Butterfly theorem

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

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.

Now, since

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

since PM = MQ.

Now,

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

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

History[edit]

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.[4]

References[edit]

  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.

External links[edit]