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 △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 △ABC is equiangular.
Since ∠A≅∠B≅∠C, AAA yields that △ABC≅△BCA. By CPCTC, ¯AB≅¯AC≅¯BC. Hence, △ABC is equilateral.
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Title | alternative proof of necessity direction of equivalent conditions for triangles (hyperbolic and spherical) |
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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 |