Average Chord Lengths in a Triangle
Abstract
Let P be a point inside a triangle T. We consider the average length of the chords of T through P, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from P to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if I is the incenter, then MT(I)=2r/π log(cotA/4cotB/4cotC/4). Our main result is the sharp inequality MT(P)≤ p/π√3log(2+√3), valid simultaneously for every triangle of perimeter p and every interior point P. Thus, among all such pairs (T,P), the largest possible average chord length occurs only when T is equilateral and P is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.
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BibTeX
@misc{rabinowitz2026average,
title = {{Average Chord Lengths in a Triangle}},
author = {Stanley Rabinowitz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11370},
eprint = {2609.11370},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11370},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Average Chord Lengths in a Triangle
AU - Stanley Rabinowitz
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11370
UR - https://arxiv.org/abs/2609.11370
AB - Let P be a point inside a triangle T. We consider the average length of the chords of T through P, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from P to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if I is the incenter, then M_T(I)=2r/π log(cotA/4cotB/4cotC/4). Our main result is the sharp inequality M_T(P)≤ p/π√3log(2+√3), valid simultaneously for every triangle of perimeter p and every interior point P. Thus, among all such pairs (T,P), the largest possible average chord length occurs only when T is equilateral and P is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.
ER -