11.6*. The perimeter of triangle $A B C$ is $2 p$. Points $M$ and $N$ are taken on sides $A B$ and $A C$ such that $M N \| B C$ and $M N$ is tangent to the inscribed circle of triangle $A B C$. Find the maximum value of the length of segment $M N$.
11.6. Let the length of the altitude dropped to side $BC$ be denoted by $h$. Since $\triangle A M N \sim \triangle A B C$, we have $M N / B C=(h-2 r) / h$, i.e., $M N=a \left(1-\frac{2 r}{h}\right)$. Since $r=S / p=a h / 2 p$, it follows that $M N=a\left(1-a / p\right)$. The maximum value of the expression $a\left(1-a / p\right)=a(p-a) / p$ is achieved when $a=p / 2$; it is equal to $p / 4$. It remains to note that there exists a triangle with perimeter $2 p$ and side $a=p / 2$ (set $b=c=3 p / 4$).