cevian轨迹与特殊椭圆曲线的尺规作图法
A cevian locus and the geometric construction of a special elliptic curve
摘要：
在以前的论文中，我们定义了一个三角形$ABC$关于一个点$P$的外接圆圆锥曲线作为圆锥曲线$\tilde C=T_{P'}^{-1}(N_{P'})$，其中$N_{P'}$是$9$ -点圆锥曲线的四边形$ABCP'$关于在无穷远处的线，$P'$是等距共轭的$P$关于$ABC$，和$T_{P'}$是仿射映射采取$ABC$到cevian三角形的$P'$。在本文中，我们确定的点的轨迹，使一定的仿射映射$\textsf{M}$采取的circumconic $\tilde C$到inconic $\mathcal{I}$，定义为唯一的圆锥切线的侧边的$ABC$在这些侧面上的点$P$的痕迹，是一个半圈。这个轨迹是一条减六点的椭圆曲线，可以用定义在三个开圆弧上的点的仿射映射族来几何构造。
英文摘要：
In a previous paper we defined the circumconic of a triangle $ABC$ with respect to a point $P$ as the conic $\tilde C=T_{P'}^{-1}(N_{P'})$, where $N_{P'}$ is the $9$-point conic for the quadrangle $ABCP'$ with respect to the line at infinity, $P'$ is the isotomic conjugate of $P$ with respect to $ABC$, and $T_{P'}$ is the affine map taking $ABC$ to the cevian triangle for $P'$. In this paper we determine the locus of points for which a certain affine map $\textsf{M}$ taking the circumconic $\tilde C$ to the inconic $\mathcal{I}$, defined to be the unique conic tangent to the sides of $ABC$ at the traces of the point $P$ on those sides, is a half-turn. This locus turns out to be an elliptic curve minus six points, which can be constructed geometrically using a family of affine maps defined for points on three open arcs of a circle.
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