On Viviani's Theorem and its Extensions

TL;DR

This paper explores extensions of Viviani's theorem, demonstrating that convex polygons can be partitioned into segments with constant distance sums, and characterizes polygons with the CVS property, including convex and concave cases and polyhedra.

Contribution

It introduces the concept of the CVS property for polygons, extends Viviani's theorem to convex and concave polygons and polyhedra, and provides conditions for the CVS property based on point configurations.

Findings

Convex polygons can be divided into segments with constant sum of distances to sides.

Three non-collinear points with equal distance sums imply the CVS property.

Extensions of Viviani's theorem to polyhedra are established.

Abstract

Viviani's theorem states that the sum of distances from any point inside an equilateral triangle to its sides is constant. We consider extensions of the theorem and show that any convex polygon can be divided into parallel segments such that the sum of the distances of the points to the sides on each segment is constant. A polygon possesses the CVS property if the sum of the distances from any inner point to its sides is constant. An amazing result, concerning the converse of Viviani's theorem is deduced; Three non-collinear points which have equal sum of distances to the sides inside a convex polygon, is sufficient for possessing the CVS property. For concave polygons the situation is quite different, while for polyhedra analogous results are deduced.