The Butterfly Theorem - 25 Proofs

According to Coxeter and Greitzer, one of the solutions to the Butterfly theorem was submitted in 1815 by W. G. Horner of Horner's method fame. Most recently a 1805 proof by William Wallace has been discovered in Wallace's family archives. Still, later it was found that Wallace's posting of a more general problem dates to 1803.

The solutions I gathered below may serve as a basis for a discussion of which proofs better clarify the gist of the problem, if at all. Why is the result true? (See the discussions on Concyclic Circumcenters: Dynamic View and A Sequel.)

Theorem

Let $M$ be the midpoint of a chord $PQ$ of a circle, through which two other chords $AB$ and $CD$ are drawn; $AD$ cuts $PQ$ at $X$ and $BC$ cuts $PQ$ at $Y$. 

Prove that $M$ is also the midpoint of $XY$.

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