center is not collinear with the line Assume that center of the circle does not lie on the line. Given a circle (in black) and a line . We wish to construct the points of intersection, and , between them (if they exist). Construct the point , which is the reflection of point across line . (See above.) Under the assumption of this case, . Construct a circle (in red). (See above, compass equivalence.) The intersections of circle and the new red circle are points and . If the two circles are (externally) tangential then . Points and are the intersection points of circle and the line . If then the line is tangential to the circle . An alternate construction, using circle inversion can also be given. Given a circle and a line . We wish to construct the points of intersection, and , between them (if they exist). Invert points and in circle to points and respectively. Under the assumption of this case, points , , and are not collinear. Find the center of the circle passing through points , , and . Construct circle , which represents the inversion of the line into circle . and are the intersection points of circles and . If the two circles are (internally) tangential then , and the line is also tangential. Circle center is collinear with the line Given the circle whose center lies on the line , find the points and , the intersection points of the circle and the line. Construct point as the other intersection of circles and . Construct point as the intersection of circles and . ( is the fourth vertex of parallelogram .) Construct point as the intersection of circles and . ( is the fourth vertex of parallelogram .) Construct point as an intersection of circles and . ( lies on .) Points and are the intersections of circles and . Thus it has been shown that all of the basic construction one can perform with a straightedge and compass can be done with a compass alone, provided that it is understood that a line cannot be literally drawn but merely defined by two points. Other types of restricted construction Renaissance mathematicians Lodovico Ferrari, Gerolamo Cardano and Niccolò Fontana Tartaglia and others were able to show in the 16th century that any ruler-and-compass construction could be accomplished with a straightedge and a