Do you see: cyclic quad? right triangle? homothety between incircle/excircle? radical axis? spiral similarity?
This is a tall order, but a great one. 106 Geometry Problems (often referring to the book by Titu Andreescu, Vlad Crisan, and Bogdan Enescu, or the classic "103 Trigonometry Problems" / "106 Geometry Problems" from the AwesomeMath series) is an for high school students targeting Olympiads (IMO, USAMO, etc.). 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
What would prove it? Congruence? Concyclicity? Equal angles? Equal products (Power of a point)? Collinearity (Menelaus)? radical axis
Redraw cleanly. Mark given equalities, angles, midpoints, tangents.
Common tricks: reflect a point across an angle bisector, draw the second intersection of two circles, construct the circumcircle of three points.