In square $ABCD$, $\overline{AC}$ and $\overline{BD}$ meet at point $E$.
Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$.
If $\overline{AF}$ meets $\overline{ED}$ at point $G$, and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$.
1. **Identify the properties of the square and the given points:**
   - Let the side length of the square \(ABCD\) be \(s\).
   - The diagonals \(\overline{AC}\) and \(\overline{BD}\) intersect at point \(E\), which is the center of the square.
   - Since \(E\) is the center, \(E\) divides each diagonal into two equal parts. Therefore, \(E\) is equidistant from all four vertices of the square.

2. **Determine the coordinates of the points:**
   - Place the square in the coordinate plane with \(A = (0, 0)\), \(B = (s, 0)\), \(C = (s, s)\), and \(D = (0, s)\).
   - The coordinates of \(E\) are \(\left(\frac{s}{2}, \frac{s}{2}\right)\).

3. **Use the given angle condition:**
   - Point \(F\) is on \(\overline{CD}\) such that \(\angle CAF = \angle FAD\). This implies that \(F\) is the midpoint of \(\overline{CD}\) because the angles are equal, and the only way for the angles to be equal in a square is if \(F\) is equidistant from \(C\) and \(D\).
   - Therefore, the coordinates of \(F\) are \(\left(0, \frac{s}{2}\right)\).

4. **Use the given length \(EG = 24\) cm:**
   - Since \(E\) is the center of the square, the length of each half-diagonal is \(\frac{s\sqrt{2}}{2}\).
   - Given that \(\overline{EG} = 24\) cm, we need to find the relationship between \(G\) and the side length \(s\).

5. **Apply mass points to find the ratio \(\frac{GD}{EG}\):**
   - Using mass points, we assign masses to points \(E\) and \(D\) such that the ratio \(\frac{GD}{EG} = \sqrt{2}\).
   - This implies that \(DE = 24 + 24\sqrt{2}\).

6. **Calculate the side length of the square:**
   - Since \(DE\) is half the diagonal of the square, we have:
     \[
     \frac{s\sqrt{2}}{2} = 24 + 24\sqrt{2}
     \]
     \[
     s\sqrt{2} = 48 + 48\sqrt{2}
     \]
     \[
     s = 24\sqrt{2} + 48
     \]

7. **Use the Angle Bisector Theorem to find \(CF\):**
   - By the Angle Bisector Theorem, we have:
     \[
     \frac{CF}{CD} = \frac{\sqrt{2}}{\sqrt{2} + 1}
     \]
   - Since \(CD = s\), we get:
     \[
     CF = s \cdot \frac{\sqrt{2}}{\sqrt{2} + 1}
     \]
   - Substituting \(s = 24\sqrt{2} + 48\):
     \[
     CF = (24\sqrt{2} + 48) \cdot \frac{\sqrt{2}}{\sqrt{2} + 1}
     \]

8. **Simplify the expression for \(CF\):**
   - Simplify the fraction:
     \[
     \frac{\sqrt{2}}{\sqrt{2} + 1} = \frac{\sqrt{2}(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \frac{2 - \sqrt{2}}{1} = 2 - \sqrt{2}
     \]
   - Therefore:
     \[
     CF = (24\sqrt{2} + 48) \cdot (2 - \sqrt{2})
     \]
   - Distribute and simplify:
     \[
     CF = 48\sqrt{2} - 24 \cdot 2 + 96 - 48\sqrt{2} = 48
     \]

The final answer is \(\boxed{48}\) cm.