Question

Let z₁, z₂, z₃ be the affixes of the vertices A, B and C respectively of a ΔABC. Prove that the triangle is equilateral if \(z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1\).

Answer 1

First, let z₁, z₂, z₃ be the affixes of the vertices A,B,C of an equilateral triangle ABC. Then, we have to prove that \(z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1\)

Since ΔABCis an equilateral triangle.

Therefore,

AB = BC = AC and ∠A = ∠B = ∠C = π/3.

Clearly, \(\vec{AC}\) can be obtained by rotating \(\vec {AB}\) in anticlockwise sense through 600.

\(\therefore\) z₃ – z₁ = (z₂ – z₁) e^iπ/3 ......(1)

Also \(\vec {BC}\) can be obtained by rotating \(\vec {BA}\) by π/3 anticlockwise