The Perimeter Winternitz Theorem in a Triangle

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.

Figures (4)

• Figure 1: Proof of the Winternitz theorem for the equilateral triangle and its perimeter analogue.
• Figure 2: The extremal triangles in Theorem \ref{['T1']} together with all their chords through the centroid $G$ that respectively separate approximately 3/10 and exactly 4/9 of the perimeter.
• Figure 3: (Left) A triangle with the three perimeter W-lines relative to vertices passing through an interior point $P$. (Right) The same triangle with the parallels from $P$ to the sides.
• Figure 4: The region in the proof of Theorem \ref{['T1']}.

Theorems & Definitions (8)

• Theorem 1: The Perimeter Winternitz Theorem in Every Triangle
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4: The Perimeter Winternitz Theorem in a Specific Triangle
• proof
• proof : Proof of Theorem \ref{['T1']}