Putnam 2018 A6

Suppose that $A,B,C,$ and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient $$\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}$$ is a rational number.