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Part of 2019 Israel Olympic Revenge

Problems(1)

Tangent from midpoint to excircle.

Source: Israel Olympic Revenge SL 2019 G4.

5/3/2022
Let ω\omega be the AA-excircle of triangle ABCABC and MM the midpoint of side BCBC. GG is the pole of AMAM w.r.t ω\omega and HH is the midpoint of segment AGAG. Prove that MHMH is tangent to ω\omega.
geometryexcircle