Two new versions of Heron's Formula

Recollect that Heron’s formula for the area of a triangle given its sides has a counterpart given the medians instead, which carries an extra factor of $\frac{4}{3}$. On the one hand, we formulate the pair of these in Linear Algebra terms, showing that they are related by a sides-to-medians involution $J$, which we find to furthermore commute with the Heron map' $H$ as visible in the expanded version of Heron. Upon further casting the pair of these in terms of mass-weighted Jacobi coordinates, we find moreover that they are placed on an exactly equal footing, the factor of $\frac{4}{3}$ having now cancelled out. This motivates the Heron–Jacobi’ version of Heron’s formulae, for mass-weighted area in terms of mass-weighted sides and mass-weighted medians respectively. On the other hand, we show that diagonalizing the Heron map $H$ provides new derivations of, firstly, the famous Hopf map, and, secondly, of Kendall’s Theorem that the space of triangles is a sphere. This occurs by the `Heron–Hopf’ version of Heron’s formula simplifying down to none other than the on-sphere condition. Thus we establish that – both an important fibre bundle model, and a foundational theorem of Shape Theory: a widely applicable Differential Geometry and Topology topic – arise together as consequences of just Heron’s formula, Stewart’s Theorem, and some elementary Linear Algebra manipulations. This working also accounts for the extra factor of 4 in the Hopf coordinate that is elsewise equal to the mass-weighted area in the 3-body problem context. It finally offers a new interpretation of the shape-theoretic ellipticity and anisoscelesness which realize the other two Hopf quantities: as eigenvectors shared by the Heron map $H$ and the sides-to-medians involution $J$.