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.
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Minevich, I., Morton, P. Synthetic foundations of cevian geometry, III: the generalized orthocenter. J. Geom. 108, 437–455 (2017). https://doi.org/10.1007/s00022-016-0350-2
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DOI: https://doi.org/10.1007/s00022-016-0350-2