prove that AP bisects the angle TAB

Question

TA and TB are tangents to a circle with centre O from an external point T, OT intersects the circle at point P

Answer 1

AB is a chord bisected by PO, the radius since OT is an angle bisector.  The arcs PA and PB are also congruent since the arc AB is bisected also. 

 

Tangent TA and chord AB intercept arc PA and is half its measure.

 

Chord PA and chord AB intercept arc PB and is half its measure.

 

Since the arcs are equal, so are the angles.

 

 I hope this makes sense.