Minimal Perimeter Triangle in Nonconvex Quadrangle: Generalized Fagnano Problem

TL;DR

This paper generalizes Fagnano's problem to nonconvex quadrilaterals, providing a geometric solution for minimal perimeter triangles with specific vertex constraints and establishing an upper bound for the classical problem.

Contribution

It introduces a new geometric generalization of Fagnano's problem for nonconvex quadrilaterals and derives an upper bound for the classical case.

Findings

Provided a geometric solution for the generalized Fagnano problem in nonconvex quadrilaterals.

Established that the minimal inscribed triangle's perimeter in an acute triangle is at most twice the length of its sides.

Abstract

In 1775, Fagnano introduced the following geometric optimization problem: inscribe a triangle of minimal perimeter in a given acute-angled triangle. A widely accessible solution is provided by the Hungarian mathematician L. Fejer in 1900. This paper presents a specific generalization of the classical Fagnano problem, which states that given a nonconvex quadrangle (having one reflex angle and others are acute angles), find a triangle of minimal perimeter with exactly one vertex on each of the sides that do not form reflex angle, and the third vertex lies on either of the sides forming the reflex angle. We provide its geometric solution. Additionally, we establish an upper bound for the classical Fagnano problem, demonstrating that the minimal perimeter of the triangle inscribed in a given acute-angled triangle cannot exceed twice the length of any of its sides