lengths of angle bisectors

In any triangle, the wa, wb, wc of the angle bisectorsMathworldPlanetmath opposing the sides a, b, c, respectively, are

wa=bc[(b+c)2-a2]b+c, (1)
wb=ca[(c+a)2-b2]c+a, (2)
wc=ab[(a+b)2-c2]a+b. (3)

Proof.  By the symmetryMathworldPlanetmath, it suffices to prove only (1).

According the angle bisector theoremMathworldPlanetmath, the bisectorMathworldPlanetmath wa divides the side a into the portions


If the angle opposite to a is α, we apply the law of cosines to the half-triangles by wa:

{2wabcosα2=wa2+b2-(abb+c)22waccosα2=wa2+c2-(cab+c)2 (4)

For eliminating the angle α, the equations (4) are divided sidewise, when one gets


from which one can after some routine manipulations solve wa, and this can be simplified to the form (1).

Title lengths of angle bisectors
Canonical name LengthsOfAngleBisectors
Date of creation 2013-03-22 18:26:50
Last modified on 2013-03-22 18:26:50
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Corollary
Classification msc 51M05
Related topic IncenterMathworldPlanetmath
Related topic AngleBisectorAsLocus
Related topic LengthsOfMedians