Answer
By the definition of a midpoint, it follows that because C is the midpoint of AB, AC and BC must be congruent. As a result, we know that the corresponding arcs AC and AB are congruent, for the length of the chord is directly proportional to the length of the corresponding arc. Since the length of the corresponding arc is directly related to the length of the angle created by the tangent line, it follows that angle CPB is congruent to CPA. Thus, by the definition if an angle bisector, line segment CP is an angle bisector.
