Butterfly Theorem

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Let $D$ be the midpoint of chord $BC$ of a circle, through which two other chords $EH$ and $FG$ are drawn. $EG$ and $HF$ intersect chord $BC$ at $I$ and $J$, respectively. The Butterfly Theorem states that $D$ is the midpoint of $IJ$.

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


See also

Midpoint