Let $a,b,c,d$ be the affixes of the vertices of the quadrilateral. Now let $m_1=\frac{a+b}2,\ m_2=\frac{b+c}2,\ m_3=\frac{c+d}2,\ m_4=\frac{d+a}2$ be the affixes of the midpoints of the sides. Furthermore, assume $O(0)$, the center of $(ABCD)$, is the origin of the plane. Let $\ell_i$ be the line dropped from $M_i$ to the opposite side of the quadrilateral.
We can see that the quadrilateral formed by the perpendicular bisectors of $AB,CD$ and $\ell_1,\ell_3$ form a parallelogram, having $O$ as one of its vertices. $\ell_1\cap\ell_3$ then has affix $m_1+m_3$. In the same manner we show that $\ell_2\cap\ell_4$ has affix $m_2+m_4$. However, $m_1+m_3=m_2+m_4=\frac{a+b+c+d}2$, which means that $\ell_1,\ell_2,\ell_3,\ell_4$ are concurrent, Q.E.D.