On the Feet of Concurrent Normals of a Conic

The main object of this paper is to investigate the curious and mutually correlated properties of three triads of conies connected with a triangle ABC. These are: (l) the three rectangular hyperbolas which pass through the in- and ex-centres I, Ij, I, I3, and have concurrent normals at these points, (2) the three parabolas which are inscribed to ABC so as to have concurrent normals at the points of contact with the sides, and (3) the three parabolas circumscribed to ABC so as to have concurrent normals at these points It is shewn that the centres of the three rectangular hyperbolas (1), and the foci of (he three inscribed parabolas (2) are the same three points p, q, r on the circum-circle ABC, and that these may be obtained parametrically as the ischolar_mains of the binary Jacobian of the triad ABC, and the pair of circular points. Several other remarkable properties connected with the points p q r are obtained.