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Introduction to Inversion
Basic Theorems about Inversion Suppose we have a circle Σ, as shown in figure 10.1, whose center T is called the center of inversion. Assume we also have a straight line l that we wish to invert with respect to Σ. Then:
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. The former situation is shown in figure 10.1; the latter situation is illustrated in figure 10.2
Σ k
l' T
Figure 10.2. Theorem A. The line l does not pass through the center of inversion T. Its inverse with respect to circle Σ is the small circle l ′, which does pass through T. l
Now assume we have a circle C and that we wish to invert C with respect to Σ. Then:
Theorem B If circle C does not pass through the center of inversion T, then C inverts into another circle C′. An example of such a situation is shown in figure 10.3.