Butterfly Theorem

Given a Chord $PQ$ of a Circle, draw any other two Chords $AB$ and $CD$ passing through its Midpoint. Call the points where $AD$ and $BC$ meet $PQ$ $X$ and $Y$. Then $M$ is the Midpoint of $XY$.

See also Chord, Circle, Midpoint

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 45-46, 1967.