Any interpretation for the fact that centroid = optimal point for maximising volume of such cuboid? (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.
 A: 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.
