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I have heard of both equilateral triangles and equiangular triangles. (For example, this sporcle quiz lists both.) Are these always equivalent, regardless of geometry?

Euclidean geometry Elliptical geometry Hyperbolic geometry

I know they are the same in Euclidean geometry (a triangle that is equilateral has three 60 degree angles), but how about elliptic or hyperbolic? Or geometries without constant curvature?

In elliptic and hyperbolic geometry, I believe an equilateral triangle is uniquely determined up to congruence by one of its angles, but I'm not sure of the proof.

If the conditions are equivalent for triangles, are there two words because they are also used to describe equilateral / equiangular polygons, which are not always the same? For example, a rectangle is equiangular but need not be equilateral, and a rhombus is equilateral but need not be equiangular. What's so special about triangles? (or tetrahedra?)

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If you don't have constant curvature, then all bets are off. You can make any set of angles and lengths into a geodesic triangle by bending the surface appropriately. –  Henning Makholm Dec 29 '11 at 22:43
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2 Answers

up vote 8 down vote accepted

The first 28 propositions in Euclid are all proved without using the parallel postulate, so they're valid in elliptic and hyperbolic geometry. Proposition 6 tells you that equiangular implies equilateral, proposition 8 the converse.

For a counterexample in a space that's not of constant curvature, just take an equilateral triangle in a Euclidean plane and then double the metric in a small circular region centered on one vertex. The geodesics are still geodesics, the angles are still all 60 degrees, but the triangle is no longer equilateral.

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In the Taxicab Plane http://en.wikipedia.org/wiki/Taxicab_geometry which obeys all of the axioms of the Euclidean plane except for the congruence axiom, one can have equilateral triangles which are not equiangular.

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