USAMO proofs — Proofs
Olympiad statements to prove in full. Write the argument out, then compare it with the model proof — structure, cases and rigor, not just the idea.
Problem 1 of 22
Geometry
Prove Euler's theorem: if and are the circumcenter and incenter of a triangle with circumradius and inradius , then . Deduce .
Write a full proof on paper. There is no answer box — when you are ready, reveal the solution and compare your argument.