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Nine points on a line are inverted in the circle (O,Z) to form nine points on a circle.

The point X between P and Q determines the ratio |QX|/|QP|. This ratio is used to dilate the 9 points to a set of 9 intermediate points, which are linked by blue segments and reflected in segment PQ. When X is animated, the linkage moves so as to represent a "slow motion inversion" of the line into the circle (and of the circle into the line).

For more, see GEOMETRY IN ACTION, Chapter 3.