# 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.

Inhalt 1 Nachweisen 2 Geschichte 3 Verweise 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. Ähnlich, let YY′ and YY″ be dropped from the point Y perpendicular to the straight lines BM and CM respectively.

Seit {displaystyle triangle MXX'sim triangle MYY',} {Anzeigestil {MX over MY}={XX' over YY'},} {displaystyle triangle MXX''sim triangle MYY'',} {Anzeigestil {MX over MY}={XX'' over YY''},} {displaystyle triangle AXX'sim triangle CYY'',} {Anzeigestil {XX' over YY''}={AX over CY},} {displaystyle triangle DXX''sim triangle BYY',} {Anzeigestil {XX'' over YY'}={DX over BY}.} From the preceding equations and the intersecting chords theorem, it can be seen that {Anzeigestil links({MX over MY}Rechts)^{2}={XX' over YY'}{XX'' over YY''},} {Anzeigestil {}={AXcdot DX over CYcdot BY},} {Anzeigestil {}={PXcdot QX over PYcdot QY},} {Anzeigestil {}={(PM-XM)cdot (MQ+XM) Über (PM+MY)cdot (QM-MY)},} {Anzeigestil {}={(PM)^{2}-(MX)^{2} Über (PM)^{2}-(MY)^{2}},} since PM = MQ.

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

Daher, 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, und 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., Fortgeschrittene euklidische Geometrie, Dover Publ., 2007 (orig. 1929). ^ Martin Celli, "A Proof of the Butterfly Theorem Using the Similarity Factor of the Two Wings", Geometrisches Forum 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, abgerufen 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, das Wolfram-Demonstrationsprojekt. Weißstein, Erich W. "Butterfly Theorem". MathWorld. Kategorien: Euclidean plane geometryTheorems about circles

Wenn Sie andere ähnliche Artikel wissen möchten Butterfly theorem Sie können die Kategorie besuchen Euclidean plane geometry.

Geh hinauf

Wir verwenden eigene Cookies und Cookies von Drittanbietern, um die Benutzererfahrung zu verbessern Mehr Informationen