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Descartes's Angle Trisection

​
a = sin(3t)
0.75
zoom
1
2
Descartes used the intersection of a circle and a parabola to trisect an angle. The equations of the circle and parabola are
2
x
+
2
y
-
13
4
x+4ay=0
and
2
y
=
1
4
x
, respectively. The coordinates of the intersection satisfy
4
3
y
=3y-a
. Since
4
3
sin
(t)=3sin(t)-sin(3t)
, by taking
a=sin(3t)
, the smaller positive root of the last equation is
sin(t)
.
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