---
tag: 0859R2V6ES
kind: statement
role: theorem
name: Angle bisector theorem
msc: [51M04]
uses: [0CTNFGYMM2, 0GJAJC6TNZ, 0NPPGHQNXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Angle_bisector_theorem
    locator: Theorem
  - url: https://en.wikipedia.org/wiki/Angle_bisector_theorem
    locator: History
same_as:
  wikidata: Q925854
  wikipedia: Angle bisector theorem
---

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $D$ be a point of the segment $BC$, and let $\angle$ denote the [[angle at a point|0GJAJC6TNZ]].
If $\angle BAD = \angle DAC$, then
$$\frac{d(B, D)}{d(D, C)} = \frac{d(A, B)}{d(A, C)}.$$
Conversely, if $D$ is a point of the segment $BC$ with this equality, then $\angle BAD = \angle DAC$.

## Remarks

In words, the bisector of the angle at $A$ cuts the opposite side in the ratio of the two adjacent sides.
In an isosceles triangle with $d(A, B) = d(A, C)$, the bisector from $A$ meets $BC$ at its midpoint.
A similar statement holds for the bisector of the exterior angle at $A$, which meets the line $BC$ outside the segment.

## History

The theorem is Proposition 3 of Book VI of Euclid's Elements.

## Explanations

### L1 human

Take the triangle with $d(A, B) = 6$ and $d(A, C) = 3$. The bisector of the angle at $A$ cuts $BC$ into two parts. The part next to $B$ is twice as long as the part next to $C$.