Prove the following:

If Γ

_{1}, Γ

_{2}, Γ

_{3}, and Γ

_{4}are circles or straight lines ( see Note below )

such that

Γ

_{1}∩Γ

_{2}= {A,K}, Γ

_{2}∩Γ

_{3}= {B,L}, Γ

_{3}∩Γ

_{4}= {C,M}, and

Γ

_{4}∩Γ

_{1}= {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.