---
tag: 056VS31A0J
kind: statement
role: theorem
name: Isosceles triangle theorem
aliases: [pons-asinorum, base-angles-theorem]
msc: [51M04]
uses: [0CTNFGYMM2, 0GJAJC6TNZ, 0NPPGHQNXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Isosceles_triangle
    locator: Terminology, classification, and examples
---

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in a Euclidean space $\mathbb{R}^n$ with $d(A, B) = d(A, C)$, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]].
Then the angles at $B$ and $C$ are equal: $\angle ABC = \angle ACB$.

## Remarks

The converse holds too: if $\angle ABC = \angle ACB$ in a triangle $ABC$, then $d(A, B) = d(A, C)$.
Mathlib's form of the statement does not need that $A$, $B$ and $C$ are not collinear.
In an equilateral triangle, the theorem applies at each vertex. So all three angles are equal.

## History

The statement is Proposition 5 of Book I of Euclid's Elements. Its Latin name is pons asinorum, "the bridge of donkeys".

## Explanations

### L1 human

An isosceles triangle has two equal sides, like the outline of a simple roof. The theorem says that the two base angles are then equal. Fold the triangle along the line from the top vertex to the midpoint of the base: the two halves match.