In a exam the following was asked and I found a proof based on a law of cosines. Is there a pure synthetic proof without trigonometry?
$ABC$ is a triangle with side lengths $CA=a,CB=b$. $D\in AB$ is a point such that $CD$ is a bisector of $ACB$ and $CD=k$. Suppose that $AD=x$ and $DB=y$. Prove that $k^2=ab-xy$.