Diverging Lines With Their Intersect On A Circle, Both Intersecting The Circle

Steps

1. Choose the circle of inversion, placing its center at the intersection point of the lines and the circle.
2. Apply circular inversion to the given objects. The lines remain invariant, and the circle is mapped to a line. Through inversion, the problem is transformed into finding circles tangent to three lines.
3. Solve the problem for three non-parallel lines. Use the method of loci of points with given properties.
4. Invert the found circles back using circular inversion.
5. The problem has four solutions.

Steps

1. The centers of the solution circles must lie on the angle bisectors formed by the intersecting lines.
2. The centers of the solution circles must also lie on a pair of parabolas that contain the centers of all circles tangent to the given circle and one of the given lines. The focus of both parabolas is the center of the circle. The directrix lines are parallel to the given line, and the distance between the parallels equals the radius of the given circle.
3. The centers of the solution circles are located at the intersections of the angle bisectors and the parabolas.
4. The problem has four solutions.

Steps

1. Draw perpendiculars to the given lines passing through the centre of the given circle and mark the four intersections.
2. Draw four lines parallel to the given lines passing through the intersections from the previous step.
3. Draw four new intersections of these lines and draw the axis of the angle between the given lines.
4. Take two opposite intersections of these lines and draw two lines that always pass through one of the intersections and through the intersection of the given lines. Plot the intersections of these new lines with the given circle. Draw two lines, each passing through one of the new intersections and through the center of the given circle.
5. Plot the intersections of these new lines with the axis of the angle of the given lines. These are the two centers of the circles we are looking for. The intersections from step 4 serve as points for the circle.
6. Plot the second axis of angle of the given lines and repeat the process with the remaining two intersections from step 3.
7. Draw two lines that always pass through one of the intersections and the intersection of the specified lines. Plot the intersections of these new lines with the specified circle.
8. Draw two lines, each passing through one of the new intersections and through the center of the given circle. Plot the intersections of these new lines with the axis of the angle of the given lines. These are the two other centers of the circles we are looking for. The intersections from step 7 serve as points for the circle
9. Construct the last two search circles given a center and a point.