Incenter Theorem

Consider a triangle and its incenter $I.$

Let $DI,$ $EI,$ and $FI$ be the distances from $I$ to the sides of the triangle. Recall that the distance from a point to a segment is perpendicular to the segment.

By the definition of an incenter, $\overline{AI}$ is the angle bisector of $\angle BAC.$ Since $I$ lies on $\overline{AI},$ it is equidistant from the angle's sides by the Angle Bisector Theorem. Similarly, since $I$ lies on $\overline{BI},$ which is the bisector of $\angle ABC,$ it is also equidistant from this angle's sides. By bringing together the above information, the following is obtained. This means that $I$ is equidistant from each of the triangle's sides.