Let $ABC$ be a fixed triangle in the plane. Let $D$ be an arbitrary point in the plane. The circle with center $D$, passing through $A$, meets $AB$ and $AC$ again at points $A_b$ and $A_c$ respectively. Points $B_a, B_c, C_a$ and $C_b$ are defined similarly. A point $D$ is called good if the points $A_b, A_c,B_a, B_c, C_a$, and $C_b$ are concyclic. For a given triangle $ABC$, how many good points can there be?

(A. Garkavyj, A. Sokolov ) 
1. **Identify the circumcenter \( O \) as a good point:**
   - The circumcenter \( O \) of \( \triangle ABC \) is always a good point because the circle centered at \( O \) passing through \( A \) will also pass through \( B \) and \( C \), making \( A_b, A_c, B_a, B_c, C_a, C_b \) concyclic.

2. **Define the isogonal conjugate \( E \) of \( D \):**
   - Let \( D \neq O \) be a good point. Define \( E \) as the isogonal conjugate of \( D \) with respect to \( \triangle ABC \). The isogonal conjugate of a point \( D \) is the point \( E \) such that the angles \( \angle ADE \) and \( \angle AOE \) are equal.

3. **Determine the reflection point \( D' \):**
   - The circumcenter of the circle through \( A_b, A_c, B_a, B_c, C_a, C_b \) must lie on the perpendicular bisectors of \( A_bB_a, B_cC_b, C_aA_c \). This point is the reflection of \( O \) over \( D \), denoted as \( D' \).

4. **Analyze the perpendicularity condition:**
   - Since \( D'A_b = D'A_c \) and \( DA_b = DA_c = DA \), we have \( DD' \perp A_bA_c \). This implies \( OD \perp A_bA_c \).

5. **Parallelism condition:**
   - The condition \( OD \perp A_bA_c \) implies that \( AE \parallel OD \) (or \( E = A \)). Similar conditions hold for the other two vertices \( B \) and \( C \).

6. **Examine the case when \( E \) is one of the vertices:**
   - If \( E \) is one of the vertices, say \( E = A \), then \( BE \parallel OD \) and \( CE \parallel OD \) cannot simultaneously hold because \( BA \) and \( CA \) are not parallel.

7. **Consider \( E \) as a point at infinity:**
   - If \( E \) is not one of the vertices, then \( AE \parallel BE \parallel CE \), meaning \( E \) is the point at infinity along line \( AE \). This is equivalent to \( D \) lying on the circumcircle of \( \triangle ABC \).

8. **Find the points \( D \) on the circumcircle:**
   - We need to find all \( D \) on the circumcircle such that \( OD \) is parallel to the reflection of \( AD \) in the angle bisector of \( \angle BAC \). By rotating \( D \) around the circle, \( OD \) traverses an angle of \( 2\pi \) while the other line traverses an angle of \( \pi \) in the opposite direction. By continuity, there are exactly three such points \( D \).

9. **Count the total number of good points:**
   - Adding the circumcenter \( O \) to the three other good points \( D \) on the circumcircle, we have a total of \( 4 \) good points.

\(\blacksquare\)

The final answer is \( \boxed{4} \)