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
This simple proof uses projective geometry.
First we note that Therefore, Since , Moreover, so as desired. .
Related Reading
http://agutie.homestead.com/FiLEs/GeometryButterfly.html
http://www.mathematik.uni-muenchen.de/~fritsch/butterfly.pdf