Angle bisector length t² = ab − mn

Dependencies: Angle bisector ⇔ equidistant from sides (the bisector + its property), Inscribed angles on same arc are equal; angle in a semicircle is right (inscribed angles subtending the same arc are equal), AA similarity (AA similarity), Intersecting chords: PA·PB = PC·PD (the equal-product theorem for two intersecting chords inside a circle).

Statement

In $\triangle ABC$, let $AD$ be the interior angle bisector of $\angle A$, with $D$ on $BC$ (by Angle bisector divides opposite side in ratio of adjacent sides, $D$ divides $BC$ into $BD = m$ and $DC = n$). Denote the two adjacent sides

and write the length of the interior bisector segment as $t = AD$. Then

That is, the square of the interior bisector length = the product of the two adjacent sides − the product of the two segments it cuts off on the opposite side. Combined with Angle bisector divides opposite side in ratio of adjacent sides ($\dfrac{m}{n} = \dfrac{c}{b}$), this lets one compute $t$ directly from $a, b, c$.