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