Less than equable Heronian triangles

Abstract

It is well known that there are precisely five integer-sided triangles which have equal area, Δ, and perimeter, P. These triangles are called equable Heronian triangles.A proof of this result was given by Whitworth [1]. Since Whitworth's time, much attention has been given to triangles whose areas are integer multiples of their perimeters, for example [2, 3]. However, as this paper will show, Heronian triangles with areas less than their perimeters have some mathematical interest.

Lattice equable quadrilaterals II -- kites, trapezoids and cyclic quadrilaterals

“…This is OEIS entry A001653 [12]. Here are the first 3 cases: We follow a strategy suggested in [6]. Consider a trapezoid with vertices OABC in counterclockwise order, such that OA is the longest side, having an acute angle at 0; see Figure 9.…”

“…In the language of [2], OA ′ C is a perimeter-dominant Heronian triangle. Heronian triangles with this property have been classified [6,2]. The complete list is given in Tables 1 and 2.…”

“…In [6], Dolan briefly outlines how equable trapezoids might be classified by reducing the problem to the study of certain triangles; [6] concludes with the words: "A nice exercise for an interested reader is then to determine precisely which triangles produce equable trapezia in this fashion". Indeed, this nice exercise gives the following result.…”

On quasi-Heronian equable triangles

“…which is the equilateral triangle. The remaining three cases from Table 1 are (u, v) = (1, 1), (1,5) and (2,2).…”

“…Recall that a Heron triangle is a triangle with integer sides and integer area. By Dolan's Theorem [1], the classification of perimeter-dominant Heronian triangles contains three exceptional examples and four infinite families determined by certain Pell, or Pell-like, equations.…”