Solved Problem 4 - Expressions for the Distances from the Vertices of the Incentral Triangle to the Incenter

RESULT


For any ΔABC, with:
  • sides a, b, and c
  • angles A, B, and C
  • internal angle bisectors AD, BE, and CF
  • incenter I, and
  • inradius r


Proof 1
By the angle bisector theorem:
Using the formula for the length of the angle bisector:

Proof 2
We can use basic trigonometry to prove the result as described below.

Let us drop a perpendicular IP from I onto AB or AC. It is trivial that IP = r. Hence, we get the following expression for AI:
We know that:
We know the popular formulae for a triangle’s inradius and area, respectively:
Thus, using the obtained expression for AI, we get:
Using the formula for the length of the angle bisector:
Taking √(b+c-a) common:

The expressions for IE and IF follow similarly.

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