I think they lie on a circle.

We have 1/z₂ = 1/2(1/z₁+1/z₃)
Simplifying, we get

z₁z₂+z₂z₃ = 2z₁z₃

or (z₂-z₃)/(z₁-z₂) = z₃/z₁.

Thus, arg(z₂-z₃/z₁-z₂) = arg(z₃/z₁).

Lets plot the points geometrically.

Say z₁ = A, z₂ = B and z₃ = C.

Now using the fact that 1/z₂ = 1/2(1/z₁+1/z₃) you can prove that

1.either |z₁|>=|z₂|>=|z₃| or |z₁|<=|z₂|<=|z₃|.Using  |1/z₂| = 1/2|1/z₁+1/z₃|. (Try to prove it).
2. z₃ never lies between z₁ and z₂. Using argz₂ = arg(z₁)+arg(z₃) - arg(z₁+z₃).
(Highlighted portion is wrong)
1. z₂ always lies between z₁ and z₃. Using argz₂ = arg(z₁)+arg(z₃) - arg(z₁+z₃).

2. arg(z₂-z₃/z₁-z₂) = arg(z₃/z₁) excludes cases when OABC is not convex (both have the same sense, clockwise or anticlockwise. This tells us CB cannot 'tilt behind' AB.  

 

Hence, OABC forms a convex quadrilateral.

And from arg(z₂-z₃/z₁-z₂) = arg(z₃/z₁) we get that angle COB and angle CBA are supplementary. Hence, OABC is a cyclic quadrilateral

That is A,B, and C lie on a circle passing through origin.

Proving that OABC is a convex quadrilateral is a critical part of the proof.