Locus of the Center of a Circle Inscribed in a Circular Segment
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 =R-(y-b), which is the equation of a parabola.
C
R
L
y=b
D
C
L
C
D
(x,y)
D
D
y-b
D
1/2
+
2
x
2
y