Complete this proof using axioms and already proven theorems.

Given: Point P is equidistant from endpoints X and Y of line XY..

Prove: P is on the perpendicular bisector of XY

Proof:

Case 1: P is on XY. By the Given, P is the midpoint of XY so it is on the perpendicular bisector.

Case 2:

1. Draw PX and PY. (On 2 points there is exactly 1 line)

2. Let M be the midpoint of XY. (Midpoint Thm)

3. Draw PM (On 2 points there is exactly 1 line)

4.

PX = PY (Given) 5. XM = YM (Definition of midpoint) 6. PM = PM (Reflexive Property of equality) 7. (SSS Postulate) 8. and (CPCTC and definition of congruency) 9. make up a Linear Pair (Definition of Linear Pair) 10. (If two angles form a linear pair then they are supplementary) 11. (Substitution using #8 and #10) 12. (Addition) 13. (Division) Similarly, you can show that 14. is a right angle. (Definition of right angle) 15. (If two lines meet to form right angles, then they are perpendicular) 16. P lies on (Step #3) Q.E.D. P lies on the perpendicular bisector of
I'm stuck after that. Any help would be greatly appreciated.