The Perimeter Winternitz Theorem in a Triangle

TL;DR

This paper extends the Winternitz theorem to perimeter ratios in triangles, identifying the exact range of possible ratios for lines through the centroid and characterizing the extremal triangles.

Contribution

It establishes the perimeter-based range of ratios for lines through the centroid in triangles, complementing the classical area-based Winternitz theorem.

Findings

Range of m is (3/10, 4/9] for all triangles.

Approaching 3/10 with scaled 5-4-1 triangles.

Achieving 4/9 with equilateral triangles.

Abstract

A variable line through the centroid G of a triangle divides the triangle into two parts each of whose lengths as a fraction of the perimeter fills a closed interval [m,1-m], with m between 0 and 1/2. We show that the range of m taken over all triangles is the interval (3/10,4/9], with 3/10 approached by scales of the triangles approaching the 5-4-1 triangle and their mid-size medians, and 4/9 attained by the equilateral triangles and the lines through G parallel to the sides. This result is the perimeter version of the classical Winternitz theorem for a triangle, asserting that, in the case of area-ratio instead of perimeter-ratio, m=4/9, and this is attained by all triangles and their lines through G and parallel to the sides.