figure is covered by the proof for the "little theorem", given above. But there exists a simple direct proof, too: Because the statement of Thomsen's theorem (the closure of the figure) uses only the terms connect, intersect and parallel, the statement is affinely invariant, and one can introduce coordinates such that (see right diagram). The starting point of the sequence of chords is One easily verifies the coordinates of the points given in the diagram, which shows: the last point coincides with the first point. Other statements of the theorem In addition to the above characterizations of Pappus's theorem and its dual, the following are equivalent statements: If the six vertices of a hexagon lie alternately on two lines, then the three points of intersection of pairs of opposite sides are collinear. Arranged in a matrix of nine points (as in the figure and description above) and thought of as evaluating a permanent, if the first two rows and the six "diagonal" triads are collinear, then the third row is collinear. That is, if are lines, then Pappus's theorem states that must be a line. Also, note that the same matrix formulation applies to the dual form of the theorem when etc. are triples of concurrent lines. Given three distinct points on each of two distinct lines, pair each point on one of the lines with one from the other line, then the joins of points not paired will meet in (opposite) pairs at points along a line. If two triangles are perspective in at least two different ways, then they are perspective in three ways. If and are concurrent and and are concurrent, then and are concurrent. Origins In its earliest known form, Pappus's Theorem is Propositions 138, 139, 141, and 143 of Book VII of Pappus's Collection. These are Lemmas XII, XIII, XV, and XVII in the part of Book VII consisting of lemmas to the first of the three books of Euclid's Porisms. The lemmas are proved in terms of what today is known as the cross ratio of four collinear points. Three earlier lemmas are used. The first of these, Lemma III, has the diagram below (which uses Pappus's lettering, with G for Γ, D for Δ, J for Θ, and L for Λ). Here three concurrent straight lines, AB, AG, and AD, are crossed by two lines, JB and JE, which concur at J. Also