Butterfly theorem

Short description: About the midpoint of a chord of a circle, through which two other chords are drawn

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

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

[math]\displaystyle{ \triangle MXX' \sim \triangle MYY', }[/math]

[math]\displaystyle{ {MX \over MY} = {XX' \over YY'}, }[/math]

[math]\displaystyle{ \triangle MXX'' \sim \triangle MYY'', }[/math]

[math]\displaystyle{ {MX \over MY} = {XX'' \over YY''}, }[/math]

[math]\displaystyle{ \triangle AXX' \sim \triangle CYY'', }[/math]

[math]\displaystyle{ {XX' \over YY''} = {AX \over CY}, }[/math]

[math]\displaystyle{ \triangle DXX'' \sim \triangle BYY', }[/math]

[math]\displaystyle{ {XX'' \over YY'} = {DX \over BY}. }[/math]

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

[math]\displaystyle{ \left({MX \over MY}\right)^2 = {XX' \over YY' } {XX'' \over YY''}, }[/math]

[math]\displaystyle{ {} = {AX \cdot DX \over CY \cdot BY}, }[/math]

[math]\displaystyle{ {} = {PX \cdot QX \over PY \cdot QY}, }[/math]

[math]\displaystyle{ {} = {(PM-XM) \cdot (MQ+XM) \over (PM+MY) \cdot (QM-MY)}, }[/math]

[math]\displaystyle{ {} = { (PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}, }[/math]

since PM = MQ.

So

[math]\displaystyle{ { (MX)^2 \over (MY)^2} = {(PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}. }[/math]

Cross-multiplying in the latter equation,

[math]\displaystyle{ {(MX)^2 \cdot (PM)^2 - (MX)^2 \cdot (MY)^2} = {(MY)^2 \cdot (PM)^2 - (MX)^2 \cdot (MY)^2} . }[/math]

Cancelling the common term

[math]\displaystyle{ { -(MX)^2 \cdot (MY)^2} }[/math]

from both sides of the resulting equation yields

[math]\displaystyle{ {(MX)^2 \cdot (PM)^2} = {(MY)^2 \cdot (PM)^2}, }[/math]

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.

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". http://mathworld.wolfram.com/ButterflyTheorem.html.