Butterfly theorem

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 $PM = MQ$ .

So

Cross-multiplying in the latter equation,

Cancelling the common term

from both sides of the resulting equation yields

hence $MX = MY$ , since MX, MY, and PM are all positive, real numbers.

Thus, $M$ is the midpoint of $XY$ .

Other proofs 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.