Cyril Parry, The Isogonal Tripolar Conic

Forum Geometricorum, 1 (2001) 33 -- 42.

Abstract:   In trilinear coordinates with respect to a given triangle ABC, we define the isogonal tripolar of a point P(p,q,r) to be the line  p: p\alpha + q\beta + r\gamma = 0. We construct a unique conic \Phi, called the isogonal tripolar conic, with respect to which  p is the polar of P for all P. Although the conic is imaginary, it has a real center and real axes coinciding with the center and axes of the real  orthic inconic. Since ABC is self-conjugate with respect to \Phi, the imaginary conic is harmonically related to every circumconic and inconic of ABC. In particular, \Phi is the reciprocal conic of the circumcircle and Steiner's inscribed ellipse. We also construct an analogous isotomic tripolar conic \Psi by working with barycentric coordinates.

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