CWMO 2014 Problem 2

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: $CP\cdot CQ=DP \cdot DQ$ .

import cse5; import olympiad; unitsize(3.5cm); dotfactor=4; pathpen=black;
real h=sqrt(55/64);
pair A=(-1,0), O=origin, B=(1,0),C=shift(-3/8,h)*O,D=shift(4/5,3/5)*O,P=circumcenter(O,A,C), Q=circumcenter(O,D,B);
D(arc(O,1,0,180),darkgreen);
D(MP("A",A,W)--MP("C",C,N)--MP("P",P,SE)--MP("D",D,E)--MP("Q",Q,E)--C--MP("O",O,S)--D--MP("B",B,E)--cycle,deepblue);
D(O);

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