10.4. Given an acute triangle $A B C$. From point $D$ on side $A B$, perpendiculars $D E$ and $D F$ are drawn to sides $A C$ and $B C$ respectively. At what position of point $D$ will the distance between points $E$ and $F$ be the smallest
Answer: $D$ is the foot of the altitude dropped from vertex $C$.

Solution. Points $E$ and $F$ lie on the circle constructed with segment $C D$ as its diameter. In this circle, the constant angle $C$ subtends the chord $E F$, so the length of chord $E F$ will be minimal if the diameter $C D$ of the circle is minimal, i.e., $C D$ is the altitude of triangle $A B C$.