Two Circles Inside Each Other, Line Tangent To The Outer Circle

Steps

2. Since the line is tangent to the outer circle k₁, the circles we are looking for must touch at their tangent point. We find this by forming a perpendicular to the given line p₁, which also passes through the centre C of the circle k₁. Let us call this tangent point F. The centres of the circles we are looking for must lie somewhere on this perpendicular line.
3. Prepare a circle for circular inversion. It can have any radius, the important thing is that it has a center at point F. Let's call it circle e.
4. Make a circular inversion for the inner given circle k₂.
5. Since the circle over which we do the circular inversion has its center at point F, we send point F to infinity. But since we are looking for something to touch point F and the inverted circle, this something must pass through infinity and also be tangent to the inverted circle. We also want this something to touch the given line p₁ at only one point, point F. But this means that even in the inverted image, this something must touch the inverted line p₁ at one point. Since this one location is supposed to be point F, which is at infinity, this something must be a line parallel to the image of the line p₁.
6. Construct two parallel lines to the line p₁ that are also tangents to the inverted circle k₂.
7. We invert the parallel lines through the circle e back. This gives the two resulting circles.

Steps

0. are given two circles inside each other and a line that is tangent to the larger circle. The circles of the solution must intersect the tangent point of line f and circle c. So we can simplify the problem to point circle circle. We will solve the problem using non-Euclideanly methods. Firstly, we will find a set of points that is equidistant from two of the three given objects, and at the end we will find their intersection.
1. Circle within a circle: Draw a line through the centers of the circles and find its intersections with the circles. Then find the midpoint between the closer intersections (I and G or F and H) and the midpoint between the more distant intersections (G and H or F and I)
2. Draw ellipses with foci at the centers of the circles and intersecting the midpoints. These ellipses are the set of all points equidistant from the two given circles.
3. A point lying on a circle: Draw a line passing through point B and the center of circle c. This line is the set of all points equidistant from point B and circle c.
4. circle and a point lying outside it: draw a line i joining the centre of the circle d and point B.
5. Find the intersection of line i and circle d. Find the centre N between this intersection and point B.
6. Draw a hyperbola with foci at point B and at the centre of circle d and passing through point N. This hyperbola is the set of all points equidistant from point B and circle d.
7. The centers of the circles that will be the solutions of the problem are the intersections of all three sets. We find these intersections and name them S₁ and S₂.
8. Draw circles with centers at S₁ and S₂ passing through point B.