The graph is
If P is a point on the plane any line we can draw from P to the intersecting point of the segment AB and the plane is perpendicular to the segment AB:
This means that between AB and any point on the plane we'll form a triangle like this one.
Since the plane is a bisector, any segment on the plane is a bisector. Therefore the distances between A and the intersecting point and B and the intersecting point are congruent. And since we have two triangles that share a side (the segment formed by P and the intersecting point) and the angles formed by these two sides are congruent (right angles becasue the bisector is perpendicular) we have that these two triangles are congruent by SAS (Side - Angle - Side).
If two triangles are congruent, then all their sides and angles are congruent. Therefore PA = PB