Orthodiagonal3

Orthodiagonal quadrangles - 3

Convex [Orthodiagonal] quadrangles q = (FGIH), have their diagonals orthogonal. The middles of their sides build a rectangle s = (BCDE) and the intersection point A of the diagonals, reflected on the sides of s, maps to the vertices of q.
Below are proved certain properties related to the cyclic quadrilateral KNPO associated to the orthodiagonal q. A discussion on this kind of quadrangles starts in the file Orthodiagonal.html .

Start with q = FGIH and construct q' = WXVU from the medial lines of q. Then draw p = POKN with vertices at the projections of A on the sides of q.
1) The medial line of NP passes through Q. Similar properties hold for the other sides of p.
2) The quadrangles k1 = APFN and m1 = VBFE are cyclic and similar.
3) Then their diagonals intersect at equal angles angle(C'D'E) = angle(PZA). This implies that ZQD'S is cyclic hence:
4) Lines YZ and C'F are parallel. Similar properties hold for the other medial lines of p and the lines joining opposite vertices of q and q'.
5) Triangles QYL and FE'I are similar with ratio 1/2. Join Y to A. YA divides QL in ratio QA/AL. Extend AY beyond Y to its double. The end point E* must be on FC'. It must be also on IE', thus E* coincides with E' and Y is the middle of AE'.
6) Line AE' is orthogonal to the polar of A (line joining the intersection points of OK, PN and OP,KN).
Some additional properties of these quadrilaterals are further discussed in the file Orthodiagonal4.html .

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