A set $A_1 , A_2 , A_3 , A_4$ of 4 points in the plane is said to be [i]Athenian[/i] set if there is a point $P$ of the plane satsifying 

(*) $P$ does not lie on any of the lines $A_i A_j$ for $1 \leq i < j \leq 4$;
(**) the line joining $P$ to the mid-point of the line $A_i A_j$ is perpendicular to the line joining $P$ to the mid-point of $A_k A_l$, $i,j,k,l$ being distinct.

(a) Find all [i]Athenian[/i] sets in the plane.
(b) For a given [i]Athenian[/i] set, find the set of all points $P$ in the plane satisfying (*) and (**)
1. **Identify the conditions for an Athenian set:**
   - Condition (*) states that point \( P \) does not lie on any of the lines \( A_iA_j \) for \( 1 \leq i < j \leq 4 \).
   - Condition (**) states that the line joining \( P \) to the midpoint of \( A_iA_j \) is perpendicular to the line joining \( P \) to the midpoint of \( A_kA_l \), where \( i, j, k, l \) are distinct.

2. **Analyze the geometric implications:**
   - For condition (**) to hold, the midpoints of the segments \( A_iA_j \) and \( A_kA_l \) must be such that the lines joining \( P \) to these midpoints are perpendicular.
   - This implies that the quadrilateral formed by \( A_1, A_2, A_3, A_4 \) must have certain symmetry properties.

3. **Consider the orthocentric system:**
   - An orthocentric system is a set of four points where one point is the orthocenter of the triangle formed by the other three points.
   - In such a system, the perpendicularity condition is naturally satisfied because the altitudes of the triangle are concurrent at the orthocenter.

4. **Verify the orthocentric system:**
   - Let \( A_1, A_2, A_3, A_4 \) be an orthocentric system with \( A_1 \) as the orthocenter of \( \triangle A_2A_3A_4 \).
   - The midpoints of the sides of \( \triangle A_2A_3A_4 \) and the midpoints of the segments joining \( A_1 \) to the vertices of \( \triangle A_2A_3A_4 \) lie on the nine-point circle of \( \triangle A_2A_3A_4 \).

5. **Determine the locus of point \( P \):**
   - The nine-point circle of \( \triangle A_2A_3A_4 \) passes through the midpoints of the sides of the triangle and the feet of the altitudes.
   - Therefore, the locus of \( P \) is the nine-point circle of \( \triangle A_2A_3A_4 \).

6. **Conclusion:**
   - The set \( A_1, A_2, A_3, A_4 \) forms an orthocentric system.
   - The locus of \( P \) is the nine-point circle of any triangle formed by these points.

The final answer is that \( \boxed{ A_1, A_2, A_3, A_4 } \) form an orthocentric system, and the locus of \( P \) is the nine-point circle of any triangle formed by these points.