Question

Show that the triangle whose vertices are the points represented by the complex numbers Z₁, Z₂ and Z₃ on the Argand diagram is equilateral if and only if 1/(Z₂ - Z₃ ) +  1/(Z₃ - Z₁) +  1/(Z₁ - Z₂) = 0 (OR) equivalently Z²₁ +Z²₂ +Z²₃ = Z₁Z₂ + Z₂Z₃ + Z₃Z₁.

Answer 1

See Fig. ABC is the equilateral triangle formed of A (Z₁),B(Z₂) and  C(Z₃).So,

Z₃ - Z₁ = (Z₂ - Z₁)(cosπ/3 + i sin π/3)

(AC is obtained from AB by a rotation anticlockwise through an angle π/3)

Lengthwise, AC = AB

This may be equivalently written in the form

The condition for Z₁, Z₂,Z₃ and to form an equilateral triangle is given in one of the two equivalent forms given by Eqs. (3) and (4). Let us prove the converse also

Assume

Multiplying above two equations we get,

Therefore, the triangle is an equilateral triangle. Let us also prove the converse from the other condition,

ω, ω² being the two imaginary cube roots of unity, Eq. (5) may be written as

Z₁ - Z₂ = ω²(Z₂ - Z₃)

Therefore

|Z₁ - Z₂| = |ω²||Z₂ - Z₃| ⇒|Z₁ - Z₂| = |Z₂ - Z₃|

Similarly, it can be proved by combining the terms differentl

|Z₁ - Z₂| = |Z₂ - Z₃|

Hence,

|Z₁ - Z₂|=|Z₂ - Z₃| = |Z₃ - Z₁|

Therefore, the triangle is an equilateral triangle.