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Guessing fixed point geometry

Source: MEMO 2024 I3

August 26, 2024
geometryFixed pointcirclecircumcircleAngle Chasing

Problem Statement

Let ABCABC be an acute scalene triangle. Choose a circle ω\omega passing through BB and CC which intersects the segments ABAB and ACAC at the interior points DD and EE, respectively. The lines BEBE and CDCD intersects at FF. Let GG be a point on the circumcircle of ABFABF such that GBGB is tangent to ω\omega and let HH be a point on the circumcircle of ACFACF such that HCHC is tangent to ω\omega. Prove that there exists a point T≠AT\neq A, independent of the choice of ω\omega, such that the circumcircle of triangle AGHAGH passes through TT.
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