Equidiagonal quadrilateral

[1] Examples of equidiagonal quadrilaterals include the isosceles trapezoids, rectangles and squares.

Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, and 5π/12.

An equivalent condition is that the bimedians of the quadrilateral (the diagonals of the Varignon parallelogram) are perpendicular.

Using the formulas for the lengths of the bimedians, the area can also be expressed in terms of the sides a, b, c, d of the equidiagonal quadrilateral and the distance x between the midpoints of the diagonals as[5]: p.19 Other area formulas may be obtained from setting p = q in the formulas for the area of a convex quadrilateral.

[7] Silvester (2006) gives further connections between equidiagonal and orthodiagonal quadrilaterals, via a generalization of van Aubel's theorem.

An equidiagonal quadrilateral, showing its equal diagonals, Varignon rhombus, and perpendicular bimedians
An equidiagonal kite that maximizes the ratio of perimeter to diameter, inscribed in a Reuleaux triangle