What is the area of a Bicycle Crescent? A bicyclist rides through a long, shallow puddle of water.
Then he makes a right-angle turn onto a side street. The rear wheel takes a shortcut, in a futile attempt to catch up with the front wheel. The wet wheels make a crescent shape on the pavement. It turns out that the area of the crescent is path-independent. In other words, the area is the same, whether the bicycle makes a sharp turn or a gradual turn. If the distance between the two axles is d, what is the area of the crescent?
A friend of mine was the first person to solve this problem. Standard integral calculus is absolutely useless here.
Mamikon's Theorem states that the area of the tangent cluster is equal to the area of the tangent sweep. Draw the appropriate picture, and you will see that the area of the Bicyclix is equal to that of a quarter circle, whose radius is d.
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