(Visualising a sample case via the image at the bottom)
Consider a plane $\frac xa+\frac yb+\frac zc=1$ so that it intercepts with the axis at $(a,0,0)$, $(0,b,0)$ and $(0,0,c)$, $a,b,c >0$. Now consider maximising the volume of such a cuboid, which it is in the first octant, with 3 faces in $x=0$, $y=0$, $z=0$, and a vertex in the plane mentioned above. Algebraically, we will always get the optimal vertex at the centroid of the triangle {$(a,0,0)$, $(0,b,0)$ and $(0,0,c)$} in the plane mentioned above.
Any possible interpretation behind this relation? I get the logic in algebra but I can’t figure out the intuition geometrically. The question sounds vague because I don’t have much intuition behind this relation so I fail to use specific words to phrase my question. It seems to be an interesting coincidence and I want to know is there any non-algebraic interpretation behind.
blue line indicating the plane. Red line indicating the cuboid. Green indicating the vertex.
Consider the simpler problem of finding the maximum volume of an inscribed cuboid (inscribed in the same manner as the original problem) in the first octant of the graph $$x+y+z=1$$ By symmetry, one would intuitively hypothesize that this would occur when the green vertex of the cuboid was at the center of the triangular face and the cuboid was a cube. This can of course be proven with AM-GM ($xyz\leq \left(\frac{x+y+z}{3}\right)^3=\frac{1}{27}$).
Since ratios of volumes are preserved under affine transformations, it follows that if we dilate the plane so that it matches with the equation $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$, then the maximum inscribed cube should be equivalent to applying that same affine transformation to the cube with sidelength $\frac{1}{3}$.
However, we also know that the location of the centroid will remain preserved after an affine transformation. Hence, it follows that this maximum-volume cube will also contain the centroid as a vertex regardless of the equation of the plane.