Strange Inequality

The problem asks us to minimize $\frac{f(a,b) f(c,d) + f(c,d) g(a,b)}{\sqrt{a^2 + b^2 + c^2 + d^2 - 2ab - 2cd}}.$

Let $\epsilon > 0$ , and set $a = 6 - \epsilon$ , $b = 7 - \epsilon$ , $c = 6 + \epsilon$ , and $d = 7 + \epsilon$ . Then all the conditions in the problem are satisfied. Also, as $\epsilon \to 0^+$ , the numerator goes to 0 and the denominator is always $\sqrt{2}$ , so the minimum (or more precisely, infimum) of the expression is 0.

Outline : completing squares shows that $f(x,y)$ is the distance between points $(x,y)$ and $(6,7)$ , and $g(x,y)$ is the distance between points $(x,y)$ and $(5,8)$ . Plot points $(a,b),(c,d),(6,7),(5,8)$ and apply Ptolemy's inequality.