This is a theorem of Cramer. See <a href="http://www.emis.ams.org/journals/MV/9634/ves96301.ps">here</a>

For the quadrilateral case the quickest proof is using <a href="http://en.wikipedia.org/wiki/Brahmagupta%27s_formula#Extension_to_non-cyclic_quadrilaterals"> Brahmagupta's formula</a>

$$Area=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2 \theta}$$ where $a,b,c,d$ are the sides, $s$ is the half perimeter and $\theta$ is half the sum of opposite angles.