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if G is the midpoint of FH, then E is the midpoint of GH, cyclic ABCD related

Source: Finland 2019 p3

September 8, 2019
cyclic quadrilateralmidpointcirclegeometry

Problem Statement

Let ABCDABCD be a cyclic quadrilateral whose side ABAB is at the same time the diameter of the circle. The lines ACAC and BDBD intersect at point EE and the extensions of lines ADAD and BCBC intersect at point FF. Segment EFEF intersects the circle at GG and the extension of segment EFEF intersects ABAB at HH. Show that if GG is the midpoint of FHFH, then EE is the midpoint of GHGH.
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