Incenter Proof

Show the incenter is equidistant from the three sides of the triangle. In the figure below, I in the incenter. Prove IX = IY = IZ.

1) Triangle ABC; Angle Bisectors AI, CI, BI; Perpendicular Segments IX and AC, IZ and CB, IY and AB(Given)

2) IX = IY; IY = IZ(Point on the angle bisector is equidistant from the sides of the angle)

3)IX = IZ(Transitive Property of Congruence)

4) I is on the angle bisector of angle C(Converse of angle bisector theorem)

5) D is equidistant from the sides of triangle ABC(Steps 2 and 3)