The inverse in a circle

In this worksheet the point P is free to move on the circle through A and B.

The point P' is on the line OP. The distance from O is 1/OP. Hence the position of P' is often described as the "inverse in a circle" of P.

Move A, B and then move P around the circle. Can you find a configuration in which P' moves along an exact straight line?

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Chris Sangwin, 23 Dec 2006, Created with GeoGebra