Bisection
Lengths

If the side lengths of a triangle are a, b, c {\displaystyle a,b,c}, the semiperimeter s = (a + b + c) / 2, {\displaystyle s=(a+b+c)/2,} and A is the angle opposite side a {\displaystyle a}, then the length of the internal bisector of angle A is

or in trigonometric terms,

If the internal bisector of angle A in triangle ABC has length t a {\displaystyle t_{a}} and if this bisector divides the side opposite A into segments of lengths m and n, then

where b and c are the side lengths opposite vertices B and C; and the side opposite A is divided in the proportion b: c.

If the internal bisectors of angles A, B, and C have lengths t a, t b, {\displaystyle t_{a},t_{b},} and t c {\displaystyle t_{c}}, then

No two non-congruent triangles share the same set of three internal angle bisector lengths.
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