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.


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