3. Triangle $A B C$ has a right angle at $C$, and $D$ is the foot of the altitude from $C$ to $A B$. Points $L$, $M$, and $N$ are the midpoints of segments $A D, D C$, and $C A$, respectively. If $C L=7$ and $B M=12$, compute $B N^{2}$.
Answer: 193
Solution: Note that $C L, B M$, and $B N$ are corresponding segments in the similar triangles $\triangle A C D \sim$ $\triangle C B D \sim \triangle A B C$. So, we have
$$
C L: B M: B N=A D: C D: A C .
$$

Since $A D^{2}+C D^{2}=A C^{2}$, we also have $C L^{2}+B M^{2}=B N^{2}$, giving an answer of $49+144=193$.