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14 Apr 2020
: Longbow Curve. In the following figure, the circle of radius
is stationary, and for every
, the point
is the midpoint of the segment
. The curve traced out by
for
is called the longbow curve. Find parametric equations for this curve.
![](data:image/png;base64,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)
: Longbow Curve. In the following figure, the circle of radius is stationary, and for every
, the point
is the midpoint of the segment
. The curve traced out by
for
is called the longbow curve. Find parametric equations for this curve.
Casey DurganLv2
27 May 2020