1
$\begingroup$

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 $k^2=ab-xy$.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.