such that $AEHF$ is a parallelogram; $X, Y$ be the common points of the line $EF$ and the circumcircle $\omega$ of triangle $ABC$; $Z$ be the point of $\omega$ opposite to $A$. Prove that $H$ is the orthocenter of triangle $XYZ$.

2023 Sharygin Geometry Olympiad, 21
Tags: othorcenter , humpty point , projection , radical axis , geometry
Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$. Prove that $P_a,P_b,P_c,P_d$ are concyclic.