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intersection of circumcircles lies on diagonal of a parallelogram

Source: 2019 Irish Mathematical Olympiad paper 2 p8

October 5, 2020
geometrycircumcircleparallelogramconcurrency

Problem Statement

Consider a point GG in the interior of a parallelogram ABCDABCD. A circle Γ\Gamma through AA and GG intersects the sides ABAB and ADAD for the second time at the points EE and FF respectively. The line FGFG extended intersects the side BCBC at HH and the line EGEG extended intersects the side CDCD at II. The circumcircle of triangle HGIHGI intersects the circle Γ\Gamma for the second time at M≠GM \ne G. Prove that MM lies on the diagonal ACAC.
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