Problem (2014CHNWesternMOp2): Let $ AB$ be the diameter of semicircle $O$ , $C, D $ be points on the arc $AB$, $P, Q$ be respectively the circumcenter of $\triangle OAC $ and $\triangle OBD $ . Prove that:$CP\cdot CQ=DP \cdot DQ$.
AlphaGeometry formalization: a b = segment; o = midpoint a b; c = on_circle o a; d = on_circle o a; p = circumcenter o a c; q = circumcenter o b d ? eqratio c p d p d q c q
HAGeo formalization: A = point; B = point; O = midpoint A B; (O) = circle_center_point O A; C = on_circle (O); D = on_circle (O); P = circumcenter O A C; Q = circumcenter O B D; Prove: equal_ratio C P D P D Q C Q
Difficulty score: 2.1