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Mathematics as the Science of Patterns - Jacob Steiner (continued)

Michael N. Fried

To start, Steiner asks what is the locus of points having the same power with respect to two given circles?  Let the centers and radii of the circles be M and m and R and r, respectively.  Then we are looking for the set of points P satisfying the relation, MP2-R2=mp2-r   or   R2-MP2=r2-mP2 (see fig. 6).  

In either case, this means that the points P satisfy the relation MP2-mP2=R2-r2, which we have just seen is a line perpendicular to the line Mm, the line joining the centers of the circles! This ‘line of equal powers’, as Steiner called it, is also known as the ‘radical axis’ of the two circles.  When the circles intersect, the radical axis is particularly easy to find, for the power of the points intersection are 0 with respect to both circles; therefore, the radical axis is the common chord of the two circles (and, of course, it follows immediately, that that line is perpendicular to the line joining the centers of the circles).  Similarly, if the circles are tangent the radical axis has to be the tangent line.  The various cases are shown in the figure below (fig. 7): 

In the cases where the radical axis lies outside the two circles it is clear that the axis can be given another interpretation, namely, the locus of all points from which the tangents to the two circles are equal since the power of a point P with respect to a circle equals the square of the tangent to the circle from P.   

        From here, Steiner moves on to three circles.  Let the centers of the circles, which we shall assume do not lie along a line, be M1, M2, M3, and let the radical axis of circles 1 and 2 be denoted l(12), of circles 2 and 3, l(23), and of circles 1 and 3, l(13) (these are all Steiner’s notations) (see fig. 8).  Suppose   l(12) and l(23) meet at point p(123).  Then the power of p(123) is the same with respect to circles 1 and 2 and also with respect to 2 and 3; therefore, the power of p(123) with respect to circles 1 and 3 must be the same, so that p (123) must also lie on l (13). 

In other words, given three circles whose centers do not all lie on a line, the radical axes all pass through one point.   That point is also known as the radical center of the three circles.  That there is a radical center means, among other things, that 1) if three circles intersect pair-wise then the three common chords intersect at a point (see fig. 9a), 2) if three circles are tangent pair-wise then the three tangents meet at a point and are equal (see fig. 9b), and, similarly, 3) if three circles are all non-intersecting then the three tangents from the radical center to the three circles are equal (see fig. 9c).  It is clear, moreover, that a circle is orthogonal to three given circles, its center will be the radical center of the three circles and its radius the length of the equal tangents.




Michael N. Fried, "Mathematics as the Science of Patterns - Jacob Steiner (continued)," Convergence (August 2010)