One Point On The Circle, Second Inside The Circle

Steps

1. draw a circle d with centre at A and radius AB
2. construct the image of the circle k through the circular inversion over the circle d
3. draw a line f which is tangent to k' and passes through B
4. represent the line f in circular inversion through the circle d, resulting in a circle k₁ which is the solution to the problem

Steps

1. draw a line p₁ passing through points B and C
2. draw the axis p₂ of line BC
3. draw the line p₃ passing through points A and B
4. at the intersection of the lines p₂ and p₃ there is a point S₁, which is the centre of the circle k₁
5. draw the circle k₁ with radius S_1B, which is the solution of the problem

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.