What is the maximum number of points of intersection of two different lines and three different circles in the same plane?

Answer 1

the three circles can cut each other in 6 different points (think of a typical Venn diagram)
A line can cut a circle in 2 point
so each line can cut all the circles in 6 points

so total = 6 + 6 + 6 = 18
BUT, let's not forget that the two lines can also intersect
so the total is 19