AoPS - Problem

Let $M$, $N$, $P$, $Q$ denote the feet of the altitudes from $E$ to $AB$, $BC$, $CD$, $DA$. $\begin{cases} EM \cdot AB = EA \cdot EB \cdot sin \angle AEB \\ EP \cdot CD = EC \cdot ED \cdot sin \angle DEC \end{cases}$ $\Rightarrow ad = EM \cdot EP = \frac{EA \cdot EB \cdot EC \cdot ED \cdot sin \angle AEB \cdot sin \angle DEC}{AB \cdot CD}$. Similarly, $bc = EN \cdot EQ = \frac{EA \cdot EB \cdot EC \cdot ED \cdot sin \angle BEC \cdot sin \angle DEA}{BC \cdot DA}$. We also have $\frac{sin \angle AEB \cdot sin \angle DEC}{AB \cdot CD} = \frac{sin \angle ACB \cdot sin \angle DAC}{AB \cdot CD} = \frac{sin \angle BAC \cdot sin \angle DAC}{BC \cdot DA} = \frac{sin \angle AEB \cdot sin \angle DEC}{BC \cdot DA}$, so $ad = bc$.