The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:[1]: p. 78
![](http://up.wiki.x.io/wikipedia/commons/thumb/f/fa/Butterfly_theorem.svg/220px-Butterfly_theorem.svg.png)
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
editA 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
editProving 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
edit- ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).
- ^ 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
- ^ [1], problem 8.
- ^ William Wallace's 1803 Statement of the Butterfly Theorem, cut-the-knot, retrieved 2015-05-07.