where, as well as in what immediately follows, Oα, &c. denote, of course, lengths or distances.
Similarly in circle A,
Oβ·Oβ′ = Oγ2·Oγ′,
and in circle B,
Oα·Oα′ = Oγ1·Oγ′.
Consequently Oγ1·Oγ′ = Oγ2·Oγ′, that is, Oγ1 = Oγ2, or the points γ1 and γ2 coincide; that is, they each coincide with γ.
We contrast this with the analytical method:—
Here it only requires to be known that an equation Ax + By + C = 0 represents a line, and an equation x² + y² + Ax + By + C = 0 represents a circle. A, B, C have, in the two cases respectively, metrical significations; but these we are not concerned with. Using S to denote the function x² + y² + Ax + By + C, the equation of a circle is S = o. Let the equation of any other circle be S′, = x² + y² + A′x + B′y + C′ = 0; the equation S-S′ = 0 is a linear equation (S − S′ is in fact = (A − A′)x + (B − B′)y + C-C), and it thus represents a line; this equation is satisfied by the coordinates of each of the points of intersection of the two circles (for at each of these points S = 0 and S′ = 0, therefore also S − S′ = 0); hence the equation S − S′ = 0 is that of the line joining the two points of intersection of the two circles, or say it is the equation of the common chord of the two circles. Considering then a third circle S″, = x² + y² + A″x + B″y + C″ = 0, the equations of the common chords are S − S′ = 0, S − S″ = 0, S′ − S″ = 0 (each of these a linear equation); at the intersection of the first and second of these lines S = S′ and S = S″, therefore also S′ = S″, or the equation of the third line is satisfied by the coordinates of the point in question; that is, the three chords intersect in a point O, the coordinates of which are determined by the equations S = S′ = S″.
It further appears that if the two circles S = 0, S′ = 0 do not intersect in any real points, they must be regarded as intersecting in two imaginary points, such that the line joining them is the real line represented by the equation S − S′ = 0; or that two circles, whether their intersections be real or imaginary, have always a real common chord (or radical axis), and that for any three circles the common chords intersect in a point (of course real) which is the radical centre. And by this very theorem, given two circles with imaginary intersections, we can, by drawing circles which meet each of them in real points, construct the radical axis of the first-mentioned two circles.
13. The principle employed in showing that the equation of the common chord of two circles is S − S′ = 0 is one of very extensive application, and some more illustrations of it may be given.