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∘ι(P)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 CPCP on the 5 points ABCPQ as the center of the affine map ΦP=TP∘K−1∘TP′∘K−1ΦP=TP∘K−1∘TP′∘K−1, where TP is the unique affine map for which TP(ABC) = DEF; TP' is defined similarly for the isotomic conjugate P′=ι(P)P′=ι(P) of P; and K is the complement map. The point Z is the point where the nine-point conic NHNH for the quadrangle ABCH and the inconic II of ABC, tangent to the sides at D, E, F, touch. This theorem generalizes the classical Feuerbach theorem.