Using coordinates, let $A=(0,a)$, $B=(b,0)$, $C=(0,c)$ and $D=(d,0)$. Let $M,N$ be the midpoints of $AB,AD$. Then $M=(\frac{b}{2},\frac{a}{2})$ and $N=(\frac{d}{2},\frac{a}{2})$. Then the equation of the line through $M$ perpendicular to $CD$ is $y=\frac{d}{c}x-\frac{bd}{2c}+\frac{a}{2}$, and the equation of the line through $N$ perpendicular to $BC$ is $y=\frac{b}{c}x-\frac{bd}{2c}+\frac{a}{2}$. Thus, the $x$-coordinate of their intersection is given by the solution to $\frac{d}{c}x-\frac{bd}{2c}+\frac{a}{2}=\frac{b}{c}x-\frac{bd}{2c}+\frac{a}{2}\implies x=0$, as desired.