Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.
1. **Identify Key Points and Properties**:
   - Let \( M \) be the midpoint of \( \overline{AC} \).
   - Let \( F \) be the common point of the \( B \)-symmedian with the circumcircle \( \odot(ABC) \).
   - The line \( \overline{FM} \) passes through the point \( B' \) on \( \odot(ABC) \) such that \( \overline{BB'} \parallel \overline{AC} \).

2. **Construct Point \( B' \)**:
   - Since \( \overline{BB'} \parallel \overline{AC} \) and \( BM = B'M \), we can construct \( B' \) by finding the point on \( \odot(ABC) \) such that \( \overline{BB'} \parallel \overline{AC} \).

3. **Use Symmedian Properties**:
   - The symmedian \( \overline{BF} \) intersects the circumcircle again at \( F \).
   - By the properties of the symmedian, \( \overline{FM} \) passes through \( B' \).

4. **Construct Line Parallel to \( \overline{BB'} \)**:
   - Draw the line through \( M \) that is parallel to \( \overline{BB'} \).

5. **Find Intersections with \( \odot(BB'F) \)**:
   - Construct the circle \( \odot(BB'F) \).
   - Find the intersections of the line through \( M \) (parallel to \( \overline{BB'} \)) with \( \odot(BB'F) \).

6. **Determine Points \( A \) and \( C \)**:
   - The intersections of the line through \( M \) with \( \odot(BB'F) \) will give us points \( A \) and \( C \).

7. **Verify the Construction**:
   - Ensure that \( M \) is indeed the midpoint of \( \overline{AC} \).
   - Verify that \( \overline{BB'} \parallel \overline{AC} \) and \( BM = B'M \).

The final answer is the construction of triangle \( \boxed{ ABC } \) with the given properties.