Inversion in a circle, K of radius R, of a line, L. Points A, B, and C are inverted in K to points a, b, and c, respectively. A circle is drawn through a, b, and c. A line perpendicular to L through q is shown, lines intersecting at F. Triangle qFA is obviously a right triangle, and therefore so is triangle qaD, since they are similar, and therefore inversion of a line always traces a circle! If you let a point on the line go far away it becomes clear that q is also on the circle.

Points A, B, C, and q can be dragged. Note how the angles shown are always 90 degrees, no matter what point is dragged.