The points A,B,C and D represent the numbers Z₁,Z₂,Z₃ and Z₄.ABCD is rhombus;AC=2BD.if Z₂=2+i

,Z₄=1-2i,find Z₁ and Z₃

B(2,1) , D(1,-2)

Mid Point (3/2,-1/2)

Write Equation of CD in symmetric form 

(x-3/2)/-1/3 = (y+1/2)/-1/3 = r 

here r= +(10)^1/2 , -(10)^1/2

A = 3/2 - ((10)^1/2)/3

D = -1/2 + ((10)^1/2)/3

z1=-3/2 +i1/2

z3=9/2-i3/2

or vice versa.

POI of diagonals: (3-i)/2.

Using concept of rotation:

2(0.5+2i)/2z₁ + i-3      =   i/2

Solving for Z₁

Z₁ = -1.5i +5.5

Now, using midpoint formula to get Z₃

Z₃ + Z₁  = 2(3-i)/2

Z₃=0.5i-2.5