[Math] Coordinates of third point in a triangle

Question

Of triangle ABC, I am given the coordinates of two points (A and B) and the angles between the side AB and each of the two other sides.

How can I get the coordinates of point C?

Answer 1

The ambiguity mentioned above can be avoided if you consider ordered angles, i.e., encode the orientation of the triangle in your data. We can then solve the problem as follows. Choose coordinates so that $A$ is $(0,1)$, $B$ is $(1,0)$ and $C$ is $(p,q)$. Then $\tan A=\dfrac q p$ and $\tan B=\dfrac q{1-p}$. We can solve to get $$p=\frac{\tan B}{\tan A+\tan B},\quad q=\frac{\tan A \tan B}{\tan A+\tan B}.$$