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 $|\mathcal{L}|$. 

[i]B. Aronov et al.[/i]
1. **Base Case Verification:**
   - For $|\mathcal{L}| = 1$, there are no triangles formed, hence no discs.
   - For $|\mathcal{L}| = 2$, there are still no triangles formed, hence no discs.
   - For $|\mathcal{L}| = 3$, exactly one triangle is formed by the three lines, and thus one disc is inscribed in this triangle. The formula $\frac{(|\mathcal{L}|-1)(|\mathcal{L}|-2)}{2}$ gives $\frac{(3-1)(3-2)}{2} = 1$, which matches our observation.

2. **Inductive Step:**
   - Assume the formula holds for $|\mathcal{L}| = n$, i.e., the number of discs intersected by no line in $\mathcal{L}$ is $\frac{(n-1)(n-2)}{2}$.
   - Consider $|\mathcal{L}| = n+1$. The $n+1$-th line intersects each of the $n$ lines in $n$ distinct points, creating $n$ segments and two half-lines.
   - Each of these $n$ segments creates a new finite region. Thus, the total number of finite regions formed by $n+1$ lines is:
     \[
     n + \frac{(n-1)(n-2)}{2} = \frac{n(n-1)}{2}
     \]
   - Each of these finite regions contains exactly one disc that is inscribed in a triangle formed by three of its sides.

3. **Lemma:**
   - Inside each convex polygon formed by the lines, there is exactly one disc that is tangent to three of its sides.
   - **Proof of Lemma:**
     - Start with a small circle tangent to one side $e_1$ of the polygon.
     - Homothetically expand the circle until it touches another side $e_2$.
     - Let $A$ be the intersection of the lines determined by $e_1$ and $e_2$. Expand the circle from $A$ until it touches a third side $e_3$.
     - The circle is now tangent to three sides, forming a triangle $\triangle MNP$.
     - If the circle is not the incircle of $\triangle MNP$, it must be an ex-circle. Continue expanding the circle until it touches another side $e_4$.
     - This process continues, increasing the radius of the circle until it must stop, at which point the circle is inscribed in a triangle formed by three sides of the polygon.
     - To prove uniqueness, assume there are two such discs. The larger disc must be inside the triangle formed by the smaller disc, leading to a contradiction. Hence, there is exactly one such disc.

4. **Conclusion:**
   - The total number of discs intersected by no line in $\mathcal{L}$ is given by the formula $\frac{(|\mathcal{L}|-1)(|\mathcal{L}|-2)}{2}$.

The final answer is $\boxed{\frac{(|\mathcal{L}|-1)(|\mathcal{L}|-2)}{2}}$