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More Concyclic Points (Posted on 2014-01-26) Difficulty: 4 of 5

  
Prove the following:

If  Γ1, Γ2, Γ3, and Γ4 are circles or straight lines ( see Note below )
such that

     Γ1∩Γ2 = {A,K},   Γ2∩Γ3 = {B,L},   Γ3∩Γ4 = {C,M}, and
     Γ4∩Γ1 = {D,N}

then

     the points A, B, C, and D are concyclic if and only if
     the points K, L, M, and N are concyclic.

Note: At most only one of the points in {A,B,C,D} is the intersection
          of two straight lines. In that case the corresponding point in
          {K,L,M,N} is the point at infinity ( ∞ ).

For extra credit, use this theorem to prove the theorem in
On the same line.
  

See The Solution Submitted by Bractals    
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