equivalent conditions for triangles

Theorem 1.

Let ABC be a triangleMathworldPlanetmath. Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  • ABC is equilateral (http://planetmath.org/EquilateralTriangle);

  • ABC is equiangular (http://planetmath.org/EquiangularTriangle);

  • ABC is regularPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/RegularTriangle).

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


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

Sufficiency: Assume that ABC is equilateral.


Since AB¯AC¯BC¯, SSS yields that ABCBCA. By CPCTC, ABC. Hence, ABC is equiangular.

Necessity: Assume that ABC is equiangular.


By the theorem on determining from angles that a triangle is isosceles, we conclude that ABC is isosceles with legs AB¯AC¯ and that BCA is isosceles with legs AC¯BC¯. Thus, AB¯AC¯BC¯. Hence, 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