If z1,z2,z3 non-zero, non-collinear complex numbers such that 2z1=1z2+1z3, then the points z1,z2,z3 lie
We have,
2z1=1z2+1z3=z3+z2z2z3
⇒z1=2z2z3z2+z3
Now, (z2−z4z1−z4)(z1−z3z2−z3)
=⎛⎜ ⎜ ⎜⎝z2−z42z2z3z2+z3−z4⎞⎟ ⎟ ⎟⎠⎛⎜ ⎜ ⎜⎝2z2z3z2+z3−z3z2−z3⎞⎟ ⎟ ⎟⎠
=z22z2z3z2+z3(z3(z2−z3)(z2−z3)(z2−z3)) [taking z4=0]
=12 (a real number).
Hence, points z1,z2,z3 and origin are concyclic
and therefore, z1,z2,z3 lie on a circle passing through the origin.