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