Complex Numbers
Geometry of Complex Numbers
Premium Question
Grade 11
Question:
$z_1, z_2, z_3$ are the vertices of an equilateral triangle. Which of the following must hold?
(1) $z_1^2 + z_2^2 + z_3^2 = 0$
(2) $z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$
(3) $|z_1| = |z_2| = |z_3|$
(4) $z_1 + z_2 + z_3 = 0$
Step-by-Step Solution
Key Concept: Use the rotation characterisation: $z_3 - z_1 = (z_2 - z_1)e^{\pm i\pi/3}$. Expand and simplify the resulting identity between $z_1, z_2, z_3$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)