Question

Please help. 10 points. Please do not guess. Guesses will be reported. Thanks

Answer 1

The intersecting secant-tangent theorem says that

$AP^2=PT\cdot TO$

where T is the point at which line segment PO touches the circle. Similarly,

$BP^2=PT\cdot TO$

so we have

$AP^2=BP^2\implies AP=BP$

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.