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]

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 line CE is parallel to DF

If CE k FD then it is easy to see that CE k FD k AB So we have GM

CE = FG

FC =DH

DE =HM

CE and we observe MG = MH

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Second case The line CE is not parallel to DF In this case we easy to show that AB also

is not parallel to CE or FD, otherwise we come back to case one

Now we take point K, L on (O) such that EK k AB k FL then we have K 6= C,

L6= D and K, M, L are collinear

EK intersects CF at U , FL intersects ED at V and AB cuts KF, EL at P, Q, respectively

From first case we get that MP = MQ And by Pascal theorem forK D F

C L E

 then we have three point U, M,V are collinear

From this, by Thales theorem we have MH

MQ =FV

FL =U E

EK =MG

MP

Since MP = MQ then we get MG = 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

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