[Paper Review] Heron triangle and rhombus pairs with a common area and a common perimeter

Yong Zhang, Junyao Peng|arXiv (Cornell University)|Jul 3, 2017

Mathematics and Applications4 references3 citations

TL;DR

This paper proves that there are infinitely many Heron triangles and θ-integral rhombi sharing both the same area and perimeter using Fermat's method of rational point generation on algebraic curves. In contrast, it shows no such integral isosceles triangle and θ-integral rhombus pairs exist due to the non-existence of rational solutions to a derived genus-2 hyperelliptic curve.

ABSTRACT

By Fermat's method, we show that there are infinitely many Heron triangle and $θ$-integral rhombus pairs with a common area and a common perimeter. Moreover, we prove that there does not exist any integral isosceles triangle and $θ$-integral rhombus pairs with a common area and a common perimeter.

Motivation & Objective

• To determine whether Heron triangles and θ-integral rhombi can share both area and perimeter.
• To investigate the existence of integral isosceles triangle and θ-integral rhombus pairs with common area and perimeter.
• To apply Fermat's method of rational point generation on algebraic curves to construct solutions.
• To analyze the underlying Diophantine equations governing area and perimeter equality.
• To prove the non-existence of solutions for the isosceles triangle case via genus-2 curve analysis.

Proposed method

• Parameterize Heron triangles using Brahmagupta’s formula with rational parameters u, v, w, setting w = 1 for homogeneity.
• Model θ-integral rhombi with rational side length p and rational sinθ, cosθ via parameter t ≥ 1.
• Derive a system of equations equating the perimeters and areas of the triangle and rhombus.
• Eliminate p to obtain a quartic Diophantine equation in u and t, leading to a curve C₁: s² = g(t).
• Apply Fermat’s method by assuming a quadratic ansatz s = rt² + qt + 2u to force rational solutions.
• Use the resulting rational point P′ on the curve to generate infinitely many rational solutions for u > √3/3.

Experimental results

Research questions

• RQ1 Are there infinitely many Heron triangle and θ-integral rhombus pairs with equal area and perimeter?
• RQ2 Can an integral isosceles triangle and a θ-integral rhombus share both area and perimeter?
• RQ3 What algebraic structure governs the existence of such pairs?
• RQ4 How does Fermat’s method of rational point generation apply to constructing solutions?
• RQ5 What role does the genus-2 hyperelliptic curve play in proving non-existence for the isosceles case?

Key findings

• There are infinitely many Heron triangle and θ-integral rhombus pairs with a common area and perimeter, proven via Fermat’s method on a rational curve.
• For u = 1, the Heron triangle (8, 15, 17) and rhombus with side 10 and sinθ = 3/5 share area 60 and perimeter 40.
• For u = 2, the Heron triangle (1804, 2040, 1732) and rhombus with side 1394 and sinθ = 528/697 share area 1,472,064 and perimeter 5,576.
• The equation W² = U⁶ − 4U⁴ + 8U² − 4 has only rational points (U, W) = (±1, ±1), leading to trivial solutions.
• The isosceles triangle case yields no nontrivial rational solutions, proving no such integral pairs exist.
• The non-existence result follows from the rank-1 Jacobian of a genus-2 curve and Magma’s Chabauty computation.