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.

-O