In a unit circle, two fixed endpoints of the diameter and a point selected on the circumference forms a triangle. The point on the circumference is rotated counterclockwise, so that the locus of the Feuerbach point forms a double-bulb-shaped curve (marked in blue on the animation). It can be shown that the area enclosed by the locus is equal to
where are positive integers and . Find the value of .
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Bizarrely, this problem and mine seem to have been posted almost simultaneously!