Proving Stewart's theorem without trig Stewart's theorem states that in the triangle shown below,
$$ b^2 m + c^2 n = a (d^2 + mn). $$

Is there any good way to prove this without using any trigonometry? Every proof I can find uses the Law of Cosines.

Geometric equivalents of the Law of Cosines are already present in Book II of Euclid, in Propositions $12$ and $13$ (the first is the obtuse angle case, the second is the acute angle case).
Here are links to Proposition $12$, Book II, and to Proposition $13$.
There is absolutely no trigonometry in Euclid's proofs.
These geometric equivalents of the Law of Cosines can be used in a mechanical way as "drop in" replacements for the Law of Cosines in "standard" proofs of Stewart's Theorem.  What in trigonometric approaches we think of as $2ab\cos\theta$ is, in Euclid, the area of a rectangle that is added to or subtracted from the combined area of two squares.  
