Lattice equable quadrilaterals I: parallelograms (Q2077247)

The authors study so called lattice equable parallelograms. They introduce such a kind of parallelograms as follows. A lattice equable parallelogram is a parallelogram satisfying the conditions: (1) it has integer sides, (2) its perimeter equals its area, and (3) its vertices lie on the integer lattice \(\mathbb{Z }_2\). The main results of the paper give a complete description of these parallelograms. The authors also investigate a special class of lattice equable parallelograms. They define a Pythagorean equable parallelogram as a lattice equable parallelogram which is circumscribed by a rectangle having integer side lengths so that the rectangle and parallelogram share a common diagonal. It is shown that there are five infinite families of such parallelograms.