l1, l2, l3 are three distinct non-concurrent lines in a plane with no two of them parallel. The number of circles for which all of l1, l2, and l3 are tangents is :

To find the number of circles for which all of l1, l2, and l3 are tangents, we can use the following steps:

1. Draw l1, l2, and l3 on a plane, ensuring that they are distinct non-concurrent lines and no two of them are parallel.

2. Take any two lines, say l1 and l2, and find the point of intersection, which we'll call P.

3. Draw a line perpendicular to l1 at point P. This line will be the tangent to a circle.

4. Repeat the same process for l2 and l3, finding the point of intersection and drawing a perpendicular line to each line at their respective points of intersection. These lines will also be tangents to circles.

5. The point where all three perpendicular lines intersect will be the center of the circle that is tangent to all three lines.

6. Therefore, the number of circles for which all of l1, l2, and l3 are tangents is equal to the number of points of intersection of the perpendicular lines.

7. Since l1, l2, and l3 are distinct non-concurrent lines, they can intersect at most at one point each. Therefore, the maximum number of points of intersection is 3.

8. Hence, the number of circles for which all of l1, l2, and l3 are tangents is 3.

Please note that this solution assumes that the lines l1, l2, and l3 are in general position and do not have any special configurations that would result in additional tangents or circles.