Circle, One Point On It And The Second Point Outside The Circle

Steps

1. Draw a circle centered at the outer point and passing through the second given point. We will make a circular inversion through it.
2. Perform a circular inversion of the given circle. We don't need the given points, the one on the circle is self-interacting and the one at the center of the inverse circle is projected to infinity.
3. Draw the tangent of the image of the given circle passing through the given point that lies on it. This line satisfies all the properties of the solution in the circular inverse. It is tangent to the circle, it passes through a point on it, and the third point is at infinity, where the line also intersects it.
4. Perform the circular inversion of the resulting line. This is projected onto the circle, which is the solution.

Steps

1. Construct a line connecting the points S and A.
2. Construct the axis of points A and B.
3. The intersection of the line and the axis is the centre of the desired circle.
4. Construct the circle.

Steps

1. We solve the problem using a homothety in which the given circle is the image of the solution circle. The center of this homothety is the given point, which is also the point of tangency between the two circles. Therefore, the centers of both circles lie on a straight line with this point.
2. Since point B must lie on the solution circle, its image under the homothety must lie on the given circle. This image lies at the intersection of the circle and the line passing through point B and the center of homothety A.
3. Draw the line through the center of the given circle and the image of point B.
4. Homothety preserves parallelism. Therefore, this line must be parallel to the line passing through point B and the center of the solution circle.
5. The problem has one solution.