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Real numbers a and b lie in (0, 1) and the points z1 = a + i, z2 = 1 + b i, z3 = 0 are the vertices of an equilateral triangle. Then a and b equal:

1. a = b = 1/2
2. a = b = 2 - sqrt(3)
3. a = b = -2 + sqrt(3)
4. a = b = sqrt(2) - 1

Correct answer: a = b = 2 - sqrt(3)

Solution

Symmetry forces a = b, and equating |z3 z1| with |z1 z2| gives a quadratic whose root in (0,1) is 2 - sqrt(3).

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