Butterfly theorem

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

The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:

Butterfly theorem

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

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

$\triangle MXX'\sim \triangle MYY',$

${MX \over MY}={XX' \over YY'},$

$\triangle MXX''\sim \triangle MYY'',$

${MX \over MY}={XX'' \over YY''},$

$\triangle AXX'\sim \triangle CYY'',$

${XX' \over YY''}={AX \over CY},$

$\triangle DXX''\sim \triangle BYY',$

${XX'' \over YY'}={DX \over BY}.$

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 too 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.

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.