### Instruction: Complex numbers $a,$ $b,$ $c$ form an equilateral triangle with side length 18 in the complex plane.  If $|a + b + c| = 36,$ find $|ab + ac + bc|.$.### Response: I know that an equilateral triangle in the complex plane can be formed by rotating one vertex by $60^\circ$ or $120^\circ$ about another vertex.
So, if I fix $a$ as the origin, I can write $b = 18e^{i\theta}$ and $c = 18e^{i(\theta \pm \pi/3)},$ where $\theta$ is some angle.
Then, the condition $|a + b + c| = 36$ becomes $|18e^{i\theta} + 18e^{i(\theta \pm \pi/3)}| = 36.$
I can simplify this by dividing both sides by 18 and using the fact that $|e^{ix}| = 1$ for any $x.$
This gives me $|e^{i\theta} + e^{i(\theta \pm \pi/3)}| = 2.$
Now, I want to find $|ab + ac + bc|.$
I can use the fact that $a = 0$ and substitute the expressions for $b$ and $c$ in terms of $e^{i\theta}.$
This gives me $|18e^{i\theta} \cdot 18e^{i(\theta \pm \pi/3)} + 18e^{i\theta} \cdot 0 + 0 \cdot 18e^{i(\theta \pm \pi/3)}|.$
I can simplify this by multiplying the coefficients and using the fact that $e^{ix} \cdot e^{iy} = e^{i(x + y)}$ for any $x$ and $y.$
This gives me $|18^2e^{i(2\theta \pm \pi/3)}|.$
I can use the fact that $|e^{ix}| = 1$ again to simplify this further.
This gives me $|18^2|.$
I can take the absolute value of a positive number by removing the bars.
This gives me $18^2.$