Circles Do Not Intersect, They Are External To Each Other, The Point Lies On One Circle
Number of solutions: 2
GeoGebra construction
Steps
- All centers of circles that are tangent to the given circle at the given point lying on it lie on the line passing through the center of the circle and the given point.
- We find the centers of circles tangent to both given circles. The centers of the solution circles lie on a pair of hyperbolas. The foci of the hyperbolas are the centers of the given circles. To construct them, we need at least one point on each hyperbola. We draw the line through the centers of the given circles and find its intersections with the circles. The midpoints of the segments defined by the found intersection points are the vertices of the hyperbolas.
- We construct the hyperbolas with foci at the centers of the given circles and passing through the identified vertices.
- The centers of the solution circles lie at the intersections of the found hyperbolas and the line. However, not every intersection is the center of a solution circle — only those for which the type of tangency (internal or external) to the common circle matches on both curves.
- The problem has two solutions.
GeoGebra construction
Steps
- Construct a circle with the centre in point A of the same size as the initial circle c, which we'll be basing circular inversion on.
- Invert circle c and d using circular inversion. Point A is at infinity.
- Construct the tangent of the new circle h, which is to be parallel with line f.
- Invert the new tangents i and j using circular inversion.
- The problem has two solutions.
GeoGebra construction
Steps
- The solution circle will be tangent to one of the given circles at the given point lying on it. The centers of all such circles lie on the line passing through the given point and the center of the given circle.
- The tangent to this given circle at the given point will also be a tangent of the solution circles.
- There are two homotheties in which the second given circle is the image of the solution circles. The centers of these homotheties are the points of tangency between these circles. The tangent of the solution circle is mapped to a parallel tangent of the given circle.
- In the homotheties, the point of tangency is mapped to the points of tangency. The lines passing through these points also pass through the homothety centers, which are the intersections of these lines with the given circle.
- The identified homothety centers are also the tangency points between the given circle and the solution circles. Thus, for each solution circle, we have two of its points. Their centers lie on the perpendicular bisectors of the segments defined by these points.
- The centers of the solution circles lie at the intersections of these bisectors and the line from step 1.
- The problem has two solutions.