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