The falling ladder is a classic calculus problem. This version is based on the sketch of the same name from the Calculus section of Sketchpad's Samples.
The problem involves a point (bucket) on a line (ladder), and investigates the path that the point traces as the y-intercept of the line goes to zero. Drag the point Bucket to vary where it is located on the line, and drag the point Slip to reposition the ladder. Pressing the X in the lower right will clear all the traces.
When the bucket is at the midpoint of the ladder, you can see its path is part of a circle. This is easy to prove. (Think of the ladder as one diagonal of a rectangle. Given what you know about the two diagonals of a rectangle, and that the length of the ladder is fixed, what can you say about the distance from the bucket to where the wall meets the floor?)
But when the bucket is not at the ladder's midpoint, the path is not part of a circle. What is its shape, and how might you prove that?