  Sides and medians are both Jacobi coordinate magnitudes, moreover then
equably entering the spherical coordinates on Kendall's shape sphere and the
Hopf coordinates. This motivates treating medians on the same footing as sides
in triangle geometry and the resulting Shape Theory. In this paper, we
consequently reformulate inequalities for the medians in terms of shape
quantities, and proceed to find inequalities on the mass-weighted Jacobi
coordinates. This work moreover identifies the $4/3$ -- powers of which occur
frequently in the theory of medians -- as the ratio of Jacobi masses.
  One of the Hopf coordinates is tetra-area. Another is anisoscelesness, which
parametrizes whether triangles are left-or-right leaning as bounded by
isoscelesness itself. The third is ellipticity, which parametrizes
tallness-or-flatness of triangles as bounded by regular triangles. Whereas
tetra-area is clearly cluster choice invariant, Jacobi coordinates,
anisoscelesness and ellipticity are cluster choice dependent but can be
`democratized' by averaging over all clusters. Democratized ellipticity
moreover trivializes, due to ellipticity being the difference of base-side and
median second moments, whose averages are equal to each other. Thus we
introduce a distinct `linear ellipticity' quantifier of tallness-or-flatness of
triangles whose democratization is nontrivial, and find inequalities bounding
this. Some of this paper's inequalities are shape-independent bounds, whereas
others' bounds depend on the isoperimetric ratio and arithmetic-to-geometric
side mean ratio shape variables.
