Question 31
Let $ABCD$ be a convex quadrilateral. Define parabolas $\mathcal{P}_A$, $\mathcal{P}_B$, $\mathcal{P}_C$, and $\mathcal{P}_D$ to have directrices $BD$, $CA$, $DB$, and $AC$, and foci $A$, $B$, $C$, and $D$, respectively. Prove that no two of these parabolas intersect more than once.
(A parabola with directrix $\ell$ and focus $P$ consists of all points $X$ for which $PX$ equals the distance from $X$ to $\ell$.)
(A parabola with directrix $\ell$ and focus $P$ consists of all points $X$ for which $PX$ equals the distance from $X$ to $\ell$.)