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The graph of a limaçon in polar coordinates is

; it can be constructed geometrically. Let

be a ray with polar angle

. Produce this line and determine point the

so that

. Then

A limaçon with

can be used to trisect angles. Choose the angle

so that one leg intersects the limaçon at

Since the triangle

is isosceles,

. Since the triangle

is also isosceles,

. So

, or

Isaac Newton used a limaçon to trisect an angle.

Reference

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Trisecting an Angle with a Limaçon