Butterfly theorem

For the "butterfly lemma" of group theory, see Zassenhaus lemma.

File:Butterfly theorem.svg

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

File:Butterfly1.svg

Proof of Butterfly theorem

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

$\mathrm{△}MX{X}^{\prime }\sim \mathrm{△}MY{Y}^{\prime },$

$\frac{MX}{MY}=\frac{X{X}^{\prime }}{Y{Y}^{\prime }},$

$\mathrm{△}MX{X}^{″}\sim \mathrm{△}MY{Y}^{″},$

$\frac{MX}{MY}=\frac{X{X}^{″}}{Y{Y}^{″}},$

$\mathrm{△}AX{X}^{\prime }\sim \mathrm{△}CY{Y}^{″},$

$\frac{X{X}^{\prime }}{Y{Y}^{″}}=\frac{AX}{CY},$

$\mathrm{△}DX{X}^{″}\sim \mathrm{△}BY{Y}^{\prime },$

$\frac{X{X}^{″}}{Y{Y}^{\prime }}=\frac{DX}{BY}.$

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

${\left(\frac{MX}{MY}\right)}^2=\frac{X{X}^{\prime }}{Y{Y}^{\prime }}\frac{X{X}^{″}}{Y{Y}^{″}},$

$=\frac{AX\cdot DX}{CY\cdot BY},$

$=\frac{PX\cdot QX}{PY\cdot QY},$

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

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

since PM = MQ.

So,

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

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

External links

• The Butterfly Theorem at cut-the-knot
• A Better Butterfly Theorem at cut-the-knot
• Proof of Butterfly Theorem at PlanetMath
• The Butterfly Theorem by Jay Warendorff, the Wolfram Demonstrations Project.