Flatline Quadrilateral

Relevant wiki: Power Mean Inequality (QAGH)

Look at the diagram above. By moving the points connected to the sides of length $a,b,c$ onto the line of side length $d$ clearly reduces the above value as it reduces $a^2+b^2+c^2$ while leaving $d^2$ unaffected.

This now gives us a new condition $a+b+c=d$ but by QM-AM we have:

$\sqrt{\dfrac{a^2+b^2+c^2}{3}} \geq \dfrac{a+b+c}{3} \implies a^2+b^2+c^2 \geq \dfrac{(a+b+c)^2}{3}=\dfrac{d^2}{3}$

With equality iff $a=b=c=\dfrac{d}{3}$ as $a+b+c=d$ .

If we don't allow degenerate quadrilaterals (i.e. a line with 4 points) then the inequality is strict as it cannot occur:

$\dfrac{a^2+b^2+c^2}{d^2} > \dfrac{\frac{d^2}{3}}{d^2}=\boxed{\boxed{\dfrac{1}{3}}}$