First slide

Geometry of complex numbers

Question

$A (z_{1}), B (z_{2}), C (z_{3})$ are the vertices of an equilateral triangle ABC, whose circumcentre is $D (z_{0}),$ then $z_{1}^{2} + z_{2}^{2} + z_{3}^{2}$ is always equal to

Moderate

A

$z_{0}^{2}$
B

$3 z_{0}^{2}$
C

Zero
D

None

Solution

$A (z_{1}), B (z_{2}), C (z_{3})$ form an equilateral triangle so $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}$
Also, $Δ A B C$ is equilateral, so circumcentre $z_{0}$ is also centroid.

            $z_{1} + z_{2} + z_{3} = 3 z_{0}$

$\Rightarrow$      $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + 2 (z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}) = 9 z_{0}^{2}$

$\Rightarrow$     $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 3 z_{0}^{2}$

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