We study the dual of Philo's shortest line segment problem which asks to find the optimal line segments passing through two given points, with a common endpoint, and with the other endpoints on a given line. The provided solution uses multivariable calculus and geometry methods. Interesting connections with the angle bisector of the triangle are explored. A generalization of the problem using $L_p$ ($p\ge 1$) norm is proposed. Particular case $p=\infty$ is studied. Interesting case $p=2$ is proposed as an open problem and related property of a symedian of a triangle is conjectured.

翻译：我们研究菲洛最短线段问题的对偶问题，该问题要求寻找通过两个给定点、具有一个公共端点且另一端点位于给定直线上的最优线段。所提供的解法运用了多元微积分与几何方法。文中探讨了该问题与三角形角平分线之间的有趣联系。提出了使用 $L_p$ ($p\ge 1$) 范数对问题进行推广，并研究了 $p=\infty$ 的特殊情形。将 $p=2$ 这一有趣情形作为开放问题提出，并猜想其与三角形类似中线相关的一个性质。