Synthetic foundations of cevian geometry, III: the generalized orthocenter

Abstract

In this paper, the third in the series, we study the properties of the generalized orthocenter H corresponding to a point P, defined to be the unique point for which the lines HA, HB, HC are parallel, respectively, to QD, QE, QF, where DEF is the cevian triangle of P and ${Q = K \circ \iota(P)}$ is the isotomcomplement of P, both with respect to a given triangle ABC. We characterize the center Z of the cevian conic ${\mathcal{C}_P}$ on the 5 points ABCPQ as the center of the affine map ${\Phi_{P} = T_{P} \circ K^{-1} \circ T_{P'} \circ K^{-1}}$, where T _P is the unique affine map for which T _P (ABC) = DEF; T _P' is defined similarly for the isotomic conjugate ${P' = \iota(P)}$ of P; and K is the complement map. The point Z is the point where the nine-point conic ${\mathcal{N}_H}$ for the quadrangle ABCH and the inconic ${\mathcal{I}}$ of ABC, tangent to the sides at D, E, F, touch. This theorem generalizes the classical Feuerbach theorem.