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Show that XD and AM meet on Gamma

Source: IMO Shortlist 2016, Geometry 2

July 19, 2017
IMO Shortlistgeometrybutterfly theoremmixtilinear incircleprojective geometrymediangeometry solved

Problem Statement

Let ABCABC be a triangle with circumcircle Γ\Gamma and incenter II and let MM be the midpoint of BC‾\overline{BC}. The points DD, EE, FF are selected on sides BC‾\overline{BC}, CA‾\overline{CA}, AB‾\overline{AB} such that ID‾⊥BC‾\overline{ID} \perp \overline{BC}, IE‾⊥AI‾\overline{IE}\perp \overline{AI}, and IF‾⊥AI‾\overline{IF}\perp \overline{AI}. Suppose that the circumcircle of △AEF\triangle AEF intersects Γ\Gamma at a point XX other than AA. Prove that lines XDXD and AMAM meet on Γ\Gamma.
Proposed by Evan Chen, Taiwan
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