A circle with diameter $23$ is cut by a chord $AC$. Two different circles can be inscribed between the large circle and $AC$. Find the sum of the two radii.
1. Let the diameter of the large circle be \( D = 23 \). Therefore, the radius \( R \) of the large circle is:
   \[
   R = \frac{D}{2} = \frac{23}{2}
   \]

2. Consider the chord \( AC \) of the large circle. Two smaller circles can be inscribed between the large circle and the chord \( AC \). Let the radii of these two smaller circles be \( r_1 \) and \( r_2 \).

3. The key observation here is that the sum of the diameters of the two inscribed circles is equal to the diameter of the large circle. This is because the two inscribed circles are tangent to each other, the chord \( AC \), and the large circle. Therefore, the sum of the radii of the two inscribed circles is equal to the radius of the large circle.

4. Mathematically, this can be expressed as:
   \[
   r_1 + r_2 = R
   \]

5. Substituting the value of \( R \) from step 1:
   \[
   r_1 + r_2 = \frac{23}{2}
   \]

Conclusion:
\[
\boxed{\frac{23}{2}}
\]