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. ∎
随便看	partial fractions of expressions and partition problems (recreational) PartialFunction partial function PartialIsometry partial isometry PartialIsometryOnHilbertSpaces partial isometry on Hilbert spaces partialLinearSpace (partial) linear space PartiallyOrderedAlgebraicSystem partially ordered algebraic system PartiallyOrderedGroup partially ordered group PartiallyOrderedRing partially ordered ring PartialMapping partial mapping PartialOrder partial order PartialOrderingInATopologicalSpace partial ordering in a topological space PartialOrderWithChainConditionDoesNotCollapseCardinals partial order with chain condition does not collapse cardinals partialTiltingModule (partial) tilting module