The incircle of a triangle $ A_1A_2A_3$ is centered at $ O$ and meets the segment $ OA_j$ at $ B_j$ , $ j \equal{} 1, 2, 3$. A circle with center $ B_j$ is tangent to the two sides of the triangle having $ A_j$ as an endpoint and intersects the segment $ OB_j$ at $ C_j$. Prove that
\[ \frac{OC_1\plus{}OC_2\plus{}OC_3}{A_1A_2\plus{}A_2A_3\plus{}A_3A_1} \leq \frac{1}{4\sqrt{3}}\]
and find the conditions for equality.
1. **Define the given elements and relationships:**
   - Let \( A_1A_2A_3 \) be a triangle with incircle centered at \( O \).
   - The incircle touches \( OA_j \) at \( B_j \) for \( j = 1, 2, 3 \).
   - A circle centered at \( B_j \) is tangent to the sides of the triangle at \( A_j \) and intersects \( OB_j \) at \( C_j \).

2. **Express \( OC_j \) in terms of the triangle's elements:**
   - The radius of the incircle is \( r \).
   - The distance \( OB_j \) is \( r \) since \( B_j \) lies on the incircle.
   - The circle centered at \( B_j \) is tangent to the sides of the triangle at \( A_j \), so the radius of this circle is also \( r \).

3. **Calculate \( OC_j \):**
   - Since \( C_j \) lies on \( OB_j \) and the radius of the circle centered at \( B_j \) is \( r \), the distance \( OC_j \) is \( 2r \).

4. **Sum the distances \( OC_1, OC_2, OC_3 \):**
   \[
   OC_1 + OC_2 + OC_3 = 2r + 2r + 2r = 6r
   \]

5. **Express the perimeter of the triangle \( A_1A_2A_3 \):**
   - Let \( p \) be the semi-perimeter of the triangle, so the perimeter is \( 2p \).

6. **Relate the area \( S \) of the triangle to the inradius \( r \) and semi-perimeter \( p \):**
   \[
   S = pr
   \]

7. **Use the AM-GM inequality and the sum of angles:**
   - The sum of the angles in a triangle is \( 180^\circ \), so:
     \[
     \sum \sin \frac{\angle A_i}{2} \leq \frac{3}{2}
     \]

8. **Combine the inequalities and expressions:**
   \[
   \frac{OC_1 + OC_2 + OC_3}{A_1A_2 + A_2A_3 + A_3A_1} = \frac{6r}{2p} = \frac{3r}{p}
   \]

9. **Use the relationship between \( r \), \( p \), and \( S \):**
   \[
   \frac{3r}{p} = \frac{3S}{p^2}
   \]

10. **Apply the AM-GM inequality:**
    \[
    \frac{3S}{p^2} \leq \frac{3}{4\sqrt{3}}
    \]

11. **Simplify the inequality:**
    \[
    \frac{3S}{p^2} \leq \frac{1}{4\sqrt{3}}
    \]

12. **Equality condition:**
    - Equality holds when \( \sum \sin \frac{\angle A_i}{2} = \frac{3}{2} \), which occurs in an equilateral triangle.

\(\blacksquare\)

The final answer is \( \boxed{ \frac{OC_1 + OC_2 + OC_3}{A_1A_2 + A_2A_3 + A_3A_1} \leq \frac{1}{4\sqrt{3}} } \)