Coxeter- Exercise 3.1.1

3.1.1 From 2.13 we know that every two points are incident with exactly one line. Dually we can say 3.11 with the slight change of exactly to at least. However, knowing that we are in the projective plane we would only have one point that is incident to two lines which would be exactly the dual of 2.13. From 3.12 we know that there are four points of which no three are collinear. By 2.13 we can construct lines from the pairs of points. By construction this is a quadrangle. Now there are 6 lines. Every two has one intersection by 3.12. Every line would intersect all of the other lines somewhere by the axioms. Therefore, every line would have at least 5 points. By construction of a quadrangle we know that three of which are district. Now we have that the lines have at least three distinct points.