[ Plane, dissected by lines ]

a) Find the number of all resulting figures.

b) Find the number of bounded figures, i.e., polygons.

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a) Let $n$ lines divide the plane into $a_{\mathrm{n}}$ parts. If we draw one more line, the number of parts will increase by $n+1$, since the new line intersects the existing lines at $n$ points. Therefore, $a_{\mathrm{n}} + 1 = a_{\mathrm{n}} + n + 1$. Since $a_{1} = 2$, we have $a_{\mathrm{n}} = 2 + 2 + 3 + \ldots + n = (n^2 + n + 2) / 2$.

b) By enclosing all the intersection points of the given lines in a circle, it is easy to verify that the number of unbounded figures is $2n$. Therefore, the number of bounded figures is $(n^2 + n + 2) / 2 - 2n = (n^2 - 3n + 2) / 2$.

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