A system coaxal with the two given circles is readily constructed by describing circles through the common points on the radical axis and any third point; the minimum circle of the system is obviously that which has the common chord of intersection for diameter, the maximum is the radical axis - considered as a circle of infinite radius.

In the case of two non-intersecting circles it may be shown that the radical axis has the same metrical relations to the line of centres.

There are several methods of constructing the radical axis in this case.

diculars PL, P'L', from P and P' to 00', the line joining the centres, then the radical axis bisects LL' (at X) and is perpendicular to 00'.

To construct circles coaxal with the two given circles, draw the tangent, say XR, from X, the point where the radical axis intersects the line of centres, to one of the given circles, and with centre X and radius XR describe a circle.