One Circle Inside Another, Point Inside Larger Circle

Number of solutions: 4

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.
We construct the hyperbola with foci at the center of the given circle and the given point, passing through the identified vertices.
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.
We construct the ellipse with foci at the center of the given circle and the given point, passing through the identified vertices.
The centers of the solution circles lie at the intersections of the hyperbola and the ellipse.
The problem has four solutions.

Choose the circle of inversion so that its center is at the given point.
Apply the inversion to both given circles.
In the chosen inversion, the solution circles are mapped to the common tangents of the images of the two circles.
Invert the found tangents back to obtain the solution circles.
The problem has four solutions.

Loci of Points Satisfying Certain Properties