equivalent conditions for triangles

The following theorem holds in Euclidean geometry, hyperbolic geometry, and spherical geometry:

Theorem 1.

Let $\\triangle ABC$ be a triangle. Then the following are equivalent:

•
$\\triangle ABC$ is equilateral (http://planetmath.org/EquilateralTriangle);
•
$\\triangle ABC$ is equiangular (http://planetmath.org/EquiangularTriangle);
•
$\\triangle ABC$ is regular (http://planetmath.org/RegularTriangle).

Note that this statement does not generalize to any polygon with more than three sides in any of the indicated geometries.

Proof.

It suffices to show that $\\triangle ABC$ is equilateral if and only if it is equiangular.

Sufficiency: Assume that $\\triangle ABC$ is equilateral.

Since $\\overline{AB}\\cong\\overline{AC}\\cong\\overline{BC}$ , SSS yields that $\\triangle ABC\\cong\\triangle BCA$ . By CPCTC, $\\angle A\\cong\\angle B\\cong\\angle C$ . Hence, $\\triangle ABC$ is equiangular.

Necessity: Assume that $\\triangle ABC$ is equiangular.

By the theorem on determining from angles that a triangle is isosceles, we conclude that $\\triangle ABC$ is isosceles with legs $\\overline{AB}\\cong\\overline{AC}$ and that $\\triangle BCA$ is isosceles with legs $\\overline{AC}\\cong\\overline{BC}$ . Thus, $\\overline{AB}\\cong\\overline{AC}\\cong\\overline{BC}$ . Hence, $\\triangle ABC$ is equilateral.∎

单词	EquivalentConditionsForTriangles
释义	equivalent conditions for triangles The following theorem holds in Euclidean geometry, hyperbolic geometry, and spherical geometry: Theorem 1. Let $\\triangle ABC$ be a triangle. Then the following are equivalent: • $\\triangle ABC$ is equilateral (http://planetmath.org/EquilateralTriangle); • $\\triangle ABC$ is equiangular (http://planetmath.org/EquiangularTriangle); • $\\triangle ABC$ is regular (http://planetmath.org/RegularTriangle). Note that this statement does not generalize to any polygon with more than three sides in any of the indicated geometries. Proof. It suffices to show that $\\triangle ABC$ is equilateral if and only if it is equiangular. Sufficiency: Assume that $\\triangle ABC$ is equilateral. Since $\\overline{AB}\\cong\\overline{AC}\\cong\\overline{BC}$ , SSS yields that $\\triangle ABC\\cong\\triangle BCA$ . By CPCTC, $\\angle A\\cong\\angle B\\cong\\angle C$ . Hence, $\\triangle ABC$ is equiangular. Necessity: Assume that $\\triangle ABC$ is equiangular. By the theorem on determining from angles that a triangle is isosceles, we conclude that $\\triangle ABC$ is isosceles with legs $\\overline{AB}\\cong\\overline{AC}$ and that $\\triangle BCA$ is isosceles with legs $\\overline{AC}\\cong\\overline{BC}$ . Thus, $\\overline{AB}\\cong\\overline{AC}\\cong\\overline{BC}$ . Hence, $\\triangle ABC$ is equilateral.∎
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