See the first two of the below (in reverse order) for my recent exposition of this.
Heron’s formula, Apollonius’ Theorem and Stewart’s Theorem generalized!
This page will shortly be replaced by its own “brief living review” pdf article introducing the various Flat Geometry works unveiled so far. The usual Heron’s formula is for the area of a triangle in terms of side-length data. The Author’s publicly-declared work began here with the 2017 preprint “Two new versions of Heron’s Formula,…
APPLIED COMBINATORICS (Widely-Applicable Mathematics Series. Volume 0)
My new freely-available book is out here: Applied Combinatorics – Concepts of Shape
Shape (In)dependent Inequalities for Triangleland’s Jacobi and Democratic-Linear Ellipticity Quantitities
Abstract Sides and medians are both Jacobi coordinate magnitudes. These furthermore enter equably into the spherical coordinates on Kendall’s shape sphere and the Hopf coordinates. This motivates treating medians on the same footing as sides in the geometry of triangles and the ensuing Shape Theory. In this article, we consequently reformulate inequalities for the medians…
Concepts of Shape 