  A rational triangle is a triangle with rational side lengths. We consider
three different families of rational triangles having a fixed side and whose
vertices are rational points in the plane. We display a one-to-one
correspondence between each family and the set of rational points of an
algebraic curve. These algebraic curves are: a curve of genus 0, an elliptic
curve, and a genus 3 curve. We study the set of rational points on each of
these curves and describe some of its rational points explicitly.
