Geometric Constructions 5

Show that :

there are three
parallel tangents to the cardioid with any given gradient; if we connect
the points of tangency to the cusp, the three segments meet at equal angles
of 2p/3; the area of the triangle formed by
the points of tangency is constant;

the tangents at the ends of any chord through the cusp of a cardioid are
at right angles;

the length of any chord through the cusp of a cardioid is constant;

Construct
the rectangle enclosed by the tangents and normal at the ends of any chord
through the cusp of a cardioid. What's so particular about the vertices
of this rectangle? (JavaSketchpad
file)

Take
five points on a cardioid with the corresponding points on the base circle
forming a regular pentagon. What properties does the figure have?

Construct
an animation displaying a cardioid sliding along two orthogonal straight
lines.

Illustrate
the principle behind the cardioid condenser.

Illustrate
the harmonic motion associated with the rotation of the cardioid about
its cusp.

Construct
the formation of the envelope of light rays, emitted from a radiant point
source on a circle, after reflection by the same circle.

Construct
an animation displaying two cardioids having the same cusp and are orthogonal
to each other.