Sharygin 2023, correspondence round, problem 23
Sharygin 2023, correspondence round, problem 23 is a plane geometry problem from the Sharygin Geometry Olympiad 2023, correspondence round, problem 23 (grades 10–11): two ellipses and the orthocenter.
Problem
An ellipse $\Gamma_1$ with foci at the midpoints of sides $AB$ and $AC$ of a triangle $ABC$ passes through $A$, and an ellipse $\Gamma_2$ with foci at the midpoints of $AC$ and $BC$ passes through $C$. Prove that the common points of these ellipses and the orthocenter of triangle $ABC$ are collinear.
Source Sharygin Geometry Olympiad 2023, correspondence round, problem 23 (grades 10–11)
The diagram
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