A [hexagon](https://artofproblemsolving.com/wiki/index.php/Hexagon) is inscribed in a [circle](https://artofproblemsolving.com/wiki/index.php/Circle). Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$.
Let $x=AC=BF$, $y=AD=BE$, and $z=AE=BD$.
[Ptolemy's Theorem](https://artofproblemsolving.com/wiki/index.php/Ptolemy%27s_Theorem) on $ABCD$ gives $81y+31\cdot 81=xz$, and Ptolemy on $ACDF$ gives $x\cdot z+81^2=y^2$.
Subtracting these equations give $y^2-81y-112\cdot 81=0$, and from this $y=144$. Ptolemy on $ADEF$ gives $81y+81^2=z^2$, and from this $z=135$. Finally, plugging back into the first equation gives $x=105$, so $x+y+z=105+144+135=\boxed{384}$.