Properties of Pythagorean quadrilaterals

Abstract

There are many named quadrilaterals. In our hierarchical classification in [1, Figure 10] we included 18, and at least 10 more have been named, but the properties of the latter have only scarcely (or not at all) been studied. However, only a few of all these quadrilaterals are defined in terms of properties of the sides alone. Two well-known classes are the rhombi and the kites, defined to be quadrilaterals with four equal sides or two pairs of adjacent equal sides respectively. The orthodiagonal quadrilaterals are defined to have perpendicular diagonals, but an equivalent defining condition is quadrilaterals where the consecutive sides a, b, c, d satisfy a2 + c2 = b2 + d2. Then it is possible to prove that the diagonals are perpendicular and that no other quadrilaterals have perpendicular diagonals (see [2, pp. 13-14]). In the same way tangential quadrilaterals can be defined to be convex quadrilaterals where a + c = b + d. Starting from this equation, it is possible to prove that these and only these quadrilaterals have an incircle (since this equation is a characterisation of tangential quadrilaterals, see [3, pp. 65-67]).