Convex [Orthodiagonal] quadrangles q = (ABCD), have their diagonals orthogonal. The middles of their sides build a rectangle s = (EFGH) and the intersection point I of the diagonals, reflected on the sides of s, maps to the vertices of q.
Project I on the sides of q at points : J, K, L, M. The following facts hold (Steiner, Werke II, p. 358) :
(1) The quadrangle r = (JKLM) is cyclic, its vertices lying on a circle (c).
(2) Lines IJ, IK, IL and IM intersect the quadrangle at J*, K*, L* and M* respectively, lying also in (c).
(3) (J*K*L*M*) is a rectangle with sides parallel to the diagonals.

The proofs are consequences of the discussion in EightPointCircle.html . There it is proved that the projections N, O, P, Q of the side-middles of q = (ABCD) are on the circle d circumscribing s = EFGH. The following remarks imply the proofs.
1] Quadrangles OSPC and LIMC are cyclic and similar. The same is true for ANRQ and AKIJ.
2] It follows that r = JKLM and w = NOPQ have their sides parallel, hence r is cyclic.
3] Angle(EJ*J) = pi - angle(J*EH)-angle(HEQ) = pi-angle(ONQ) = pi-angle(LKJ). Last because of 2].
4] It follows that J*, L, K, J are concyclic, implying assertion (2).
5] angle(J*K*K) = angle(J*JK) = angle(EQN) = angle(EHN), proving last assertion (3).

To see the circles referred to in the assertions together with some additional remarks look at the file Orthodiagonal2.html .