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:^[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

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

\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, it can be easily 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 $PM = MQ$ .

Now,

{(MX)^{2} \over (MY)^{2}}={(PM)^{2}-(MX)^{2} \over (PM)^{2}-(MY)^{2}}.

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

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

↑ 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
↑ , problem 8.
↑ William Wallace's 1803 Statement of the Butterfly Theorem, cut-the-knot, retrieved 2015-05-07.

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.

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