Equiangular Triangle is Equilateral

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Let $\triangle ABC$ be equiangular.

Then $\triangle ABC$ is an equilateral triangle.


Let $\triangle ABC$ be equiangular.

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

Without loss of generality, 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.


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