Concurrence of Three Area-Bisecting Cevians of a Triangle in Three Geometries

Abstract

An area-bisecting cevian of a triangle is a cevian that divides the triangle into two regions with the same area. It is known that three area-bisecting cevians of a triangle are concurrent in each of three geometries: Euclidean, spherical, and hyperbolic. We give a simple proof of this fact.

Acknowledgment

This note relies on a brilliant idea found in the website Cut-The-Knot managed by Alexander Bogomolny. The authors regret that he passed away two years ago. The authors dedicate this short note to him.