We consider two circles $k_1(M_1, r_1)$ and $k_2(M_2, r_2)$ with $z = M_1M_2 > r_1+r_2$ and a common outer tangent with the tangent points $P_1$ and $P2$ (that is, they lie on the same side of the connecting line $M_1M_2$). We now change the radii so that their sum is $r_1+r_2 = c$ remains constant. What set of points does the midpoint of the tangent segment $P_1P_2$  run through,  when $r_1$ varies from $0$ to $c$?
1. **Define the circles and their properties:**
   We have two circles \( k_1(M_1, r_1) \) and \( k_2(M_2, r_2) \) with the distance between their centers \( z = M_1M_2 > r_1 + r_2 \). This ensures that the circles do not intersect and have a common outer tangent. The sum of the radii is constant, \( r_1 + r_2 = c \).

2. **Set up the coordinates and radii:**
   Let the center of the first circle be \( M_1(0, 0) \) with radius \( r_1 = \lambda \), and the center of the second circle be \( M_2(a, 0) \) with radius \( r_2 = c - \lambda \). Here, \( a \) and \( c \) are fixed, with \( a > c \).

3. **Determine the tangent points:**
   Let \( P_1(\lambda \cos \alpha, \lambda \sin \alpha) \) be the point of tangency on the first circle. The equation of the common tangent line can be written as:
   \[
   \cos \alpha \cdot x + \sin \alpha \cdot y = \lambda
   \]
   For this line to be tangent to the second circle, we need:
   \[
   \cos \alpha = \frac{2\lambda - c}{a}
   \]

4. **Find the coordinates of the tangent points:**
   The coordinates of the tangent points \( P_1 \) and \( P_2 \) are:
   \[
   P_1 \left( \frac{\lambda(2\lambda - c)}{a}, \frac{\lambda \sqrt{a^2 - (c - 2\lambda)^2}}{a} \right)
   \]
   \[
   P_2 \left( \frac{a^2 + (c - 2\lambda)(\lambda - c)}{a}, \frac{(c - \lambda) \sqrt{a^2 - (c - 2\lambda)^2}}{a} \right)
   \]

5. **Calculate the midpoint of the tangent segment \( P_1P_2 \):**
   The midpoint \( M \) of the segment \( P_1P_2 \) is given by:
   \[
   M \left( \frac{\frac{\lambda(2\lambda - c)}{a} + \frac{a^2 + (c - 2\lambda)(\lambda - c)}{a}}{2}, \frac{\frac{\lambda \sqrt{a^2 - (c - 2\lambda)^2}}{a} + \frac{(c - \lambda) \sqrt{a^2 - (c - 2\lambda)^2}}{a}}{2} \right)
   \]
   Simplifying the coordinates of \( M \):
   \[
   M \left( \frac{a^2 - c(c - 2\lambda)}{2a}, \frac{c \sqrt{a^2 - (c - 2\lambda)^2}}{2a} \right)
   \]

6. **Eliminate the parameter \( \lambda \) to find the locus:**
   To find the locus of the midpoint \( M \), we eliminate \( \lambda \) from the equations. The resulting equation is:
   \[
   4x^2 + 4y^2 - 4ax + a^2 = c^2
   \]
   This represents a circle with the center at \( (a/2, 0) \) and radius \( c/2 \).

7. **Determine the range of \( x \) and \( y \):**
   The range of \( x \) is:
   \[
   x \in \left[ \frac{a^2 - c^2}{2a}, \frac{a^2 + c^2}{2a} \right]
   \]
   Since the tangent points lie on the same side of the line \( M_1M_2 \), \( y \) must be positive:
   \[
   y > 0
   \]

The final answer is the locus of the midpoint of the tangent segment \( \boxed{ P_1P_2 } \) is part of the circle \( 4x^2 + 4y^2 - 4ax + a^2 = c^2 \) with \( x \in \left[ \frac{a^2 - c^2}{2a}, \frac{a^2 + c^2}{2a} \right] \) and \( y > 0 \).