Question

The three angle bisectors of ΔABC meet at P. Drag the vertices around to form different triangles, and then make a conjecture.

Which segments appear to be equal in length? Check all that apply.

XP = YP for an acute triangle
XP = ZP for an acute triangle
YP = ZP for an obtuse triangle
ZP = XP for a right triangle
XP = YP for an obtuse triangle

Answer 1

All of them. In fact XP = YP = ZP for every triangle. P is the incenter (intersection of the angle bisectors), so its perpendicular distances to the three sides (the segments XP, YP, ZP) are equal (they are all the inradius). Hence each listed equality holds (in acute, obtuse, and right cases).