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. ∎

Title	equivalent conditions for triangles
Canonical name	EquivalentConditionsForTriangles
Date of creation	2013-03-22 17:12:46
Last modified on	2013-03-22 17:12:46
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	10
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 51-00
Related topic	Triangle
Related topic	IsoscelesTriangle
Related topic	EquilateralTriangle
Related topic	EquiangularTriangle
Related topic	RegularTriangle