A Desarguesian dual for Nagel's middlespoint (Q914152)

Nagel's middlespoint of a triangle was defined by Nagel in 1836 as follows. For a given triangle ABC, let \(S_ A\), \(S_ B\), \(S_ C\) be the midpoints of BC, CA, AB, and \(I_ a\), \(I_ b\), \(I_ c\) the centres of the excircles; then the lines \(S_ AI_ a\), \(S_ BI_ b\), \(S_ CI_ c\) meet at M, the middlespoint of ABC. In the present paper the author derives a dual of this point, namely a straight line m. This is obtained from Desargues' theorem; since the triangles \(I_ aI_ bI_ c\) and \(S_ AS_ BS_ C\) are perspective from the middlespoint M, they are perspective from a line m, called the middlesline of the triangle ABC. The author shows how the middlesline is related to the Gergonne point G of ABC.