An Investigation of the Butterfly Problem
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
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.