A straight ladder starts upright against a vertical wall, and slides down until it is horizontal such that the top is always along the wall and the bottom on the floor. What shape does its midpoint trace out?
The ladder always forms a right triangle with the wall whose hypotenuse has a fixed length. Since the length of the median to the hypotenuse in a right triangle is always half the hypotenuse, the distance from the midpoint of the ladder to the corner where the wall meets the floor remains fixed at half the length of the ladder. Thus, the path traced out by the midpoint is a quarter circle whose center is the corner where the wall and floor meet and whose radius is half the length of the ladder.