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.

∎

单词	AlternativeProofOfNecessityDirectionOfEquivalentConditionsForTriangleshyperbolicAndSpherical
释义	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. ∎
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