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equivalent conditions for triangles∗
Wkbj79†
2013-03-21 23:08:15
The following theorem holds in Euclidean geometry, hyperbolic geometry,
and spherical geometry:
Theorem 1. Let 4ABC be a triangle. Then the following are equivalent:
• 4ABC is equilateral;
• 4ABC is equiangular;
• 4ABC is regular.
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 4ABC is equilateral if and only if it is equiangular.
Sufficiency: Assume that 4ABC is equilateral.
A
B
C
∗ hEquivalentConditionsForTrianglesi
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h39536i Privacy setting: h1i hTheoremi h51-00i
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∼ AC =
∼ BC, SSS yields that 4ABC ∼
Since AB =
= 4BCA. By CPCTC,
∠A ∼
= ∠B ∼
= ∠C. Hence, 4ABC is equiangular.
Necessity: Assume that 4ABC is equiangular.
A
B
C
By the theorem on determining from angles that a triangle is isosceles, we
conclude that 4ABC is isosceles with legs AB ∼
= AC and that 4BCA is isosceles with legs AC ∼
= BC. Thus, AB ∼
= AC ∼
= BC. Hence, 4ABC is equilateral.
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