Line Tangent to Circle and Point on Line

Steps

1. By dilation, we shrink the given circle into a single point—its center. We imagine a solution circle, which enlarges by the radius of the given circle. Since the given line must also be tangent to this enlarged circle, it must be shifted by the radius of the given circle. The given point is shifted together with it.
2. The center of the desired solution circle lies on the perpendicular bisector of the segment connecting the dilated point A′ and the center of the original circle.
3. The center must also lie on the perpendicular to the line p passing through the original point A.
4. The center of the solution circle is located at the intersection of these two lines.
5. The problem has one solution.

Steps

1. Choose the circle of inversion so that its center is the point of tangency between the circle and the line.
2. Apply circular inversion to the given circle and the given point. The given line remains invariant under inversion.
3. Through inversion, the problem is transformed into finding a circle tangent to two parallel lines and passing through a point lying on one of the lines. Draw a perpendicular through this point to both lines. The center of the desired circle lies halfway between the two lines.
4. Construct the circle with the center found in the previous step.
5. Invert the constructed circle back to the original setting using the circular inversion.
6. The problem has one solution.

Steps

1. Choose the circle of inversion so that its center is the given point.
2. Apply the inversion to the given circle. The line remains invariant under inversion, and the given point is mapped to infinity.
3. The desired solution circle is mapped to a tangent to the image of the given circle, which is parallel to the given line.
4. Construct the tangent and invert it back to obtain the solution circle.
5. The problem has one solution.

Steps

1. The centers of all circles tangent to a given circle and passing through a given point lie on a hyperbola. The foci of this hyperbola are the given point and the center of the given circle. To construct the hyperbola, we also need one additional point. To find it, locate its vertex, which lies on the line defined by the foci. The midpoint between the intersection of this line with the circle and point A gives a point on the hyperbola.
2. Construct the hyperbola based on the information from the previous step.
3. The centers of all circles tangent to the given line at the given point lie on the perpendicular to the line passing through that point.
4. The center of the desired solution circle lies at the intersection of the hyperbola and the perpendicular line.
5. The problem has one solution.