Given a triangle $ABC$. Let $S$  be the circle passing through $C$, centered at $A$. Let $X$ be a variable point on $S$ and let $K$ be the midpoint of the segment $CX$ . Find the locus of the midpoints of $BK$, when $X$ moves along $S$.

(I. Gorodnin)
1. **Identify the Circle \( S \) and Point \( K \)**:
   - Given a triangle \( ABC \), let \( S \) be the circle passing through \( C \) and centered at \( A \). 
   - Let \( X \) be a variable point on \( S \). 
   - Let \( K \) be the midpoint of the segment \( CX \).

2. **Properties of \( K \)**:
   - Since \( K \) is the midpoint of \( CX \), \( K \) lies on the circle with diameter \( AC \). This is because the midpoint of any chord of a circle lies on the circle with the same diameter.

3. **Circle \( \Gamma \)**:
   - Let \( \Gamma \) be the circle with diameter \( AC \). 
   - Since \( \angle AKC = 90^\circ \) for any \( X \) on \( S \), \( K \) lies on \( \Gamma \).

4. **Foot of the Altitude \( D \)**:
   - Let \( D \) be the foot of the altitude from \( C \) to \( AB \). 
   - Since \( \angle ADC = 90^\circ \), \( D \) also lies on \( \Gamma \).

5. **Homothety Centered at \( B \)**:
   - Consider a homothety centered at \( B \) with a coefficient of \( \frac{1}{2} \). Denote this homothety by \( \mathcal{H} \).
   - Under \( \mathcal{H} \), the point \( K \) maps to \( M \), the midpoint of \( BK \).

6. **Image of Points under Homothety**:
   - The point \( C \) maps to the midpoint of \( BC \), denoted as \( G \).
   - The point \( A \) maps to the midpoint of \( AB \), denoted as \( N \).
   - The point \( D \) maps to a point \( P \) such that \( \frac{BD}{BP} = 2 \).

7. **Circle \( \omega \)**:
   - The points \( N \), \( G \), and \( P \) lie on a circle \( \omega \) with radius half of \( \Gamma \)'s radius.
   - Since \( \Gamma \) has a radius of \( \frac{1}{2}AC \), \( \omega \) has a radius of \( \frac{1}{4}AC \).

8. **Locus of \( M \)**:
   - Since \( N \), \( G \), and \( P \) are fixed points, \( M \) moves along a fixed circle \( \omega \).

Conclusion:
The locus of the midpoints of \( BK \) when \( X \) moves along \( S \) is a fixed circle with radius \( \frac{1}{4}AC \).

The final answer is the locus of the midpoints of \( \boxed{ BK } \) is a fixed circle with radius \( \frac{1}{4}AC \).