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Chapter 10
Theorem L Inversion preserves angles (That is, if two curves intersect at a given angle, their inverses intersect at the same angle.)
Theorem M A circle, its inverse, and the center of inversion are colinear.
Theorem N By the proper choice of the center of inversion T, two circles that are not in contact can be inverted into two concentric circles.
Theorem P (Iwata’s theorem). If four circles can be inverted into four circles of equal radii, r′, whose centers form the vertices of a rectangle, then 1 1 1 1 + = + , r1 r3 r2 r4 where r 1, r 2, r 3, r4 are the radii of the original circles (see figure 10.15).
Proofs of Theorems A–E To prove the basic theorems is not difficult. The first part of theorem A follows directly from the definition of inversion: We may take a line passing through the center of inversion to be the horizontal line drawn in figure 10.1. Under inversion, points P and P ′ are merely swapped, as are all the other points and their inverses on the line, but by construction all of them remain on the same line. Thus the line inverts into itself. The second part of theorem A is proved by figure 10.5. Therefore, we have again
Theorem A A straight line passing through the center of inversion inverts into itself. A straight line not passing through the center of inversion inverts into a circle that passes through the center of inversion.