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.