To solve the problem of finding the number of circles that can touch three straight lines \( L_1, L_2, L_3 \) in a plane, we can follow these steps:
### Step-by-Step Solution:
1. **Understanding the Configuration**:
We start by visualizing the three lines \( L_1, L_2, L_3 \). These lines can be arranged in various ways (intersecting, parallel, etc.), but for the purpose of this problem, we will assume they are not parallel and intersect each other.
**Hint**: Draw three non-parallel lines that intersect each other to visualize the scenario.
2. **Identifying Possible Circles**:
When three lines intersect, they create different regions in the plane. We need to determine how many distinct circles can be drawn that touch all three lines.
**Hint**: Consider the angles formed by the intersections of the lines and how circles can be positioned in relation to these angles.
3. **Positioning the Circles**:
For three lines, there are typically four distinct regions where circles can be placed:
- One circle can be placed in the interior region formed by the intersection of the three lines.
- Three additional circles can be placed in the exterior regions, one in each of the three corners created by the angles between the lines.
**Hint**: Visualize or sketch the circles in each of these regions to ensure they touch all three lines.
4. **Counting the Circles**:
After analyzing the possible placements, we find that:
- 1 circle can be placed in the interior region.
- 3 circles can be placed in the exterior regions.
Therefore, the total number of circles \( n \) that can touch all three lines is \( 1 + 3 = 4 \).
**Hint**: Sum the number of circles from both interior and exterior placements to arrive at the final count.
5. **Conclusion**:
Thus, the value of \( n \), the number of circles that can touch all three lines \( L_1, L_2, L_3 \), is \( 4 \).
**Final Answer**: \( n = 4 \)
