This is a basic theorem and there should be formal a proof somehere in internet. Let's try something a little clumsy but it MAY work:
1) Extend OP in both directions. Let M and N be the points where extended OP intersects the circle.
2) We know that:
|MP|*|PN|= |BP|*|PC| (This can be shown by similarity of triangles)
=> |MP|*|PN|= |PC|^2 (Can you see the Euclidean identity in MCN triangle?)
=> angle(MCN) = 90 degrees.
=> MN is the diameter
3) Repeat the same steps for OQ, which will reveal that O is the center of THE circle.