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Prove that the lines $EG, DF$ and $BC$ are concurrent

Source: Moldova TST 2023

April 8, 2023
geometry

Problem Statement

Let ABCABC be an acute triangle with orthocenter H H and AB<AC.AB<AC. Let Ω1\Omega_1 be a circle with diameter ACAC and Ω2\Omega_2 a circle with diameter AB. AB. Line BHBH intersects Ω1\Omega_1 in points D D and EE such that EE is not on segment BH.BH. Line CH CH intersects Ω2\Omega_2 in points F F and GG such that GG is not on segment CH.CH. Prove that the lines EG,DFEG, DF and BCBC are concurrent.
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