Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The dual of Philo's shortest line segment problem

Yagub N. Aliyev

TL;DR

The paper analyzes the dual of Philo's shortest line segment problem by fixing a line and two points and seeking a position for a third point that minimizes the sum of two distances to the line. Using multivariable calculus combined with geometric arguments, it shows the minimizer occurs at a stationary point $f_x=f_y=0$ corresponding to an angle-bisector configuration, with the minimum equal to $|CD|+|CE|$ under broad conditions; a explicit $p=\infty$ solution is provided and cases with no minimizer are discussed. It also extends the framework to general $l_p$ norms, discusses when medians or symmedians fail to yield solutions for certain $p$, and outlines open problems and potential higher-dimensional generalizations. The work highlights deep connections between angle-bisector geometry and extremal distance problems, offering both concrete results and avenues for future exploration.

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.

The dual of Philo's shortest line segment problem

TL;DR

The paper analyzes the dual of Philo's shortest line segment problem by fixing a line and two points and seeking a position for a third point that minimizes the sum of two distances to the line. Using multivariable calculus combined with geometric arguments, it shows the minimizer occurs at a stationary point corresponding to an angle-bisector configuration, with the minimum equal to under broad conditions; a explicit solution is provided and cases with no minimizer are discussed. It also extends the framework to general norms, discusses when medians or symmedians fail to yield solutions for certain , and outlines open problems and potential higher-dimensional generalizations. The work highlights deep connections between angle-bisector geometry and extremal distance problems, offering both concrete results and avenues for future exploration.

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 () norm is proposed. The particular case is also studied. It is shown that in the cases and the median and the symedian, respectively, of a triangle do not always give a solution for the corresponding optimization problems. The general case and related problems are proposed as open problems.
Paper Structure (4 sections, 3 theorems, 61 equations, 11 figures, 1 table)

This paper contains 4 sections, 3 theorems, 61 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results.
  3. Conclusion
  4. Declarations

Key Result

Theorem 2.1

Let $CDE$ be a triangle with acute angles at the vertices $D$ and $E$. Let $CK$ be the angle bisector of the triangle $CDE$. Drop the perpendiculars $KL$ and $KM$ to the sides $CD$ and $CE$, respectively. Take the points $A$ and $B$ on the sides $CD$ and $CE$, respectively, such that $|AC|=|LD|$, $|

Figures (11)

  • Figure 1: Philo's line and the dual problem
  • Figure 2: Inequality $|C_1 D_1 |+|C_1 E_1 |>|CD|+|CE|$
  • Figure 3: $|K_1L_1|=|K_2M_1|.$
  • Figure 4: One possible order of points on line $D_1E_1$.
  • Figure 5: Another order of points on line $D_1E_1$.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Theorem 2.3