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Prove tangent meet at circumcircle (ABC)

Source: BMO SL 2019, G3

November 8, 2020
geometrycircumcircletangent

Problem Statement

Let ABCABC be a scalene and acute triangle with circumcenter OO. Let ω\omega be the circle with center AA, tangent to BCBC at DD. Suppose there are two points FF and GG on ω\omega such that FG⊥AOFG \perp AO, ∠BFD=∠DGC\angle BFD = \angle DGC and the couples of points (B,F)(B,F) and (C,G)(C,G) are in different halfplanes with respect to the line ADAD. Show that the tangents to ω\omega at FF and GG meet on the circumcircle of ABCABC.
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