Equiangular Triangle is Equilateral

Theorem
Let $\triangle ABC$ be equiangular.

Then $\triangle ABC$ is an equilateral triangle.

Proof
Let $\triangle ABC$ be equiangular.

By definition of equiangular polygon, any two of the internal angles of $\triangle ABC$ are equal.

, let $\angle ABC = \angle ACB$.

Then by Triangle with Two Equal Angles is Isosceles, $AB = AC$.

As the choice of equal angles was arbitrary, it follows that any two sides of $\triangle ABC$ are equal.

Hence all $3$ sides of $\triangle ABC$ are equal.

Hence the result by definition of equilateral triangle.

Also see

 * Equilateral Triangle is Equiangular, of which this is the converse.