An Investigation of the Butterfly Problem
Jackie Ruff

We are going to consider a problem that looks deceptively simple.  After working on it for quite a while, I looked for guidance.  I found an outline of a proof in an old volume of The Mathematical Magazine*.  Even with the outline it seemed an educational endeavor to fill in the steps until the proof made sense to me.

We are given chord PQ with midpoint at M in circle O.  Two other chords AB and CD intersect this chord at its midpoint.  Lines AD and CB intersect PQ at points X and Y respectively.  The task is to prove that XM = YM.  

You may click here for a Geometer's Sketchpad file so that you can move points A and/or C to convince yourself of the truth of this statement. The proof with accompanying diagrams is given below.

*Bankoff, Leon. The Metamorphosis of the Butterfly Problem.
The Mathematics Magazine 1987;60(4):195-210.