Investigation of The Butterfly Problem


Nathan Wisdom

Through the midpoint M of any chord PQ of a circle, any chords AB and CD are drawn; lines AD and BC meet PQ at points X and Y.

Prove that M is the midpoint of XY.

This problem is known as The butterfly problem or The butterfly Theorem because of :

But do not let the looks fool you. It is a difficult problem!!!

OPEN GSP File (Page 3) for the Proof

Here is the Crux of the proof: Use the Inscribed Angle Theorem to get congruent angles, then prove that the two yellow triangles are congruent and we are done.