Locus of the Center of a Circle Inscribed in a Circular Segment

The trajectory of the center of a moving circle inscribed in a circular segment is parabolic.
This Demonstration shows this result with a horizontal chord ("base line").
Proof: Let be a circle with its center at the origin and radius with a horizontal chord given by and let be the small circle inscribed on the circular segment bounded by and . Since the circles and are tangent, the tangent point and the centers of the circles are collinear. Let be the coordinates of the center of . The radius of is equal to . Hence the distance from the origin to the center of is , which is the equation of a parabola.

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