西维安几何的综合基础，III：广义正交中心
Synthetic foundations of cevian geometry, III: The generalized orthocenter
摘要：
在本文中，第三个系列，我们定义了广义正交中心$H$对应于一个点$P$，关于三角形$ABC$，作为唯一的点，其中的线$HA, HB, HC$分别平行于$QD, QE, QF$，其中$DEF$是$P$的cevian三角形和$Q=K \circ ι(P)$是$P$的$isotomcomplement$，都关于$ABC$。本文证明了一个推广的费尔巴哈定理，并将第二部分中定义的cevian圆锥$\mathcal{C}_P$的中心$Z$刻画为仿射映射$Φ_P = T_P \circ K^{-1} \circ T_{P'} \circ K^{-1}$的中心，其中$T_P$是唯一的仿射映射，对于$T_P(ABC)=DEF$，$T_{P'}$的定义类似于$P$的等距共轭$P'=ι(P)$，$K$是补映射.仿射映射$Φ_P$固定$Z$，并将四边形$ABCH$的九点二次曲线$\mathcal{N}_H$（相对于无穷远处的直线）取为非二次曲线$\mathcal{I}$，非二次曲线定义为在点$D, E, F$处与$ABC$的边相切的唯一二次曲线。因此，点$Z$是九点二次曲线$\mathcal{N}_H$和非二次曲线$\mathcal{I}$接触的点。这个定理推广了通常的费尔巴哈定理，并且在点$P$不在中位数上的所有情况下都成立，无论所涉及的二次曲线是椭圆、抛物线还是双曲线，并且当$Z$是无限点时也成立。我们还确定的轨迹点$P$的广义正交中心$H$与$ABC$的顶点重合，这个轨迹原来是三个二次曲线减去六个点的联盟。我们所有的证明都是合成的，并结合仿射和投影参数。
英文摘要：
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 $DEF$ is the cevian triangle of $P$ and $Q=K \circ ι(P)$ is the $isotomcomplement$ of $P$, both with respect to $ABC$. 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 $Φ_P = T_P \circ K^{-1} \circ T_{P'} \circ K^{-1}$, where $T_P$ is the unique affine map for which $T_P(ABC)=DEF$; $T_{P'}$ is defined similarly for the isotomic conjugate $P'=ι(P)$ of $P$; and $K$ is the complement map. The affine map $Φ_P$ 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.
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