In a right-angled triangle $ABC$ with hypotenuse $AB$, draw lines through point $P$ on $AB$ parallel to the legs. Let the intersection points be $M$ and $N$. How should point $P$ be chosen so that the length of segment $MN$ is minimized?
A $C M P N$ (possibly degenerate) quadrilateral is a rectangle because its opposite sides are parallel, and the angle at $C$ is a right angle. Since the diagonals of a rectangle are equal, $M N = C P$. Therefore, the length of the $M N$ segment is minimal when $C P$ is minimal. The latter is obviously minimal when $P$ coincides with the foot $T$ of the altitude from $C$ in the triangle $A B C$.

# $1986-11-395-1 . e p s$ 

Since the triangle is right-angled, $T$ is always on the interior of the segment $A B$, so it satisfies the conditions of our problem. Therefore, the length of the $M N$ segment is minimal when $P$ coincides with the foot of the altitude from the vertex $C$.