Synthetic foundations of cevian geometry, III: The generalized orthocenter

In this paper, the third in the series, we define the generalized orthocenter $H$ corresponding to a point $P$, with respect to triangle $ABC$, as the unique point for which the lines $HA,HB,HC$ are parallel, respectively, to $QD, QE, QF$,where$DEF$istheceviantriangleof$P$and$Q=K \circ \iota(P)$isthe$isotomcomplement$of$P$,bothwithrespectto$ABC$. We prove a generalized Feuerbach Theorem, and characterize the center $Z$ of the cevian conic ${\mathcal{C}}_P$, defined in Part II, 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 $T_P(ABC)=DEF$; $T_{P^{\mathrm{\prime }}}$ is defined similarly for the isotomic conjugate $P^{\mathrm{\prime }}=\iota (P)$ of $P$; and $K$ is the complement map. The affine map ${\mathrm{\Phi }}_P$ fixes $Z$ and takes the nine-point conic ${\mathcal{N}}_H$ for the quadrangle $ABCH$ (with respect to the line at infinity) to the inconic $\mathcal{I}$, defined to be the unique conic which is tangent to the sides of $ABC$ at the points $D,E,F$. The point $Z$ is therefore the point where the nine-point conic ${\mathcal{N}}_H$ and the inconic $\mathcal{I}$ touch. This theorem generalizes the usual Feuerbach theorem and holds in all cases where the point $P$ is not on a median, whether the conics involved are ellipses, parabolas, or hyperbolas, and also holds when $Z$ is an infinite point. We also determine the locus of points $P$ for which the generalized orthocenter $H$ coincides with a vertex of $ABC$; this locus turns out to be the union of three conics minus six points. All our proofs are synthetic, and combine affine and projective arguments.