Given,
z1,z2,z3 are three complex numbers joined to form an equilateral triangle the coordinates of z1 are (1,1)
and that of ,z2 are (−1,1).
and that of z3 are (x,y) (say)
There's no harm in considering complex numbers as coordinates in a real plane.After all, there isn't much distinction while representing graphically.Now using the distance formula,we can write
(x−1)2+(y−1)2=(x+1)2+(y+1)2=8
Solving we get
x+y=1−(i)x2+y2=8−2=6−(ii)
Solving for x & y we get
x=±√3y=±√3
∴z3=(√3+i√3) or (√3−i√3)