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
 * $$ \triangle MXX' \sim \triangle MYY',$$
 * $$ {MX \over MY} = {XX' \over YY'}, $$


 * $$ \triangle MXX \sim \triangle MYY,$$
 * $$ {MX \over MY} = {XX \over YY}, $$


 * $$ \triangle AXX' \sim \triangle CYY'',$$
 * $$ {XX' \over YY''} = {AX \over CY}, $$


 * $$ \triangle DXX'' \sim \triangle BYY',$$
 * $$ {XX'' \over YY'} = {DX \over BY}. $$

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


 * $$ \left({MX \over MY}\right)^2 = {XX' \over YY' } {XX \over YY}, $$


 * $$ {} = {AX \cdot DX \over CY \cdot BY}, $$


 * $$ {} = {PX \cdot QX \over PY \cdot QY}, $$


 * $$ {} = {(PM-XM) \cdot (MQ+XM) \over (PM+MY) \cdot (QM-MY)}, $$


 * $$ {} = { (PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}, $$

since $PM = MQ$.

So


 * $$ { (MX)^2 \over (MY)^2} = {(PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}. $$

Cross-multiplying in the latter equation,


 * $$ {(MX)^2 \cdot (PM)^2 - (MX)^2 \cdot (MY)^2} = {(MY)^2 \cdot (PM)^2 - (MX)^2 \cdot (MY)^2} . $$

Cancelling the common term


 * $$ { -(MX)^2 \cdot (MY)^2} $$

from both sides of the resulting equation yields


 * $$ {(MX)^2 \cdot (PM)^2} = {(MY)^2 \cdot (PM)^2}, $$

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.