Angle between two segments described using complex numbers

Question

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?

Answer 1

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,

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

Answer 2

Assuming that the order of the points is $A-B-C$ going counter-clockwise around the triangle, we first of all put $C$ in the origin (so $c = 0$). Then $a$ is a complex number with absolute $|a|$ and argument $\alpha$, and $b$ is a complex number with absolute $|b|$ and argument $\beta$. If you multiply $a$ with a number $z$ whose absolute is $|b|/|a|$ and argument is $\phi = \beta - \alpha$, you get a number with absolute $|b|$ and argument $\beta$ (hence, this is $b$). This number is given by

$$b = az\,, \quad z = \frac{|b|}{|a|}e^{i\phi}\,.$$

If you now replace $a$ by $a - c$ and $b$ by $b - c$ in this derivation, you shift $C$ back to an arbitrary place and the formula becomes

$$(b - c) = (a - c)\frac{|b - c|}{|a - c|}e^{i\phi}\,,$$

where $\phi$ is the angle between these two segments. This gives you

$$e^{i\phi} = \frac{b - c}{a - c}\frac{|a - c|}{|b - c|}\,.$$