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.

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.

$\blacksquare$

Also see

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

Sources

• 1968: M.N. Aref and William Wernick: Problems & Solutions in Euclidean Geometry ... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.13$: Corollary
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs: Problem Set $\text A.5$: $32$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): polygon
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): polygon