Geometry Problem 804, Step by step Illustration

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## Saturday, September 15, 2012

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## Saturday, September 15, 2012

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Butterfly Theorem Proof

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Online Geometry theorems, problems, solutions, and related topics.

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From perspective of G, cross ratio R(A,F,E,B)=R(A,C,N,B)

ReplyDeleteFrom perspective of D, cross ratio R(A,F,E,B)=R(A,M,C,B)

Hence, (AB*NC)/(AN*BC) = (AB*MC)/(AC*MB)

=> NC / AN = MC / MB

=> MC=NC

qed

To W Fung

DeleteI note that your solution never use or require points A,B,D,E,F,G located on the same circle. Does that mean that problem gives extra information or I may miss something here ?. I appreciate if you can give all of us details explanation.

Peter Tran

To peter tran,

ReplyDeleteFrom my solution, "From perspective of G, cross ratio R(A,F,E,B)=R(A,C,N,B)"

The perspectivity holds when G lies on the circle of (AFEB). This should be the properties of the perspectivity. (the perspectivity is preserved under inversion)

For more information, please see:

http://www.imomath.com/index.php?options=331

W Fung

Thank you for the explanation.

ReplyDeletePeter Tran