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# alternative proof of necessity direction of equivalent conditions for triangles (hyperbolic and spherical)

The following is a proof that, in hyperbolic geometry and spherical geometry, an equiangular triangle $\triangle ABC$ is automatically equilateral (and therefore regular). It better parallels the proof of sufficiency supplied in the entry equivalent conditions for triangles and is slightly shorter than the proof of necessity supplied in the same entry.

###### Proof.

Assume that $\triangle ABC$ is equiangular.

Since $\angle A\cong\angle B\cong\angle C$, AAA yields that $\triangle ABC\cong\triangle BCA$. By CPCTC, $\overline{AB}\cong\overline{AC}\cong\overline{BC}$. Hence, $\triangle ABC$ is equilateral.

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## Mathematics Subject Classification

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