Solved Problem 2 - Expressions for the Side Lengths of the Incentral Triangle


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

Applying the angle bisector theorem: 
Applying the cosine rule in ΔCDE:
The above expression looks a bit messy, but we can simplify it using the cosine rule in ΔABC:
This is much better, but we can improve it further.

Let us use the projection formulae:
The expressions for EF and FD follow similarly.

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