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Non-zero complex numbers z1 and z2 satisfy z1² - 2*z1*z2 + 2*z2² = 0. Considering the triangle with vertices at the origin and the points representing z1 and z2, what is its geometrical nature?

1. an isosceles right angled triangle
2. a right angled triangle which is not isosceles
3. an equilateral triangle
4. an isosceles triangle which is not right angled

Correct answer: an isosceles right angled triangle

Solution

z1/z2 = 1 +/- i = sqrt(2)*exp(+/- i*pi/4), meaning |z1| = sqrt(2)|z2| and the angle at the origin is 45 deg, producing an isosceles right-angled triangle.

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