A circle of radius r touching both sides of a parabola having focal length a. A line segment tangent to the circle is split in two equal segments of length d. What is d in terms of r and a?

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geometry puzzles to train your brain

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If you roll a circle down a parabola, small circles can roll all the way down and up, whereas large circles get stuck between the sides at some point. What is the transition radius? In other words, what’s the maximum circle that keeps on rolling?