# [Math] Angle between two segments described using complex numbers

complex numbers

Assume we have two segments, $AC$ and $BC$. We can represent points $A$, $B$ and $C$ on the complex plane with three complex numbers, respectively $a,b$ and $c\in\mathbb{C}$. My question is: is there a nice formula for the angle between these two segments?

You can do a translation of the points:

$$A' = A - C \\ B' = B - C \\ C' = C - C = 0$$

Now $C'$ is the origin, you can get the arguments of $A'$ and $B'$ and subtract them:

$$\alpha = \arg(B') - \arg(A')$$

So $\alpha$ is the angle between $A'C'$ and $B'C'$. But since translations preserve angles, $\alpha$ is also the angle between $AC$ and $BC$. To sum up,

$$\alpha = \arg(B - C) - \arg(A - C)$$