Butterfly Theorem
Let be the midpoint of chord of a circle, through which two other chords and are drawn. and intersect chord at and , respectively. The Butterfly Theorem states that is the midpoint of .
Proof
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Link to a good proof :
http://agutie.homestead.com/FiLEs/GeometryButterfly.html
Also another nice proof by Darij Grinberg can be found here:
http://www.cip.ifi.lmu.de/~grinberg/Butterfly.zip