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 similar property of a symedian of a triangle is conjectured.

翻译：本研究探讨菲洛最短线段问题的对偶问题：给定两点与一条直线，求以该两点为公共端点、另一端点位于给定直线上的最优线段组。解决方案运用了多元微积分与几何方法，深入探讨了该问题与三角形角平分线的有趣关联，并提出了三角形类似中线可能具有相似性质的猜想。