# 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 (http://planetmath.org/EquilateralTriangle) (and therefore regular (http://planetmath.org/RegularTriangle)). It better 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.

Title alternative proof of necessity direction of equivalent conditions for triangles (hyperbolic and spherical) AlternativeProofOfNecessityDirectionOfEquivalentConditionsForTriangleshyperbolicAndSpherical 2013-03-22 17:12:55 2013-03-22 17:12:55 Wkbj79 (1863) Wkbj79 (1863) 5 Wkbj79 (1863) Proof msc 51-00