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Amicable Heronian Parallelograms

Iwan Praton, Weiran Zeng

Abstract

A convex polygon is Heronian if its side lengths and its area are integers. Two polygons are amicable if the area of one is equal to the perimeter of the other, and vice versa. We show that there are infinitely many pairs of amicable Heronian parallelograms, and we give necessary and sufficient conditions for a Heronian parallelogram to be part of an amicable pair.

Amicable Heronian Parallelograms

Abstract

A convex polygon is Heronian if its side lengths and its area are integers. Two polygons are amicable if the area of one is equal to the perimeter of the other, and vice versa. We show that there are infinitely many pairs of amicable Heronian parallelograms, and we give necessary and sufficient conditions for a Heronian parallelogram to be part of an amicable pair.
Paper Structure (2 theorems, 3 equations, 2 figures)

This paper contains 2 theorems, 3 equations, 2 figures.

Key Result

Lemma 1

Let $b$ and $s$ be positive integers, and let $A$ be any positive integer with $A\leq bs$. Then there is a Heronian parallelogram with base $b$, side $s$, and area $A$. Similarly, suppose $b$ and $h$ are positive integers, and let $u$ be any positive integer with $u\geq h$. Then there is a Heronian

Figures (2)

  • Figure 1: Three parallelograms with the same sides
  • Figure 2: Three parallelograms with the same base and height

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 2
  • proof