that a quadrilateral whose two pairs of opposite sides are equal in length is a parallelogram. Please do not use any axioms of synthetic geometry but can use vector geometry.
Suppose four distinct complex numbers $z_1$, $z_2$, $z_3$, $z_3$, such that $|z_2-z_1|= |z_4-z_3|$ and $|z_3-z_2|= |z_1-z_4|$, prove that $z_1+z_3=z_2+z_4$.