Orthodiagonal2

Orthodiagonal quadrangles - 2

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.
All these statements are shown in Orthodiagonal.html . Here are displayed the relevant circles and some additional lines.

Two additional remarks as exercises:
a) The medial lines of segments KJ, LM pass trhough the middles W, V of IA and IC respectively.
b) These medials intersect at U, lying on the line joining the middles X, Y of the diagonals of the original quadrangle.
For the proofs see the file Orthodiagonal3.html .

Some additional facts on [orthodiagonal quadrilaterals] are discussed in the file Orthogonal_Diagonals.html .

The file OrthodiagonalFromCyclic.html contains the inverse procedure, of constructing the orthodiagonal quadrilateral q, given the cyclic quadrilateral r.

See in the file QuadModuli.html for an interesting application of a special kind of orthodiagonal quadrilaterals, namely those that have equal diagonals.

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