The dual of Philo's shortest line segment problem

TL;DR

This paper explores the dual of Philo's shortest line segment problem, providing solutions using calculus and geometry, and investigates generalizations involving different norms, revealing interesting geometric connections and open problems.

Contribution

It introduces the dual problem to Philo's shortest line segment problem, offers solutions using calculus and geometry, and explores generalizations with $l_p$ norms, including special cases and open questions.

Findings

Optimal line segments passing through two points with a common endpoint are characterized.

Connections between the problem and triangle angle bisectors are established.

Generalizations with $l_p$ norms reveal cases where classical triangle centers do not solve the optimization.

Abstract

We study the dual of Philo's shortest line segment problem and find the optimal line segments passing through two given points, with a common endpoint, and with the other endpoints on a given line. This problem is dual, in a point-and-line-exchanging sense, to a famous problem of antiquity used to solve the problem of duplicating the cube. The provided solution uses multivariable calculus and elementary 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. The particular case $p=\infty$ is also studied. It is shown that in the cases $p=0$ and $p=2$ the median and the symedian, respectively, of a triangle do not always give a solution for the corresponding optimization problems. The general case $p\ne 1$ and related problems are proposed as open problems.