Orthodiagonal quadrilateral

In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

[6] A convex quadrilateral is orthodiagonal if and only if its Varignon parallelogram (whose vertices are the midpoints of its sides) is a rectangle.

[7] If the normals to the sides of a convex quadrilateral ABCD through the diagonal intersection intersect the opposite sides in R, S, T, U, and K, L, M, N are the feet of these normals, then ABCD is orthodiagonal if and only if the eight points K, L, M, N, R, S, T and U are concyclic; the second eight point circle.

A related characterization states that a convex quadrilateral is orthodiagonal if and only if RSTU is a rectangle whose sides are parallel to the diagonals of ABCD.

[6] There are several metric characterizations regarding the four triangles formed by the diagonal intersection P and the vertices of a convex quadrilateral ABCD.

Denote by m1, m2, m3, m4 the medians in triangles ABP, BCP, CDP, DAP from P to the sides AB, BC, CD, DA respectively.

An orthodiagonal quadrilateral (yellow). According to the characterization of these quadrilaterals, the two red squares on two opposite sides of the quadrilateral have the same total area as the two blue squares on the other pair of opposite sides.
An orthodiagonal quadrilateral ABCD (in blue). The Varignon parallelogram (in green) formed by the midpoints of the edges of ABCD is a rectangle. Additionally, the four midpoints (grey) and the four feet of the maltitudes (red) are cocyclic on the 8-point-circle .
A second 8-point circle can be constructed from an orthodiagonal quadrilateral ABCD (in blue). The lines perpendicular to each side through the intersection of the diagonals intersect the sides in 8 different points, which are all cocyclic.
is an orthodiagonal quadrilateral, and are rectangles whose sides are parallel to the diagonals of the quadrilateral.
is an orthodiagonal quadrilateral. and are Pascal points formed by the circle , is Pascal-points circle which defines the rectangle . and are Pascal points formed by the circle , is Pascal-points circle which defines the rectangle .