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Geometric Construction 12

Given a circle O with radius r and a point A,
construct a point B on OA so OA . OB=r^{2}. The point B
is called the **inverse** of A with respect to the circle.
Construct a script which constructs the inversion of a circle given its
center and a point on the circumference.

Construct an animation making the chain of circles move about the center.

Construct an animation illustrating Steiner's Porism: For any two (nonconcentric)
circles one inside another, if circles are drawn successively touching
them and one another so the last one touches the first, then it will always
happen whatever the position of the first circle.

Investigate the various properties associated with Steiner's Porism:

Investigate the various properties in the configuration of Steiner's porism
for the chain of three circles.

Investigate the various properties in the configuration of Steiner's
porism for the chain of four circles.

Construct an animation displaying a "star" inscribed in an ellipse moving
while the intersections of the diagonals lie on another fixed ellipse with
major axis perpendicular to that of the first one.

Construct the Poncelet's Porism in Poincare's Model.