  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
$J$.
