Let $\mathcal{L}$ be a finite collection of lines in the plane in general position (no two lines in $\mathcal{L}$ are parallel and no three are concurrent). Consider the open circular discs inscribed in the triangles enclosed by each triple of lines in $\mathcal{L}$. Determine the number of such discs intersected by no line in $\mathcal{L}$, in terms of $|\mathcal{L}|$.
Let $n=|\mathcal{L}|$. We can think that all intersetions are vertices of a planar graph. We have $n-1$ vertices and so $n-2$ edges on each line. So we have $V ={n(n-1)\over 2}$ vertices and $E = n(n-2)$ edges. So the number of bounded faces $F $ (which are all convex polygons) we get from Euler formula $$V - E +F = 1\implies F = {(n-1)(n-2)\over 2}$$ Now, we can always draw a circle in a convex polygon that touches 3 sides of it and does not intersect other side: among all triples of (extended) sides in given polygon take a circle with the smallest radius. So the number of circles is the same as $F $.
Edit: If some three sides determine excircle then we enlarge this circle with a homothety so that this enlarged circle does touch sides which determine new circle as incircle of a triangle.
Any idea?