The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows: Let M be the midpoint of a chord PQ of a circle, 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 .

Proof

A formal proof of the theorem is as follows: Let the perpendiculars XX′ and XX″ be dropped from the point X on the straight lines AM and DM respectively. Similarly, let YY′ and YY″ be dropped from the point Y perpendicular to the straight lines BM and CM respectively. Since From the preceding equations and the intersecting chords theorem, it can be seen that since . So, Cross-multiplying in the latter equation, Cancelling the common term from both sides of the resulting equation yields hence , since MX, MY, and PM are all positive, real numbers. Thus, M is the midpoint of XY . Other proofs too exist, including one using projective geometry.

History

Proving the butterfly theorem was posed as a problem by William Wallace in The Gentleman's Mathematical Companion (1803). Three solutions were published in 1804, and in 1805 Sir William Herschel posed the question again in a letter to Wallace. Rev. Thomas Scurr asked the same question again in 1814 in the Gentleman's Diary or Mathematical Repository.