  A rational triangle is a triangle with rational sides and rational area. A
Heron triangle is a triangle with integral sides and integral area. In this
article we will show that there exist infinitely many rational
parametrizations, in terms of s, of rational triangles with perimeter 2s(s+1)
and area s(s^2-1). As a corollary, there exist arbitrarily many Heron triangles
with all the same area and the same perimeter. The proof uses an elliptic K3
surface Y. Its Picard number is computed to be 18 after we prove that the
Neron-Severi group of Y injects naturally into the Neron-Severi group of the
reduction of Y at a prime of good reduction. We also give some constructions of
elliptic surfaces and prove that under mild conditions a cubic surface in
projective three-space can be given the structure of an elliptic surface by
cutting it with the family of hyperplanes through a given line L. Some of these
constructions were already known but appear to have lacked proof in the
literature until now.
