Butterfly theorem

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.

Contents 1 Proof 2 History 3 References 4 External links 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.

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 the intersecting chords theorem, it can be seen that {displaystyle left({MX over MY}right)^{2}={XX' over YY'}{XX'' over YY''},} {displaystyle {}={AXcdot DX over CYcdot BY},} {displaystyle {}={PXcdot QX over PYcdot 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.

So {displaystyle {(MX)^{2} over (MY)^{2}}={(PM)^{2}-(MX)^{2} over (PM)^{2}-(MY)^{2}}.} Cross-multiplying in the latter equation, {displaystyle {(MX)^{2}cdot (PM)^{2}-(MX)^{2}cdot (MY)^{2}}={(MY)^{2}cdot (PM)^{2}-(MX)^{2}cdot (MY)^{2}}.} Cancelling the common term {displaystyle {-(MX)^{2}cdot (MY)^{2}}} from both sides of the resulting equation yields {displaystyle {(MX)^{2}cdot (PM)^{2}}={(MY)^{2}cdot (PM)^{2}},} hence MX = MY, since MX, MY, and PM are all positive, real numbers.

Thus, 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 ^ [1], problem 8. ^ William Wallace's 1803 Statement of the Butterfly Theorem, cut-the-knot, retrieved 2015-05-07. External links Wikimedia Commons has media related to Butterfly theorem. 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. Categories: Euclidean plane geometryTheorems about circles