Let us generalize Problem 20. Suppose we wish to create a geometric structure from k points in the plane and that we consider a set of points to be a line if it contains exactly r points.
(a) How many possible lines are there?
(b) How many lines are parallel to a given line?
(c) How many lines intersect a given line at exactly one point?
Problem 20
We are going to construct a geometric structure from a set of lines using a finite number of points in the plane. Suppose that we have four points a,b,c,d and that we consider a set of points to be a line if it contains exactly two points.
(a) How many possible lines are there?
(b) How many lines are parallel to the line ab?
(c) How many lines intersect the line ab?
(d) Suppose we want a geometric structure with no parallel lines (because they remind us of bottom less pits-see Burkhard
Polster's paper "YEA WHY TRY HER RAW WET HAT"). We might start with the line ab and all lines that intersect it; however, we would need to be sure that no two of these lines are parallel. How many lines can our geometric structure have if no two are parallel? And, which lines can we choose for our structure?
(e) Now suppose further that we want exactly two lines to intersect at a point. Which lines can we choose for our structure?
(f) Try to draw the points and lines of this geometric structure.