  Triangles with integer length sides and integer area are known as Heron
triangles. Taking rescaling freedom into account, one can apply the same name
when all sides and the area are rational numbers. A perfect triangle is a Heron
triangle with all three medians being rational, and it is a longstanding
conjecture that no such triangle exists. However, Buchholz and Rathbun showed
that there are infinitely many Heron triangles with two rational medians, an
infinite subset of which are associated with rational points on an elliptic
curve $E(\mathbb{Q})$ with Mordell-Weil group $\mathbb{Z}\times
\mathbb{Z}/2\mathbb{Z}$, and they observed a connection with a pair of Somos-5
sequences. Here we make the latter connection more precise by providing
explicit formulae for the integer side lengths, the two rational medians, and
the area in this infinite family of Heron triangles. The proof uses a combined
approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT)
maps in the plane, from several different viewpoints: complex analysis, real
dynamics, and reduction modulo a prime.
