For an arbitrary triangle $ABC$.
If given the length of side $AB=15$ and $AC=12$ and the length of the angle bisector that connects angle $A$ to side $BC = 10$. What is the length of $BC$.
I considered using the angle bisector theorem or the length of the angle bisector formula but both times that I've reversed my findings to double-check my work my results have been off.
Best Answer
Following your ideas, using angle bisector theorem:
$BC$ is split in the ratio $4k$ to $5k$.
From the length of the angle bisector formula:
$$180-20k^2=10^2 \implies k=2$$ $$BC=18$$