Question

If z1,z2,z3,z4 be the vertices of rhombus ABCD in argand plane and angle CBA = 60 degree then prove that 2z2=z1(1+i root3)+z3(1-i root3) and 2z4=z1(1-i root 3) + z3(1+i root 3).

Answer 1

Thank you for asking this question. Here is your answer:

ΔABC is an equilateral triangle.

|z₃ - z₂| = |t₂-t₁| = |t₃-t₁|

Now we will use the rotation of complex numbers:

t₂ - t₁  = ( t₃ - t₁)/ |t₃ - t₁| e ^-iπ/3

t₂ = t₁ (1-e ^ -iπ/3)  + t₃ e ^ -iπ/3

t₂ = t₁ (1+i√3/2) + t₃ (1 - i√3/2)

2t₂ = t₁ (1+i√3) + t₃(1+i√3)

Please see the attached image to understand this question better.

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