Let $C_1 , C_2$ be two circles in the plane intersecting at two distinct points. Let $P$ be the midpoint of a variable chord $AB$ of $C_2$ with the property that the circle on $AB$ as diameter meets $C_1$ at a point $T$ such that $P T$ is tangent to $C_1$ . Find the locus of $P$ .
1. **Define the centers and intersection points:**
   Let \( O_1 \) and \( O_2 \) be the centers of circles \( C_1 \) and \( C_2 \), respectively. Let \( E \) and \( F \) be the points where \( C_1 \) and \( C_2 \) intersect.

2. **Consider the midpoint \( P \) of chord \( AB \) of \( C_2 \):**
   Let \( P \) be the midpoint of a variable chord \( AB \) of \( C_2 \). The circle with \( AB \) as its diameter intersects \( C_1 \) at point \( T \).

3. **Condition for tangency:**
   Given that \( PT \) is tangent to \( C_1 \) at \( T \), we know that \( PT \) is perpendicular to the radius of \( C_1 \) at \( T \). This implies that \( PT \) is perpendicular to \( O_1T \).

4. **Use the power of a point theorem:**
   The power of point \( P \) with respect to circle \( C_1 \) is given by:
   \[
   PT^2 = PO_1^2 - r_1^2
   \]
   where \( r_1 \) is the radius of \( C_1 \).

5. **Relate the power of point \( P \) to the geometry of the circles:**
   Since \( P \) is the midpoint of \( AB \) and \( AB \) is a chord of \( C_2 \), the circle with diameter \( AB \) has its center at \( P \) and radius \( \frac{AB}{2} \). The circle intersects \( C_1 \) at \( T \), and \( PT \) is tangent to \( C_1 \).

6. **Determine the locus of \( P \):**
   Since \( P \) is the midpoint of \( AB \) and \( PT \) is tangent to \( C_1 \), \( P \) must lie on the radical axis of \( C_1 \) and \( C_2 \). The radical axis of two intersecting circles is the line passing through their intersection points \( E \) and \( F \).

7. **Conclusion:**
   The locus of \( P \) is the line segment joining \( E \) and \( F \), which is the radical axis of \( C_1 \) and \( C_2 \).

The final answer is the line segment joining \( \boxed{ E } \) and \( F \).