The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:[1]: p. 78 

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

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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

 
 
 
 
 
 
 
 

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

 
 
 
 
 

since PM = MQ.

So,

 

Cross-multiplying in the latter equation,

 

Cancelling the common term

 

from both sides of the resulting equation yields

 

hence MX = MY, since MX, MY, and PM are all positive, real numbers.

Thus, M is the midpoint of XY.

Other proofs too exist,[2] including one using projective geometry.[3]

History

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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.[4]


References

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  1. ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).
  2. ^ Martin Celli, "A Proof of the Butterfly Theorem Using the Similarity Factor of the Two Wings", Forum Geometricorum 16, 2016, 337–338. http://forumgeom.fau.edu/FG2016volume16/FG201641.pdf
  3. ^ [1], problem 8.
  4. ^ William Wallace's 1803 Statement of the Butterfly Theorem, cut-the-knot, retrieved 2015-05-07.
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  • 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.