1. Classify the 20 lines

There are two groups:

• Odd-indexed lines: $L_1, L_3, \dots, L_{19}$

  These are $10$ distinct lines, all parallel to each other.

• Even-indexed lines: $L_2, L_4, \dots, L_{20}$

  These are $10$ distinct lines, all passing through a fixed point $P$.

We want the maximum number of distinct intersection points of pairs of lines.

2. Intersections among odd-indexed lines

Since all $10$ odd-indexed lines are parallel, no two of them intersect.

So, number of intersection points from this group: $0$

3. Intersections among even-indexed lines

All $10$ even-indexed lines pass through the same point $P$.

Any pair of these lines intersects at $P$, but this gives only one distinct intersection point.

So, number of distinct intersection points from this group: $1$

4. Intersections between an odd-indexed line and an even-indexed line

Each odd-indexed line is parallel to the other odd-indexed lines, and each even-indexed line passes through $P$.

To maximize the number of distinct intersection points:

• none of the odd-indexed lines should pass through $P$,
• no even-indexed line should be parallel to the odd family.

Then each odd line intersects each even line in exactly one point.

Number of such pairs: $10 \times 10 = 100$

Now check whether these $100$ intersection points are distinct.

• For a fixed even line, its intersections with the $10$ parallel odd lines are distinct.
• For two different even lines, their intersection with a given odd line are different because the even lines meet only at $P$, and we ensured the odd lines do not pass through $P$.

Hence all these $100$ points can be made distinct.

So, number of distinct intersection points from mixed pairs: $100$

5. Total maximum number of distinct intersection points

Adding all contributions: $0 + 1 + 100 = 101$

6. Final Answer

The maximum number of points of intersection is $\boxed{101}$