Constructing a Parabola from Tangent Circles

Let be a circle that does not intersect a horizontal line . Use the sliders to change the radius of or the distance of to . The locus of the centers of the circles that are tangent to both and is a parabola.

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The synthetic (as opposed to analytic) definition of a parabola is that a point on is at equal distances from a fixed point (the focus) and a fixed line (the directrix).
Let have center and radius and let be the line parallel to on the other side of at distance from . Consider a circle of radius and center that is tangent to both and . Then and the distance from to is also , so the centers of the tangent circles lie on a parabola with focus and directrix .

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