Four points are joined in order: A(0, 0), B(4, 2), C(0, 6) and D(-4, 2). What kind of quadrilateral is ABCD?

$AB=\sqrt{4^2+2^2}=\sqrt{20}=2\sqrt{5}$ and $AD=\sqrt{(-4{)}^2+2^2}=2\sqrt{5}$; $CB=\sqrt{(4-0{)}^2+(2-6{)}^2}=\sqrt{32}=4\sqrt{2}$ and $CD=\sqrt{16+16}=4\sqrt{2}$. The equal sides meet at A and again at $C$, so the equal pairs are adjacent, which is exactly a kite. In a parallelogram the equal sides would be opposite $(AB=CD)$, and here $2\sqrt{5}$ is not $4\sqrt{2}$, so it is neither a parallelogram nor a rhombus.