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)