Orthodiagonal4

Orthodiagonal quadrangles - 4

Convex [Orthodiagonal] quadrangles q = (ABCD), have their diagonals orthogonal. A discussion on their properties starts with the file Orthodiagonal.html . Here are some additional remarks based in that discussion.
1) The side lengths of q are dependent through the equation: b^2 + d^2 = a^2 + c^2 .
2) The opposite sides of the inscribed cyclic quadrangle q' = EFGH intersect at J, K, on the diagonals of q and the line JK is parallel to line ML, where M, L are the intersection points of the opposite sides of q.
3) Both lines ML and JK are orthogonal to the line joining the intersection point I of the diagonals of q with the center N of the circumcircle of q'.
4) The distance of I from LM is double its distance from JK.
5) The diagonals of q are bisectors of the angle formed by lines IL, IM.

In the file refered above it is shown that the intersections E*, F*, ... etc. of EI, FI, ... etc. with the opposite sides of q are on the circumcircle of q'. Thus (see CyclicProjective.html ) the lines G*H, H*G intersect on a point X of the polar of I with respect to the circumcircle of q'. The same hapens for lines EF*, E*F. They intersect on a point Y of the polar of I too. But (B,D,I,Z) =-1 is a harmonic tetrad. Analogous statement holds for the intersection point W of AC with LM. Thus, LM coincides with the polar of I with respect to the circumcircle of q'. The other statements follow easily from this and the properties discussed in the references cited above.

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