Simple Proof of the Butterfly Theorem

I “discovered” a simple proof of the butterfly theorem in classical geometry in 2012 or so.

Statement

Let M be the midpoint of a chord PQ of a conic, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly. Then M is the midpoint of XY. ¹

Proof

Let Y’ be a reflection of X through M. Then \((PXMQ)\bar{\bar{\wedge}}(PDBQ)\bar{\bar{\wedge}}(PMYQ)\bar{\wedge}(PRMQ)\). Therefore, Y’ is equal to Y.

1. https://en.wikipedia.org/wiki/Butterfly_theorem ↩