# Butterfly theorem

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

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

${\displaystyle \triangle MXX'\sim \triangle MYY',\,}$
${\displaystyle {MX \over MY}={XX' \over YY'},}$
${\displaystyle \triangle MXX''\sim \triangle MYY'',\,}$
${\displaystyle {MX \over MY}={XX'' \over YY''},}$
${\displaystyle \triangle AXX'\sim \triangle CYY'',\,}$
${\displaystyle {XX' \over YY''}={AX \over CY},}$
${\displaystyle \triangle DXX''\sim \triangle BYY',\,}$
${\displaystyle {XX'' \over YY'}={DX \over BY},}$

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

${\displaystyle \left({MX \over MY}\right)^{2}={XX' \over YY'}{XX'' \over YY''},}$
${\displaystyle {}={AX\cdot DX \over CY\cdot BY},}$
${\displaystyle {}={PX\cdot QX \over PY\cdot QY},}$
${\displaystyle {}={(PM-XM)\cdot (MQ+XM) \over (PM+MY)\cdot (QM-MY)},}$
${\displaystyle {}={(PM)^{2}-(MX)^{2} \over (PM)^{2}-(MY)^{2}},}$

since PM = MQ.

Now,

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

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

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.