Is an equilateral triangle the same as an equiangular triangle, in any geometry?

The first 28 propositions in Euclid are all proved without using the parallel postulate, so they're valid in euclidean 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.

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.