intersect at point . In this case coordinates are introduced, such that (see diagram). have the coordinates . From the parallelity of the lines one gets and the parallelity of the lines yields . Hence line has slope and is parallel line . Case 2: (little theorem). In this case the coordinates are chosen such that . From the parallelity of and one gets and , respectively, and at least the parallelity . Proof with homogeneous coordinates Choose homogeneous coordinates with . On the lines , given by , take the points to be for some . The three lines are , so they pass through the same point if and only if . The condition for the three lines and with equations to pass through the same point is . So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so . Equivalently, are collinear. The proof above also shows that for Pappus's theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a (commutative) field. German mathematician Gerhard Hessenberg proved that Pappus's theorem implies Desargues's theorem. In general, Pappus's theorem holds for some projective plane if and only if it is a projective plane over a commutative field. The projective planes in which Pappus's theorem does not hold are Desarguesian projective planes over noncommutative division rings, and non-Desarguesian planes. The proof is invalid if happen to be collinear. In that case an alternative proof can be provided, for example, using a different projective reference. Dual theorem Because of the principle of duality for projective planes the dual theorem of Pappus is true: If 6 lines are chosen alternately from two pencils with centers , the lines are concurrent, that means: they have a point in common. The left diagram shows the projective version, the right one an affine version, where the points are points at infinity. If point is on the line than one gets the "dual little theorem" of Pappus' theorem. If in the affine version of the dual "little theorem" point is a point at infinity too, one gets Thomsen's theorem, a statement on 6 points on the sides of a triangle (see diagram). The Thomsen figure plays an essential role coordinatising an axiomatic defined projective plane. The proof of the closure of Thomsen's