Skip to content

Interior Angle Bisectors#

Definition: Interior Angle Bisectors

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

![[res/Interior Angle Bisector.svg]]

Characterizations#

The Angle Bisector Theorem

Let \(\triangle ABC\) be a [[Triangles|triangle]] and let \(L\) be a point on the side \(AC\).

The [[Line Segments|line segment]] \(BL\) is the [[Angle Bisectors#Interior Angle Bisector|interior angle bisector]] of \(\angle ABC\) if and only if

\[ \frac{AL}{AB} = \frac{CL}{CB} \]

![[res/The Angle Bisector Theorem.svg]]

Proof

TODO

Properties#

Theorem: Length of Interior Angle Bisectors

The length \(l\) of an [[Angle Bisectors#Interior Angle Bisector|interior angle bisector]] in a [[Triangles|triangle]] is given by the lengths of its adjacent sides and and the lengths of the segments into which it divides the opposite side.

\[ l^2 = ab - mn \]

![[res/Length of Interior Angle Bisector.svg]]

Proof

TODO

Theorem: Tangentiality of Triangles

Every [[Polygons|triangle]] is a [[Tangential Polygons|tangential polygon]].

Proof

TODO

Theorem: Inradius in a Triangle

The [[Tangential Polygons|inradius]] \(r\) of a [[Polygons|triangle]] with [[Euclidean Geometry|sides]] \(a,b,c\) and [[Polygons|semiperimeter]] \(s\) is given by

\[r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}\]
Proof

TODO