Given four numbers $a, b, c, d$, among all quadrilaterals $ABCD$ with $AB=a, BC=b, CD=c, DA=d$, $[ABCD]$ is maximal if and only if $ABCD$ is cyclic.

Among all such quadrilaterals, only the angles can be different. According the previous theorem, $[ABCD]$ is maximal if and only if $\cos(\frac{A+C}2)=0$, if and only if $\frac{A+C}2=90^\circ, $ if and only if $A+C=180^\circ$,if and only if $ABCD$ is cyclic.