5. The three angle bisectors of a triangle meet at the center of the inscribed circle.

Let O be the point where the bisectors of angle A and angle B meet. Then, since O is on the bisector of the angle at A, by the last theorem, the perpendicular distance from O to F is the same as the perpendicular distance from O to D. Since O is also on the bisector of the angle at B, the perpendicular distance from O to D is the same as the perpendicular distance from O to E. If we put these facts together, we conclude that the perpendiculat distance from O to F is the same as the perpendicular distance from O to E because they are both the same as the perpendicular distance from O to D. By the converse result in the last theorem, O is also on the perpendicular bisector of the angle at C. We conclude that O is on all three angle bisectors. Since the three distances, OD, OE, and OF, are all the same, D, E, and F all lie on the same circle centered at O. Since the perpendicular distances from O to the three sides of the triangle are the shortest distances by Theorem 1 of this section, we conclude that all three sides of the triangle are tangent to the circle centered at O going through D, E, and F.