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The family of Euclidean triangles having some fixed perimeter and area can beidentified with a subset of points on a nonsingular cubic plane curve, i.e., anelliptic curve; furthermore, if the perimeter and the square of the area arerational, then the curve has rational coordinates and those triangles withrational side lengths correspond to rational points on the curve. We firstrecall this connection, and then we develop hyperbolic analogs. There areinteresting relationships between the arithmetic on the elliptic curve (rankand torsion) and the family of triangles living on it. In the hyperbolicsetting, the analogous plane curve is a quartic with two singularities atinfinity, so the genus is still 1. We can add points geometrically by realizingthe quartic as the intersection of two quadric surfaces. This allows us toconstruct nontrivial examples of rational hyperbolic triangles having the sameinradius and perimeter as a given rational right hyperbolic triangle.

Rational Hyperbolic Triangles and a Quartic Model of Elliptic Curves