三线形极线 垂直于PP'的点的轨迹是一条五次曲线

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436.Let P be a point in the plane of a given triangle ABC, P' the

isogonally conjugated point of P with respect to ABC, L1 the line PP', L2 the trilinear polar (or harmonical) of P with respect to ABC. Show that the locus of the points P such that L1 en L2 are perpendicular is a quintic curve, with nodes at the vertices of ABC, passing through the isotropic points, through the incentre and the three excentres, through the centroid, through the ortho­ centre and through the vertices of the pedal triangle of the orthocentre.

(0. BOTTEMA & M.C. VAN HOORN)

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077_2001-02_04.pdf第24页Problem 436
左边有一个几何图，待补充到1#

Let $p$ be a point in the plane of a given triangle $ABC$, $P'$ the isogonally conjugated point of $p$ with respect to $ABC$, $L_1$ the line $PP'$, $L_2$ the trilinear polar (or harmonical) of $p$ with respect to $ABC$. Show that the locus of the points $p$ such that $L_1$ en $L_2$ are perpendicular is a quintic curve, with nodes at the vertices of $ABC$, passing through the isotropic points, through the incentre and the three excentres, through the centroid, through the orthocentre and through the vertices of the pedal triangle of the orthocentre.

设 $p$ 为给定三角形 $ABC$ 平面上的一点，$P'$ 为 $p$ 关于 $ABC$ 的等角共轭点，$L_1$ 为直线 $PP'$，$L_2$ 为 $p$ 关于 $ABC$ 的三线性极线。 证明点 $p$ 的轨迹使得 $L_1$ 和 $L_2$ 垂直的是一个五次曲线，该曲线在 $ABC$ 的顶点处有节点，经过各向同性点，经过内心和三个旁心，经过重心，经过垂心并经过垂心的垂足三角形的顶点。

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设点 $P$ 的三线坐标为 $p : q : r$。则 $L_1$ 的三线方程为\[\begin{vmatrix}x&y&z\\p&q&r\\p^{-1}&q^{-1}&r^{-1}\end{vmatrix}=0\]即\[x \left(\frac{q}{r}-\frac{r}{q}\right)+y \left(\frac{r}{p}-\frac{p}{r}\right)+z \left(\frac{p}{q}-\frac{q}{p}\right)=0\]
$L_2$ 的三线方程为\[\frac xp+\frac yq+\frac zr=0.\]根据《近代的三角形几何学》第3节, 两直线成直角的条件为$l l^{\prime}+m m^{\prime}+n n^{\prime}=\sum\left(m n^{\prime}+m^{\prime} n\right) \cos A$
因此$L_1\perp L_2$等价于\[\frac{\frac{q}{r}-\frac{r}{q}}{p}+\frac{\frac{r}{p}-\frac{p}{r}}{q}+\frac{\frac{p}{q}-\frac{q}{p}}{r}=\sum\left(\frac{\frac{p}{q}-\frac{q}{p}}{q}+\frac{\frac{r}{p}-\frac{p}{r}}{r}\right) \cos A\]
即\[0=\sum p\left(\frac1{q^2}-\frac1{r^2} \right)\cos A\]
即$0=\sum p^3\left(q^2-r^2\right)\cos A$, 确实是一条5次曲线