Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Berele, Allan, Catoiu, Stefan
Institution: DePaul University
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7140466602
Cites:
Cited by:
Related to: