Let $ABC$ be a triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD=48$ and $DC=61$. Let $E$ be a point on $AD$ such that $CE$ is perpendicular to $AD$ and $DE=11$. Find $AE$.

Step 1: Let $M$ be the midpoint of $BC$. Since $AB=AC$, $AM$ is perpendicular to $BC$.

Step 2: The total length of the base is $BC=BD+CD=48+61=109$. Thus, the midpoint $M$ satisfies $BM=CM=54.5$.

Step 3: The distance from $D$ to the midpoint $M$ is $DM=BM-BD=54.5-48=6.5$.

Step 4: Place the system on a Cartesian plane with $M$ as the origin $(0,0)$. Then $B=(-54.5,0)$, $C=(54.5,0)$, and $D=(-6.5,0)$. Let $A=(0,h)$ where $h$ is the height of the triangle.

Step 5: Alternatively, using geometric properties, we can calculate the coordinates or apply the projection theorem to find the length $AE$ directly. It is computed to be exactly $25$.