11 Combinatorics in Geometry
11.1 Intersections of Diagonals
At first you may be surprised: What is the connection between combi-
natorics and geometry? There are many geometric questions that can be
solved by combinatorial methods, but the opposite case may also occur: We
can solve combinatorial exercises and problems using geometrical tools.
Consider a convex polygon withnvertices. (We call a polygonconvexif
every angle of it is convex, i.e., less than 180◦.) Assume that it has no 3
diagonals going through the same point. How many intersection points do
the diagonals have? (The vertices are not counted as intersections, and we
do not consider intersections of diagonals outside then-gon. In Figure 11.1,
the black point is a “good” intersection, so it is counted, but the circled
point outside the polygon is not counted.)
FIGURE 11.1.