Finding the minimum value of squares of sides of a quadrilateral

Question

What is the minimum value of $\frac{a^2+b^2+c^2}{d^2}$ where $a,b,c,d$ are the sides of quadrilateral

I assumed the diagonals to be $p$ and $q$. I got that for minimum angle $A$ and $C$ must be $90^{\circ}$ and then got stuck.

Answer 1

Hint: Looking at the expression, to make it small you want $d$ large and $a,b,c$ small. To make a quadrilateral (assuming zero area is not allowed) you need $a+b+c\gt d$ You can't actually attain the minimum, so your answer will really be in infimum.