Orthogonal_Diagonals

Orthodiagonal quadrangles

Given a rectangle s = (ABCD) and a point E inside it, consider the quadrangle q = (FGHI), whose vertices result by reflecting E on the sides of the rectangle.
Prove that:
(1) The diagonals of q intersect at E, are orthogonal and of double length, compared to their parallel rectangle-side.
(2) The vertices of the rectangle are the middles of the sides of q.
(3) The area of the q is the double of the area of s.
(4) All quadrangles, having orthogonal diagonals [orthodiagonal] can be constructed by this method.
(5) The middles of the sides of a quadrilateral with orthogonal diagonals lie on a circle.
(6) The medial lines of the sides of q build another orthodiagonal quadrangle q'=(JKLM), with diagonals parallel to those of q. In fact K, M are on the medial line of segment IG. Thus, line [KM] coincides with that medial line.
(7) The intersection points of the diagonals of q and q' are symmetric with respect to the center O of s. In fact, one diagonal of the rectangle with sides the diagonals of q and q' is the line joining the middles of the diagonals of q.
(8) Repeat the construction of (6) for q', resulting to a third orthodiagonal quadrilateral q'' = PQRS.
(9) q'' is similar to q.
(10) q' and q'' are perspective to q with respect to a point U lying on the line EV, V being the intersection of the diagonals of q''. The vertices of q' and q'' lie on lines through U. q'' is anti-homothetic to q.
(11) The projections of E on the sides of q lie on a circle with center T. T is the middle of EU.
(12) Repeating the procedure of (7) to q'' we get q(3) and continuing that way we produce a sequence of orthodiagonal quadrilaterals q(n). Setting q(0) = q, q(1) = q' and q(2) = q'' we have a sequence consisting of two subsequences {q(2n)} and {q(2n+1)} of similar orthodiagonal quadrilaterals with parallel sides and converging to the point U.
Some properties of these quadrilaterals, related to (11), are examined in the file Orthodiagonal.html .

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

http://users.math.uoc.gr/~pamfilos/