Properties of equidiagonal quadrilaterals (Q2878869)

The paper under review is a continuation of a series of papers, written by the same author in the same journal, that deal with characterizations of special kinds of quadrilaterals. These include orthodiagonal, tangential, extangential, and bicentric quadrilaterals, and rectangles, trapezoids, and kites. In the paper under review, the author considers equidiagonal quadrilaterals, i.e., quadrilaterals whose diagonals have equal lengths. He gives eight characterizations of these figures that involve side lengths, angles, area, distance between midpoints of the diagonals, bimedians, the Varignon parallelogram. The author observes some kind of duality between a convex quadrilateral \(Q\) and the triangle version of its van Aubel's quadrilateral \(Q^*\), i.e., the quadrilateral whose vertices are the centroids of the four equilateral triangles constructed outwardly on the sides of \(Q\); he shows that \(Q\) (respectively, \(Q^*\)) is equidiagonal if and only if \(Q^*\) (respectively, \(Q\)) is orthodiagonal. This resembles an earlier duality that he had established between \(Q\) and its Varignon parallelogram \(Q'\) in an earlier paper, where he had proved that \(Q\) (respectively, \(Q'\)) is equidiagonal if and only if \(Q'\) (respectively, \(Q\)) is orthodiagonal. He then answers the questions regarding what parallelograms, rhombi, trapezoids, and cyclic quadrilaterals are equidiagonal. Finally, he uses Euler's polynomial relation among the lengths of the sides and the diagonals to express the (squares of the) diagonals of an equidiagonal quadrilateral with given side lengths as a root of a cubic equation.