Triangles in arrangements of lines

A set of n lines in the projective plane divides the plane into a certain number of polygonal cells. We show that if we insist that all of these cells be triangles, then there are at most $\text{1}\text{3}\text{n(n ? 1) + 4 ?}\text{2}\text{7}\text{n}$ of them. We also observe that if no point of the plane belongs to more than two of the lines and n is at least 4, some of the cells must be either quadrangles or pentagons. We further show that for $\text{n ≧ 4}$, there is a set of n lines which divides the plane into at least $\text{1}\text{3}\text{n(n ? 3) + 4}$ triangles.