Complex numbers $a,$ $b,$ $c$ form an equilateral triangle with side length 18 in the complex plane.  If $|a + b + c| = 36,$ find $|ab + ac + bc|.$
Note that given complex numbers $a$ and $b$ in the plane, there are two complex numbers $c$ such that $a,$ $b,$ and $c$ form an equilateral triangle.  They are shown as $c_1$ and $c_2$ below.

[asy]
unitsize(1 cm);

pair A, B;
pair[] C;

A = (2,-1);
B = (0,0);
C[1] = rotate(60,B)*(A);
C[2] = rotate(60,A)*(B);

draw(C[1]--A--C[2]--B--cycle);
draw(A--B);

label("$a$", A, SE);
label("$b$", B, NW);
label("$c_1$", C[1], NE);
label("$c_2$", C[2], SW);
[/asy]

Then for either position of $c,$
\[\frac{c - a}{b - a}\]is equal to $e^{\pm \pi i/6}.$  Note that both $z = e^{\pm \pi i/6}$ satisfy $z^2 - z + 1 = 0.$  Thus,
\[\left( \frac{c - a}{b - a} \right)^2 - \frac{c - a}{b - a} + 1 = 0.\]This simplifies to
\[a^2 + b^2 + c^2 = ab + ac + bc.\]Then
\[(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 3(ab + ac + bc).\]Hence,
\[|ab + ac + bc| = \frac{|a + b + c|^2}{3} = \frac{36^2}{3} = \boxed{432}.\]
Note that given complex numbers $a$ and $b$ in the plane, there are two complex numbers $c$ such that $a,$ $b,$ and $c$ form an equilateral triangle.  They are shown as $c_1$ and $c_2$ below.

[asy]
unitsize(1 cm);

pair A, B;
pair[] C;

A = (2,-1);
B = (0,0);
C[1] = rotate(60,B)*(A);
C[2] = rotate(60,A)*(B);

draw(C[1]--A--C[2]--B--cycle);
draw(A--B);

label("$a$", A, SE);
label("$b$", B, NW);
label("$c_1$", C[1], NE);
label("$c_2$", C[2], SW);
[/asy]

Then for either position of $c,$
\[\frac{c - a}{b - a}\]is equal to $e^{\pm \pi i/6}.$  Note that both $z = e^{\pm \pi i/6}$ satisfy $z^2 - z + 1 = 0.$  Thus,
\[\left( \frac{c - a}{b - a} \right)^2 - \frac{c - a}{b - a} + 1 = 0.\]This simplifies to
\[a^2 + b^2 + c^2 = ab + ac + bc.\]Then
\[(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 3(ab + ac + bc).\]Hence,
\[|ab + ac + bc| = \frac{|a + b + c|^2}{3} = \frac{36^2}{3} = \boxed{432}.\]
The final answer is $The final answer is:

\[432\]$. I hope it is correct.