Let the circles $S_1$ and $S_2$ meet at the points $A$ and $B$. A line through $B$ meets $S_1$ at a point $D$ other than $B$ and meets $S_2$ at a point $C$ other than $B$. The tangent to $S_1$ through $D$ and the tangent to $S_2$ through $C$ meet at $E$. If $|AD|=15$, $|AC|=16$, $|AB|=10$, what is $|AE|$?

$ 
\textbf{(A)}\ 20
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 25
\qquad\textbf{(D)}\ 26
\qquad\textbf{(E)}\ 31
$
1. **Identify the given information and the relationships between the points:**
   - Circles \( S_1 \) and \( S_2 \) intersect at points \( A \) and \( B \).
   - A line through \( B \) intersects \( S_1 \) at \( D \) and \( S_2 \) at \( C \).
   - Tangents to \( S_1 \) at \( D \) and to \( S_2 \) at \( C \) meet at \( E \).
   - Given lengths: \( |AD| = 15 \), \( |AC| = 16 \), \( |AB| = 10 \).

2. **Use the properties of tangents and angles:**
   - The tangents from a point to a circle are equal in length.
   - The angles formed by the tangents and the radii to the points of tangency are right angles.

3. **Establish the similarity of triangles:**
   - Consider triangles \( \triangle ABD \) and \( \triangle ACE \).
   - Since \( \angle DAB = \angle CDE \) (both are angles subtended by the same arc \( AB \) in circles \( S_1 \) and \( S_2 \) respectively), and \( \angle ADB = \angle AEC \) (both are right angles formed by the tangents), we have:
     \[
     \triangle ABD \sim \triangle ACE
     \]
     by AA (Angle-Angle) similarity criterion.

4. **Set up the proportion based on the similarity of triangles:**
   - From the similarity, we have:
     \[
     \frac{AD}{AE} = \frac{AB}{AC}
     \]
   - Substitute the given lengths:
     \[
     \frac{15}{AE} = \frac{10}{16}
     \]

5. **Solve for \( AE \):**
   \[
   \frac{15}{AE} = \frac{10}{16} \implies 15 \cdot 16 = 10 \cdot AE \implies 240 = 10 \cdot AE \implies AE = \frac{240}{10} = 24
   \]

6. **Conclusion:**
   - The length \( |AE| \) is 24.

The final answer is \( \boxed{24} \)