# The Cardioid as the Envelope of Pencil Curves

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Draw a circle and divide the circumference into and even number of equally spaced points.  In this example I have used 44:

$1,2,3, \dots , 44$

Draw following chords between numbers in such a way that when viewed as a sequemce, the second point of each pair moves twice as fast around the circle.:
$1 \rightarrow 2,\ 2 \rightarrow 4,\ 4 \rightarrow 8, \dots,\ 22 \rightarrow 44$
and
$23 \rightarrow 2,\ 24 \rightarrow 4,\ 25 \rightarrow 8, \dots,\ 44 \rightarrow 44$

and on completion, sketch the curve which owns each of the chords as a tangent: