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equivalent conditions for triangles
The following theorem holds in Euclidean geometry, hyperbolic geometry, and spherical geometry:
Theorem 1.
Let $\triangle ABC$ be a triangle. Then the following are equivalent:

$\triangle ABC$ is equilateral;

$\triangle ABC$ is equiangular;

$\triangle ABC$ is regular.
Note that this statement does not generalize to any polygon with more than three sides in any of the indicated geometries.
Proof.
It suffices to show that $\triangle ABC$ is equilateral if and only if it is equiangular.
Sufficiency: Assume that $\triangle ABC$ is equilateral.
Since $\overline{AB}\cong\overline{AC}\cong\overline{BC}$, SSS yields that $\triangle ABC\cong\triangle BCA$. By CPCTC, $\angle A\cong\angle B\cong\angle C$. Hence, $\triangle ABC$ is equiangular.
Necessity: Assume that $\triangle ABC$ is equiangular.
By the theorem on determining from angles that a triangle is isosceles, we conclude that $\triangle ABC$ is isosceles with legs $\overline{AB}\cong\overline{AC}$ and that $\triangle BCA$ is isosceles with legs $\overline{AC}\cong\overline{BC}$. Thus, $\overline{AB}\cong\overline{AC}\cong\overline{BC}$. Hence, $\triangle ABC$ is equilateral. ∎
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