# equivalent conditions for triangles

###### 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 EquivalentConditionsForTriangles 2013-03-22 17:12:46 2013-03-22 17:12:46 Wkbj79 (1863) Wkbj79 (1863) 10 Wkbj79 (1863) Theorem msc 51-00 Triangle IsoscelesTriangle EquilateralTriangle EquiangularTriangle RegularTriangle