Angle Bisectors

Interior Angle Bisectors

Definition: Interior Angle Bisectors

An interior angle bisector of a triangle is a cevian which divides the angle at a vertex into two angles of equal measure.

Characterizations

The Angle Bisector Theorem

Let $△ABC$ be a triangle and let $L$ be a point on the side $AC$.

The line segment $BL$ is the interior angle bisector of $\angle ABC$ if and only if

$$\frac{AL}{AB}=\frac{CL}{CB}$$

PROOF

TODO

Properties

Theorem: Length of Interior Angle Bisectors

The length $l$ of an interior angle bisector in a triangle is given by the lengths of its adjacent sides and and the lengths of the segments into which it divides the opposite side.

$$l^2=ab-mn$$

PROOF

TODO

Theorem: Tangentiality of Triangles

Every triangle is a tangential polygon.

PROOF

TODO

Theorem: Inradius in a Triangle

The inradius $r$ of a triangle with sides $a,b,c$ and semiperimeter $s$ is given by

$r=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}}$

PROOF

TODO