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Heron triangles and the hunt for unicorns

Andrew N. W. Hone

Abstract

A Heron triangle is one that has all integer side lengths and integer area, which takes its name from Heron of Alexandria's area formula. From a more relaxed point of view, if rescaling is allowed, then one can define a Heron triangle to be one whose side lengths and area are all rational numbers. A perfect triangle is a Heron triangle with all three medians being rational. According to a longstanding conjecture, no such triangle exists, so perfect triangles are as rare as unicorns. However, if perfect is the enemy of good, then perhaps it is best to insist on only two of the medians being rational. Buchholz and Rathbun found an infinite family of Heron triangles with two rational medians, which they were able to associate with the set of rational points on an elliptic curve $E(\mathbb{Q})$. Here we describe a recently discovered explicit formula for the sides, area and medians of these (almost perfect) triangles, expressed in terms of a pair of integer sequences: these are Somos sequences, which first became popular thanks to David Gale's column in Mathematical Intelligencer.

Heron triangles and the hunt for unicorns

Abstract

A Heron triangle is one that has all integer side lengths and integer area, which takes its name from Heron of Alexandria's area formula. From a more relaxed point of view, if rescaling is allowed, then one can define a Heron triangle to be one whose side lengths and area are all rational numbers. A perfect triangle is a Heron triangle with all three medians being rational. According to a longstanding conjecture, no such triangle exists, so perfect triangles are as rare as unicorns. However, if perfect is the enemy of good, then perhaps it is best to insist on only two of the medians being rational. Buchholz and Rathbun found an infinite family of Heron triangles with two rational medians, which they were able to associate with the set of rational points on an elliptic curve . Here we describe a recently discovered explicit formula for the sides, area and medians of these (almost perfect) triangles, expressed in terms of a pair of integer sequences: these are Somos sequences, which first became popular thanks to David Gale's column in Mathematical Intelligencer.
Paper Structure (6 sections, 1 theorem, 28 equations, 4 figures, 3 tables)

This paper contains 6 sections, 1 theorem, 28 equations, 4 figures, 3 tables.

Table of Contents

  1. Pythagorean triples
  2. Heron triangles and unicorns
  3. Heron triangles with one rational median
  4. Triangles with two rational medians
  5. Intermezzo: Somos-5 sequences
  6. Heron triangles with two rational medians

Key Result

Theorem 6.1

For each integer $n\geq 1$, the terms in the pair of Somos-5 sequences $(S_n)$ and $(T_n)$ in (s5seq) provide a Heron triangle with two rational medians, having integer side lengths given by with $\gcd(a,b,c)=1$, rational median lengths and integer area

Figures (4)

  • Figure 1: The (15,13,14) Heron triangle from the two Pythagorean triples (9,12,15) & (5,12,13).
  • Figure 2: Triangle with one labelled median
  • Figure 3: Some orbits of the map (\ref{['qrts5']}) in the positive quadrant.
  • Figure 4: The curve (\ref{['uvcurve']}) in the $(U,V)$-plane.

Theorems & Definitions (1)

  • Theorem 6.1