Saturday, September 15, 2012

Butterfly Theorem Proof

Geometry Problem 804, Step by step Illustration
Click the figure to see the interactive illustration

Butterfly Theorem Proof.

Posted by Antonio Gutierrez at 12:15 PM
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4 comments:

  1. W fungSeptember 15, 2012 at 5:23 PM

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

    ReplyDelete
    Replies
    1. Peter TranSeptember 17, 2012 at 4:31 PM

      To W Fung
      I 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

      Delete
  2. W FungSeptember 17, 2012 at 10:24 PM

    To peter tran,

    From 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

    ReplyDelete
  3. Peter TranSeptember 19, 2012 at 12:00 AM

    Thank you for the explanation.

    Peter Tran

    ReplyDelete

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