Draw a perpendicular from P to AB. Label the intersection C. We are given that PA = PB, so PA ≅ PB by the definition of . We know that angles PCA and PCB are right angles by the definition of . PC ≅ PC by the . So, triangle ACP is congruent to triangle BCP by HL, and AC ≅ BC by . Since PC is perpendicular to and bisects AB, P is on the perpendicular bisector of AB by the definition of perpendicular bisector.
Equal lengths mean the segments are congruent: since PA = PB, PA ≅ PB by the definition of congruent segments. The other steps use different facts: perpendicular lines form right angles, and matching sides of congruent triangles are congruent by CPCTC.
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