One Circle Inside Another, Third Has Internal Tangency with the Outer and External Tangency with the Inner
Number of solutions: 4
GeoGebra construction
Steps
- We first focus on one pair of tangent circles. The centers of the solution circles lie on a straight line and an ellipse. The line passes through the centers of both circles. The centers of the given circles also serve as the foci of the ellipse. The ellipse passes through the point of tangency of the two circles.
- We now find the loci of centers of circles tangent to the second pair of tangent circles. This time, the centers of the solution circles lie on a straight line and a hyperbola. The line passes through the centers of both circles. The foci of the hyperbola are the centers of the given circles, and the hyperbola passes through their point of tangency.
- The centers of the solution circles lie at the intersections of the identified lines, ellipse, and hyperbola. However, not every intersection of these curves is a center of a solution circle — only those for which the type of tangency (internal or external) to the shared circle matches on both curves.
- The problem has four solutions.
GeoGebra construction
Steps
- We choose the reference circle for the inversion. Its center is chosen at the point of tangency of the given circles.
- We apply the circular inversion to the given objects. The circle passing through the point of tangency is transformed into a line.
- The first three solution circles are transformed into circles tangent to the images of the given circles. We find them using sets of points with given properties.
- The last solution circle is transformed into tangent to a circle parallel to the images of the given circles.
- We invert the images of the solution circles back using the same inversion.
- The problem has four solutions.