There are several methods of constructing the radical axis in this case. One of the simplest is: Let P and P′ (fig. 5) be the points of contact of a common tangent; drop perpendiculars PL, P′L′, from P and P’ to OO′, the line joining the centres, then the radical axis bisects LL’ (at X) and is perpendicular to OO′. To prove this let AB, AB¹ be the tangents from any point on the line AX. Then by Euc. i. 47, AB² = AO² - OB² = AX² + OX² + OP²; and OX² = OD² - DX² = OP² + PD² - DX². Therefore AB² = AX² - DX² + PD². Similarly AB′² = AX² - DX² + DP′². Since PD = PD′, it follows that AB = AB′.
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. Then circles having the intersections of tangents to this circle and the line of centres for centres, and the lengths of the tangents as radii, are members of the coaxal system.
In the case of non-intersecting circles, it is seen that the minimum circles of the coaxal system are a pair of points I and I′, where the orthogonal circle to the system intersects the line of centres; these points are named the “limiting points.” In the case of a coaxal system having real points of intersection the limiting points are imaginary. Analytically, the Cartesian equation to a coaxal system can be written in the form x² + y² + 2ax ± k² = 0, where a varies from member to member, while k is a constant. The radical axis is x = 0, and it may be shown that the length of the tangent from a point (0, h) is h² ± k², i.e. it is independent of a, and therefore of any particular member of the system. The circles intersect in real or imaginary points according to the lower or upper sign of k², and the limiting points are real for the upper sign and imaginary for the lower sign. The fundamental properties of coaxal systems may be summarized:—
1. The centres of circles forming a coaxal system are collinear;
2. A coaxal system having real points of intersection has imaginary limiting points;
3. A coaxal system having imaginary points of intersection has real limiting points;
4. Every circle through the limiting points cuts all circles of the system orthogonally;
5. The limiting points are inverse points for every circle of the system.
The theory of centres of similitude and coaxal circles affords elegant demonstrations of the famous problem: To describe a circle to touch three given circles. This problem, also termed the “Apollonian problem,” was demonstrated with the aid of conic sections by Apollonius in his book on Contacts or Tangencies; geometrical solutions involving the conic sections were also given by Adrianus Romanus, Vieta, Newton and others. The earliest analytical solution appears to have been given by the princess Elizabeth, a pupil of Descartes and daughter of Frederick V. John Casey, professor of mathematics at the Catholic university of Dublin, has given elementary demonstrations founded on the theory of similitude and coaxal circles which are reproduced in his Sequel to Euclid; an analytical solution by Gergonne is given in Salmon’s Conic Sections. Here we may notice that there are eight circles which solve the problem.