Number of circles touching 3 parallel lines

Question

I came across this question in my problem book where equations of 3 lines were given which were parallel and we were asked the number of circles touching all these 3 lines . I think the answer should be 0 but solution given in book says there will be 4 circles. Please help me

Answer 1

From the way I'm hearing it, it sounds like there should be four classes of circles satisfying this property. If $L_1, L_2, L_3$ are parallel lines and $L_2$ lies between $L_1, L_3$ then the collection $\{\{s\cong S^1|\forall i=1,2,3: |s\cap L_i|=f(i)\}|\forall j=1,3: f(j)\in \{1,2\}, f(2)=2\}$ has cardinality 4. A circle must intersect $L_2$ twice, but can intersect $L_1$ and $L_3$ once or twice. This gives four possibilities.

A circle can be attached to the side lines by one point and cross the middle line twice on opposite sides of the circle this defines one class of circles. Another class is of circles that are larger than these and are cut by all three lines so that the circle intersects each line twice. There are two other similar classes where the circle intersects one side line once and the other two lines twice.