Parallel Lines With A Circle Between Them

Steps

1. Draw a line perpendicular to parallel lines. Name the intersection points A and B.
2. Create an axis to line segment AB. Name the intersection point C. The centers of the resulting circles will lie on the axis because it is equidistant from both parallels.
3. From the center the given circle construct another circle (d). This circle will have the radius of the given circle, to which we'll add the distance between A and C (or B and C, the distance is the same). Because we know, that the resulting circle will have the radius AC, we draw this circle to obtain points with a given distance from the given circle.
4. The intersections of the axis of the line segment AC and the circle d are the centres of the two resulting circles. From these, draw the circles k1 and k2. These will have radius AC.
5. Just as we created the circle d by adding the radius of the given circle and the distance AC, we should find the other two intersections by subtracting AC from the radius of the given circle. But the result is negative, so we multiply it by -1 to get the radius of circle e. This, like circle d, will start from point D. Name the intersections s3 and s4.
6. The intersections of the axis of line AB and circle e are the remaining two centres of the resulting circles. Draw circles k3 and k4 with radius AC starting from points s3 and s4. This gives the remaining two results.

Steps

0. We are given two parallel lines and a circle between them that has no common points with them.
1. The problem can be solved by translation. The solution must be a circle whose diameter will be the distance of the two lines. We first find centre of the translated circle - any point whose distance from the two lines is the same.
2. Draw the circle we are looking for.
3. Centres of the solutions will lie on the axis of the two given lines, draw it.
4. Two centres of the solutions will lie the |radius of the given circle + the radius of the solution circle| away from the centre of the given circle. The other two centres will lie |radius of the solution - radius of the given| away from the centre of the assignment. To take these values into the compass, we draw the given circle concentric with the circle from the second step.
5. Pick up the radius of the solution circle + the assignment circle. Draw a circle cantered on the centre of the given circle and find the intersection with the axis of the given lines. These are the centres of the first two solutions
6. The centres and radius of the first two solutions are know known. Draw the first two solutions.
7. Pick up the radius of the solution circle - the circle of the problem. Draw a circle with the centre at the centre of the given circle and find the intersection with the axis of the given lines. These are the centres of the other two solutions
8. The centres and radius of the second two solutions are know known. Draw the first two solutions.