Point On The Circle

Steps

1. To find the solution, we use circular inversion with the given point as the center of the inverting circle.
2. Apply the inversion to the given line and the given circle. The given point is mapped to infinity.
3. In the inverted setting, the solutions appear as tangents to the image of the circle that are parallel to the image of the line.
4. Invert the found tangents back to obtain the solution circles.
5. The problem has two solutions

Steps

1. Construct line p₁ passing through the given point A and center S of the given circle.
2. Draw a parabola with line p as its directrix and point A as its focus. Name the points of intersection of the parabola and the line p₁ S₁ and S₂.
3. The solution are the two circles with centres and points S₁ and S₂.

Steps

1. Draw a line through the given point and through the centre of the given circle.
2. Draw a random point on the line, a perpendicular to the given line passing through this point, and use it to draw a circle centered at a point tangent to the line.
3. Draw the intersections of this circle with the line from step 1, and connect them with tangent points.
4. Draw the parallel lines of these line segments passing through the given point.
5. At the intersections of these parallels with the given line, draw perpendiculars to the given line. Label the intersections of these perpendiculars with the line from step 1. These are the centers of the solutions.
6. Draw circles from these two points passing through the given point. These are the two solutions.