Given two fixed points $O$ and $G$ in the plane. Find the locus of the vertices of triangles whose circumcenters and centroids are $O$ and $G$ respectively.
1. **Understanding the given points and their relationships:**
   - We are given two fixed points \( O \) (the circumcenter) and \( G \) (the centroid) of a triangle.
   - We need to find the locus of the vertices of triangles whose circumcenters and centroids are \( O \) and \( G \) respectively.

2. **Using the properties of the Euler line:**
   - In any triangle, the orthocenter \( H \), circumcenter \( O \), and centroid \( G \) are collinear and lie on the Euler line.
   - The centroid \( G \) divides the segment \( OH \) in the ratio \( 2:1 \), i.e., \( HG = 2 \cdot OG \).

3. **Determining the orthocenter \( H \):**
   - Given \( O \) and \( G \), we can determine the position of \( H \) using the relationship \( HG = 2 \cdot OG \).
   - Let \( \vec{O} \) and \( \vec{G} \) be the position vectors of \( O \) and \( G \) respectively. Then, the position vector of \( H \) is given by:
     \[
     \vec{H} = \vec{G} + 2(\vec{G} - \vec{O}) = 3\vec{G} - 2\vec{O}
     \]

4. **Constructing the triangle:**
   - Let \( \omega_R \) be the circle with center \( O \) and radius \( R \).
   - Choose an arbitrary point \( P_R \) on \( \omega_R \). This point \( P_R \) will be one of the vertices of the triangle.
   - Draw the line passing through \( P_R \) and \( H \). Let this line intersect \( \omega_R \) at another point \( Q_R \).

5. **Finding the other vertices:**
   - Draw the perpendicular bisector of the line segment \( HQ_R \). This bisector will intersect \( \omega_R \) at two points, say \( B \) and \( C \).
   - The triangle \( P_RBC \) will have \( O \) as its circumcenter and \( G \) as its centroid.

6. **Verification:**
   - Since \( P_R \) is on the circumcircle \( \omega_R \), and \( B \) and \( C \) are determined by the perpendicular bisector of \( HQ_R \), the triangle \( P_RBC \) satisfies the given conditions.
   - The vertices of the triangle must lie on the circle \( \omega_R \).

Conclusion:
The locus of the vertices of the triangles whose circumcenters and centroids are \( O \) and \( G \) respectively is the circle with center \( O \) and radius \( R \).

The final answer is the circle with center \( \boxed{ O } \) and radius \( R \).