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