Butterfly theorem

The butterfly theorem is a classical result in Euclidean geometry, 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.

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.

Now, 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, it can be easily seen that

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

${} = {AX.DX \over CY.BY},$

${} = {PX.QX \over PY.QY},$

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

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

since PM = MQ.

Now,

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

So, it can be concluded that MX = MY, or M is the midpoint of XY.

An alternate proof can be found using projective geometry.^[2]

History

Proving the butterfly theorem was posed as a problem by William Wallace in The Gentlemen'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 Gentlemen's Diary or Mathematical Repository.^[3]

References

1. Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).
2. , problem 8.
3. William Wallace's 1803 Statement of the Butterfly Theorem, cut-the-knot, retrieved 2015-05-07.

Bibliography

H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967.

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.
• Weisstein, Eric W., "Butterfly Theorem", MathWorld.