11.17. From a point $M$ on the circumcircle of triangle $A B C$, perpendiculars $M P$ and $M Q$ are dropped to the lines $A B$ and $A C$. For what position of point $M$ is the length of segment $P Q$ maximized?
11.17. Points $P$ and $Q$ lie on the circle with diameter $A M$, so $P Q=$ $=A M \sin P A Q=A M \sin A$. Therefore, the length of segment $P Q$ is maximized when $A M$ is the diameter of the circumscribed circle.