Constructing a Parabola from Tangent Circles

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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.
Contributed by: Samuel Lesser (January 2018)
Inspired by: Matthew Hoek
Open content licensed under CC BY-NC-SA
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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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