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Sides and medians are both Jacobi coordinate magnitudes, moreover thenequably entering the spherical coordinates on Kendall's shape sphere and theHopf coordinates. This motivates treating medians on the same footing as sidesin triangle geometry and the resulting Shape Theory. In this paper, weconsequently reformulate inequalities for the medians in terms of shapequantities, and proceed to find inequalities on the mass-weighted Jacobicoordinates. This work moreover identifies the $4/3$ -- powers of which occurfrequently in the theory of medians -- as the ratio of Jacobi masses.  One of the Hopf coordinates is tetra-area. Another is anisoscelesness, whichparametrizes whether triangles are left-or-right leaning as bounded byisoscelesness itself. The third is ellipticity, which parametrizestallness-or-flatness of triangles as bounded by regular triangles. Whereastetra-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 ellipticitymoreover trivializes, due to ellipticity being the difference of base-side andmedian second moments, whose averages are equal to each other. Thus weintroduce a distinct `linear ellipticity' quantifier of tallness-or-flatness oftriangles whose democratization is nontrivial, and find inequalities boundingthis. Some of this paper's inequalities are shape-independent bounds, whereasothers' bounds depend on the isoperimetric ratio and arithmetic-to-geometricside mean ratio shape variables.

Shape (In)dependent Inequalities for Triangleland's Jacobi and Democratic-Linear Ellipticity Quantitities