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another geometry problem with sharky-devil point

Source: The francophone mathematical olympiads P1

June 27, 2020
geometryincircleFrancophonesharky-devil

Problem Statement

Let ABCABC be an acute triangle with AC>ABAC>AB, Let DEFDEF be the intouch triangle with D∈(BC)D \in (BC),E∈(AC)E \in (AC),F∈(AB)F \in (AB),, let GG be the intersecttion of the perpendicular from DD to EFEF with ABAB, and X=(ABC)∩(AEF)X=(ABC)\cap (AEF). Prove that B,D,GB,D,G and XX are concylic
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