Challenge Level

Consider any convex quadrilateral $Q$ made from four rigid rods
with flexible joints at the vertices so that the shape of $Q$ can
be changed while keeping the lengths of the sides constant. If the
diagonals of the quadrilateral cross at an angle $\theta$ in the
range $(0 \leq \theta < \pi/2)$, as we deform $Q$, the angle
$\theta$ and the lengths of the diagonals will change.

Using the results of the two problems on quadrilaterals
Diagonals for Area and
Flexi Quads prove that the area of $Q$ is proportional to
$\tan\theta$.