Synthetic proof of a bisector length 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 $k^2=ab-xy$.

It depends on considering Stewart's theorem a result in synthetic geometry or trigonometry.
It directly follows from the law of cosines, so the problem boils down to considering the law of cosines as a result in synthetic geometry or trigonometry. In fact, it can be proved only through the definition of the cosine function (not its algebraic or analytic properties) and a simple dissection argument (a variation on the usual proof of the Pythagorean theorem), hence I would say trigonometry is not really involved.
For a pure synthetic proof you may look at samarth srivastava's answer to this similar question.
