Arrangements of lines with a large number of triangles

An arrangement of lines is constructed by choosing n diagonals of the regular 2/i-gon. This arrangement is proved to form at least n(n - 3)/3 triangular cells. FI. The number of lines in j/is denoted by n(s/). If no point belongs to more than two of these lines Li, the arrangement is called simple. With an arrangement si there is associated a 2-dimensional cell-complex into which the lines of .s/decompose Y\. It is well known that in a simple arrangement si the number of cells (or polygons) of that complex is (n2 — n + 2)/2 (w = n(s/)). We shall denote by Pj(si) the number of y'-gons among the cells of (the complex associated with) si. 2. Constructions. Let us denote by P{O) a fixed point on the circle #with centre C. For any real a, let P(a) be the point obtained by rotating P(O) around C, with angle a. Further denote by L(a) the straight hne P(a) P(tr - 2a). In case a = it - 2a (mod 2it), L(a) is the line tangent to #at P(a). (See Figure 1.)

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