Butterfly II

From a "mina_world" problem on sci.math

Given a circle (C) centered in O, a point P outside the circle, and a line (L) at P perpendicular to OP.
Draw any two secants PCD and PEF to the circle.
Line DE intersects line (L) in A, and line CF intersects (L) in B.
Prove that P is always the midpoint of AB Hint

Details (includes a JavaSketchpad applet, patience when loading the page the first time)

And the butterfly ??
Consider a similar problem when P is inside the circle and then ... The Butterfly theorem

 

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