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I want to prove Pappus' Theorem using affine combinations. The theorem states that given two lines $l_1$ and $l_2$ in the plane and six points $A_i,B_i,C_i \in l_i (i=1,2)$ show that the points $A_3= B_1 C_2 \cap B_2 C_1$, $B_3 = A_1 C_2 \cap A_2 C_1$ and $C_3= A_1B_2 \cap A_2B_1$ are collinear.

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Most elegant proofs that I know of either use projective geometry or only proof an Euclidean special case of Pappus' Theorem, i.e. one using parallel lines or similar. You have the full projective generality of the theorem, but are still asking for an affine proof. You could simply choose coordinates for some points and let a computer algebra system compute those of the other points and check the collinearity. But some intermediate results might get really large, so I'd not consider this particularly elegant. – MvG Jan 29 '13 at 18:56

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