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common line of ellipses pass through orthocenter

Source: Sharygin 2023 - P23 (Grade-10-11)

March 4, 2023
geometryellipsede longchamps pointcollinearitySharygin Geometry OlympiadSharygin 2023

Problem Statement

An ellipse Γ1\Gamma_1 with foci at the midpoints of sides ABAB and ACAC of a triangle ABCABC passes through AA, and an ellipse Γ2\Gamma_2 with foci at the midpoints of ACAC and BCBC passes through CC. Prove that the common points of these ellipses and the orthocenter of triangle ABCABC are collinear.
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