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：我们研究了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和几何方法。文中探讨了其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。研究了 $p=\infty$ 的特殊情况。$p=2$ 的有趣情况被提出作为一个开放问题，并猜想其与三角形类似中线的一个相关性质。