Given the binary operation $\star$ defined in the plane as follows: for any two points $A$ and $B$, $C = A \star B$ is the third vertex of the equilateral triangle $ABC$ oriented positively.

We aim to determine the relative position of three points $I$, $M$, and $O$ such that:
$$I \star (M \star O) = (O \star I) \star M.$$

To solve this, consider the properties of the operation $\star$:

1. Equilateral Triangles: The operation $\star$ produces the third vertex of an equilateral triangle oriented positively. This implies that if $P = X \star Y$, the triangle $XYP$ is equilateral with a counterclockwise orientation.

2. Orientation and Triangle Properties:
- For the operation $I \star (M \star O)$, let $N = M \star O$, meaning $I \star N$ finds the third point of the equilateral triangle completing vertex $I$ with base $N$.
- Similarly, $(O \star I) \star M$ results in a point where the triangles are also equilateral.

3. Properties of Rotations:
- Each $\star$ operation corresponds geometrically to a rotation of the plane by $60^\circ$ counterclockwise about the point $I$, followed by translating the point $M$.

For these expressions to be equal, $\triangle IMO$ must satisfy specific geometric properties:

- Isosceles Triangle: For both paths of $\star$ operations, the configurations lead to a requirement: each point must subtend the same base with an equal angle, which implies the isosceles nature with $OI = OM$.

- **Angle $\angle IOM = \frac{2\pi}{3}$**: The operations must satisfy rotation symmetry to maintain equality, implying a rotation by $120^\circ$ to cycle through each vertex.

Thus, these geometric constraints firmly conclude:

$$I \star (M \star O) = (O \star I) \star M \text{ if and only if } \triangle IMO \text{ is positively oriented, is isosceles with } OI = OM \text{, and } \angle IOM = \frac{2\pi}{3}.$$

The configuration ensuring the operation equivalence is:
$$\boxed{\text{The triangle } \triangle IMO \text{ is positively oriented, isosceles with } OI = OM, \text{ and } \angle IOM = \frac{2\pi}{3}.}$$