SHARYGIN 2023 Problem 23

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.