### Instruction: Let $a$ and $b$ be nonzero complex numbers such that
\[|a| = |b| = |a + b|.\]Find the sum of all possible values of $\frac{a}{b}.$.### Response: I recognize that this problem involves complex numbers and their absolute values, which are related to their magnitudes and arguments.
I also notice that the given condition implies that $a$, $b$, and $a + b$ are all on the same circle centered at the origin with radius $|a| = |b|$.
I wonder if I can use some geometric intuition to find the possible values of $\frac{a}{b}$.
I recall that dividing two complex numbers is equivalent to subtracting their arguments and dividing their magnitudes.
Since the magnitudes of $a$ and $b$ are equal, the magnitude of $\frac{a}{b}$ will always be $1$.
So I only need to focus on the arguments of $a$, $b$, and $\frac{a}{b}$.
I draw a diagram of the complex plane and label the points $a$, $b$, and $a + b$ on the circle.
I see that the angle between $a$ and $b$ must be $120^{\circ}$, because that is the only way to make $a + b$ also lie on the circle.
I also see that there are two possible orientations for $a$ and $b$: either $a$ is counterclockwise from $b$, or $b$ is counterclockwise from $a$.
In the first case, the argument of $a$ is $60^{\circ}$ more than the argument of $b$, so the argument of $\frac{a}{b}$ is $60^{\circ}$.
In the second case, the argument of $a$ is $60^{\circ}$ less than the argument of $b$, so the argument of $\frac{a}{b}$ is $-60^{\circ}$.
Therefore, the possible values of $\frac{a}{b}$ are $e^{i\pi/3}$ and $e^{-i\pi/3}$.
To find their sum, I use the formula $e^{i\theta} = \cos\theta + i\sin\theta$.
I get $\cos(\pi/3) + i\sin(\pi/3) + \cos(-\pi/3) + i\sin(-\pi/3)$.
I simplify this using the fact that cosine is even and sine is odd.
I get $2\cos(\pi/3)$.
I recall that $\cos(\pi/3) = \frac{1}{2}$.
I get $1$.