Isosceles triangle theorem

Statement

Let $ABC$ be a triangle in a Euclidean space $\mathbb{R}^n$ with $d(A, B) = d(A, C)$, where $d$ is the Euclidean distance. 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.