Two Circles With Inner Contact, Point Between The Circles

GeoGebra construction

Download GeoGebra file

Steps

1. We choose the reference circle for the inversion. We take the point of tangency of the two given circles as its center.
2. We perform a circular inversion of the given objects. Since the point of tangency lies on both given circles, they are transformed into lines.
3. We find circles tangent to the images of the original circles and passing through the image of the given point. For finding their centers, we can use, for example, loci of points with given properties. We have found two such circles. These are the images of the solution circles.
4. The image of the third solution circle is a line parallel to the images of the original circles and passing through the image of the given point.
5. We invert the found images of the solution circles back using the original inversion.
6. The problem has three solutions.

GeoGebra construction

Download GeoGebra file

Steps

1. First, we find the centers of circles tangent to the inner given circle and passing through the given point. These centers lie on a hyperbola. The foci of the hyperbola are the center of the given circle and the given point. To construct it, we need to know at least one of its points. We draw the line passing through the center of the given circle and the given point and find its intersections with the circle. The midpoints of the segments defined by these intersections and the given point are the vertices of the hyperbola.
2. We construct the hyperbola with foci at the center of the given circle and the given point, passing through the identified vertices.
3. Next, we find the centers of circles tangent to the outer given circle and passing through the given point. These centers lie on an ellipse. The foci of the ellipse are the center of the given circle and the given point. To construct it, we need to know at least one of its points. We draw the line passing through the center of the given circle and the given point and find its intersections with the circle. The midpoints of the segments defined by these intersections and the given point are the vertices of the ellipse.
4. We construct the ellipse with foci at the center of the given circle and the given point, passing through the identified vertices.
5. The centers of the solution circles lie at the intersections of the hyperbola and the ellipse.
6. The problem has three solutions.

GeoGebra construction

Download GeoGebra file

Steps

We have two internally tangent circles and a point A lying between them. Draw any circle e with its center at point A. Use a circle inversion to map the circles given in the problem inside circle e. Draw tangents to the new circles from the point 3. Use a circle inversion to map these tangents outside circle e. The resulting circles are the solution to the problem.