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 from a to a is to the segment.
By the definition of an incenter,
AI is the of
∠BAC. Since
I lies on
AI, it is equidistant from the angle's sides by the .
DI=EI
Similarly, since
I lies on
BI, which is the bisector of
∠ABC, it is also equidistant from this angle's sides.
EI=FI
By bringing together the above information, the following is obtained.
DI=EIandEI=FI⇕DI=EI=FI
This means that
I is equidistant from each of the triangle's sides.