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Another important theorem in both euclidean and Projective Geometry is the Theorem of Desargues. There are two parts of this much used theorem.
Desargues' Theorem (I). Suppose that A, B and C are distinct noncollinear points with
l (A, A') || l (B, B') || l (C,C') and l (A,B) || l (A', B') and l (A,C) || l (A', C'). Then l (B,C) || l (B',C').
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Desargues' Theorem (II). Suppose that A, B and C are distinct noncollinear points with  and l (A,B) || l(A',B') and l(A,C) || l(A',C'). Then l(B,C) || l(B',C')
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PROBLEM 23: Give a proof of part I and II of Desargues Theorem for the euclidean plane. Again, you'll need to use some simple facts about similar figures.
Solution
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References: Batten, Bennett, Beutelspacher & Rosenbaum, Polster
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