Isosceles with a 60° angle ⇒ equilateral

Dependencies: Base angles of an isoceles triangle are equal, Triangle interior angles sum to 180°, Isoceles converse: equal base angles ⇒ equal sides.

Statement

Let $\triangle ABC$ be an isosceles triangle with legs $AB = AC$ (apex at $A$). If any one of its interior angles equals $60^\circ$, then all three sides are equal — i.e. $\triangle ABC$ is equilateral.

Note that "the $60^\circ$ angle" could be either the apex angle $\angle A$, or a base angle $\angle B$ (or symmetrically $\angle C$). Both cases must be handled separately.