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Do you see: cyclic quad? right triangle? homothety between incircle/excircle? radical axis? spiral similarity?
What would prove it? Congruence? Concyclicity? Equal angles? Equal products (Power of a point)? Collinearity (Menelaus)? 106 geometry problems
If stuck for 20 min, switch to coordinates/complex numbers (but only if allowed in contest – IMO accepts pure synthetic or analytic). Do you see: cyclic quad
Common tricks: reflect a point across an angle bisector, draw the second intersection of two circles, construct the circumcircle of three points. 106 geometry problems
Redraw cleanly. Mark given equalities, angles, midpoints, tangents.