Another synthetic proof of the butterfly theorem using Pascal theorem

This article give a new synthetic proof of the butterfly theorem, based on the use of Pascal and Thales theorem.. Butterfly theorem.[r]

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Nguyen Dang Khoa April 17, 2020

Abstract

This article give a new synthetic proof of the butterfly theorem, based on the use of Pascal and Thales theorem

Butterfly theorem Let M be the midpoint of a chord AB of a circle (O) Through M two other chords CD and EF are drawn If C and F are on opposite sides of AB, and CF, DE intersect AB at G and H respectively, then M is also the midpoint of GH

Proof We have two cases of this theorem First case The lineCEis parallel toDF

IfCE kFD then it is easy to see thatCEkFDkAB So we have GM CE = FG

FC = DH DE =

HM

CE and we observeMG=MH

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Second case The lineCE is not parallel toDF In this case we easy to show thatABalso is not parallel toCEorFD, otherwise we come back to case one

Now we take pointK,Lon(O)such thatEKkABkFLthen we haveK6=C, L6=DandK,M,Lare collinear

EK intersectsCF atU,FLintersectsEDatV andABcutsKF,ELatP,Q, respectively

From first case we get thatMP=MQ And by Pascal theorem for

K D F

C L E

then we have three pointU,M,V are collinear From this, by Thales theorem we have MH

MQ = FV FL =

U E EK =

MG MP

SinceMP=MQthen we getMG=MH, as desired

References

[1] A Bogomolny, Butterfly theorem,Interactive Mathematics Miscellany and

Puz-zles,

http://www.cut-the-knot.org/pythagoras/Butterfly.shtml

[2] M Celli, A proof of the butterfly theorem using the similarity factor of the two

wings,Forum Geom., 16 (2016) 337–338

[3] C Donolato, A proof of the butterfly theorem using Ceva’s theorem,Forum Geom.,

16 (2016) 185–186

[4] Q.H Tran, Another synthetic proof of the butterfly theorem using the midline in

trian,Forum Geom., 16 (2016) 345–346

Nguyen Dang Khoa: Hung Vuong high school for Gifted students, Phu Tho, Viet Nam

E-mail address:khoanguyen17112003@gmail.com