Butterfly theorem

In Euclidean geometry, the butterfly theorem is a classical result which can be stated as follows:^{1: p. 78}

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.

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.

$${\displaystyle {\begin{aligned}\triangle MXX'&\sim \triangle MYY',\\[2pt]\therefore {{\overline {MX}} \over {\overline {MY}}}&={{\overline {XX'}} \over {\overline {YY'}}}.\\[8pt]\triangle MXX''&\sim \triangle MYY'',\\[2pt]\therefore {{\overline {MX}} \over {\overline {MY}}}&={{\overline {XX''}} \over {\overline {YY''}}}.\\[8pt]\triangle AXX'&\sim \triangle CYY'',\\[2pt]\therefore {{\overline {XX'}} \over {\overline {YY''}}}&={{\overline {AX}} \over {\overline {CY}}}.\\[8pt]\triangle DXX''&\sim \triangle BYY',\\[2pt]\therefore {{\overline {XX''}} \over {\overline {YY'}}}&={{\overline {DX}} \over {\overline {BY}}}.\end{aligned}}}$$

From the preceding equations and the intersecting chords theorem, it can be seen that

$${\displaystyle {\begin{aligned}\left({{\overline {MX}} \over {\overline {MY}}}\right)^{2}&={{\overline {XX'}} \over {\overline {YY'}}}\cdot {{\overline {XX''}} \over {\overline {YY''}}},\\[2pt]&={{\overline {AX}}\cdot {\overline {DX}} \over {\overline {CY}}\cdot {\overline {BY}}},\\[2pt]&={{\overline {PX}}\cdot {\overline {QX}} \over {\overline {PY}}\cdot {\overline {QY}}},\\[2pt]&={{\bigl (}{\overline {PM}}-{\overline {XM}}{\bigr )}\cdot {\bigl (}{\overline {MQ}}+{\overline {XM}}{\bigr )} \over {\bigl (}{\overline {PM}}+{\overline {MY}}{\bigr )}\cdot {\bigl (}{\overline {QM}}-{\overline {MY}}{\bigr )}},\\[2pt]&={{\overline {PM}}^{2}-{\overline {MX}}^{2} \over {\overline {PM}}^{2}-{\overline {MY}}^{2}},\end{aligned}}}$$

since PM = MQ. Thus,

$${\displaystyle {{\overline {MX}}^{2} \over {\overline {MY}}^{2}}={{\overline {PM}}^{2}-{\overline {MX}}^{2} \over {\overline {PM}}^{2}-{\overline {MY}}^{2}}.}$$

Cross-multiplying the latter equation and cancelling out common terms,

$${\displaystyle {\begin{aligned}{\overline {MX}}^{2}\cdot {\overline {PM}}^{2}-{\overline {MX}}^{2}\cdot {\overline {MY}}^{2}&={\overline {MY}}^{2}\cdot {\overline {PM}}^{2}-{\overline {MX}}^{2}\cdot {\overline {MY}}^{2},\\[4pt]\Rightarrow \quad {\overline {MX}}^{2}\cdot {\overline {PM}}^{2}&={\overline {MY}}^{2}\cdot {\overline {PM}}^{2},\\[4pt]\Rightarrow \quad {\overline {MX}}^{2}&={\overline {MY}}^{2},\\[4pt]\Rightarrow \quad {\overline {MX}}&={\overline {MY}}.\end{aligned}}}$$

Thus, M is the midpoint of XY.

Other proofs exist,² including one using projective geometry.³

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. Reverend Thomas Scurr asked the same question again in 1814 in the Gentleman's Diary or Mathematical Repository.⁴