ngents to the circles are equal is a straight line.
Consider the case of two circles, and in the first place suppose them to
intersect in two real points A and B. Then by Euclid iii. 36 it is seen
that the line joining the points A and B is the locus of the
intersection of equal tangents, for if P be any point on AB and PC and
PD the tangents to the circles, then PA.PB = PC squared = PD squared, and therefore PC
= PD. Furthermore it is seen that AB is perpendicular to the line
joining the centres, and divides it in the ratio of the squares of the
radii. The line AB is termed the "radical axis." 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.

[Illustration: Fig. 5]

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, AB squared =
AO squared - OB squared = AX squared + OX squared + OP squared; and OX squared = OD squared - DX squared = OP squared + PD squared - DX squared.
Therefore AB squared = AX squared - DX squared + PD squared. Similarly AB' squared = AX squared - DX squared + DP' squared.
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 inter