Three Circles Intersecting In A Common Point

Steps

0. Three circles intersecting at one point are given.
1. Draw a circle ω centered at the intersection of the circles P, which will be the control circle of the circular inverse we will use to solve the problem.
2. In this circular inversion, the point P appears at infinity, the images of the circles k₁, k₂ and k₃ are the lines p₁, p₂ and p₃. The images of the solutions sought will be the circles tangent to these lines.
3. The centers of the images of the solution circles are the intersections of the axes of the angles of the lines p₁, p₂, p₃.
4. The images of the solution circles are the circles k₄', k₅', k₆', and k₇' that touch the lines p₁, p₂, and p₃.
5. Show the circles k₄', k₅', k₆' and k₇' in circular inversion. This gives the solution circles.
6. The problem has a total of four solutions.

Steps

1. Draw three circles that intersect at one point.
2. Construct three hyperbolas whose foci are the individual circles and which pass through their intersection.
3. Construct two ellipses whose foci are the centers of the given circles and which pass through their intersection (can be done with any of the given circles).
4. The intersection of the three hyperbolas (which is not the intersection of the circles at the same time) is one solution, the other three interior ones always have their centers at the intersection of the ellipse and the hyperbola.