This article give a new synthetic proof of the butterfly theorem, based on the use of Pascal and Thales theorem.. Butterfly theorem.[r]
(1)Another synthetic proof of the butterfly theorem using Pascal theorem
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
(2)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
http://www.cut-the-knot.org/pythagoras/Butterfly.shtml