The product of the distance from the green point (the center of the hypocycloid) to the blue point (on the hypocycloid), and the distance from the green point to the red point (on the inverse curve) is constant (and equal to the square of the radius of the green circle).
For more information, films, and interactive material, see
http://tinyurl.com/62yuqc
http://tinyurl.com/4oreze
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