Synthetic foundations of cevian geometry, III: the generalized orthocenter

摘要

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 _(? ι)(P) is the isotomcomplement of P, both with respect to a given triangle ABC. We characterize the center Z of the cevian conic CP on the 5 points ABCPQ as the center of the affine map Φ_P = T_P ? K~(-1) ? T_(P') ? 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'= ι(P) of P; and K is the complement map. The point Z is the point where the nine-point conic N_H for the quadrangle ABCH and the inconic I of ABC, tangent to the sides at D,E, F, touch. This theorem generalizes the classical Feuerbach theorem.