On quasi-Heronian equable triangles

Aebi, Christian

doi:10.4171/em/512

Cited by 1 publication

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“…We should also remark that it is not surprising that there are relatively fewer equable parallelograms on the Eisenstein lattice than there are on the integer lattice. It was already observed that for triangles, up to Euclidean transformations, there are only two equable triangles on the Eisenstein lattice, while there are five on the integer lattice; see [5,6,7] and the Appendix in [2].…”

Section: Introductionmentioning

confidence: 99%

Lattice equable quadrilaterals I: Parallelograms

Aebi

Cairns

2021

Enseign. Math.

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This paper studies equable parallelograms whose vertices lie on the integer lattice. Using Rosenberger’s Theorem on generalised Markov equations, we show that the g.c.d. of the side lengths of such parallelograms can only be 3, 4 or 5, and in each of these cases the set of parallelograms naturally forms an infinite tree all of whose vertices have degree 4, bar the root. The paper then focuses on what we call Pythagorean equable parallelograms. These are lattice equable parallelograms whose complement in a circumscribing rectangle consists of two Pythagorean triangles. We prove that for these parallelograms the shortest side can only be 3, 4, 5, 6 or 10, and there are five infinite families of such parallelograms, given by solutions to corresponding Pell-like equations.

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