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The intersecting secant-tangent theorem says that
where T is the point at which line segment PO touches the circle. Similarly,
so we have
OP is of course as long as itself, and AO and BO are radii of the same circle so they have the same lengths.
This means triangles APO and BPO are congruent, which means angles APO and BPO are congruent, so angle APB is bisected by PO.