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 $\mathrm{△}ABC$ 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 $\mathrm{△}ABC$ is equiangular.

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

∎

Title	alternative proof of necessity direction of equivalent conditions for triangles (hyperbolic and spherical)
Canonical name	AlternativeProofOfNecessityDirectionOfEquivalentConditionsForTriangleshyperbolicAndSpherical
Date of creation	2013-03-22 17:12:55
Last modified on	2013-03-22 17:12:55
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	5
Author	Wkbj79 (1863)
Entry type	Proof
Classification	msc 51-00