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Introduction to Inversion
Theorem D is easily derived by referring to figure 10.6. The radius r′ of the inverse circle is (TP ′ − TQ′)/2. From the definition of inversion we can write this as r′ =
1 ⎞ k 2 ⎛ TQ − TP ⎞ k2 ⎛ 1 − ⎜ ⎟ = ⎜ ⎟. 2 ⎝ TP TQ ⎠ 2 ⎝ TP ⋅ TQ ⎠
(9)
However, we also have r = (TQ − TP)/2, and so if we call d the distance from T to the center of circle C, equation (9) can be written as r′ =
k2 r. (d − r )(d + r )
Recognizing that d may be greater or less than r, but that r′ must always be positive, we recast this as
Theorem D r′ =
k2 r. |d 2 − r 2 |
This result may also be obtained analytically by completing the square in equation (8) and writing the equation of the inverse circle analogously to equation (3): 2
2
2
2
k2 f ⎞ k 2g ⎞ ⎛ ⎛ ⎛ k 2g ⎞ ⎛ k2 f ⎞ k4 . ⎜x ′ − ⎟ + ⎜ y′ − ⎟ =⎜ ⎟ +⎜ ⎟ − ⎝ ⎝ ⎝ c ⎠ ⎝ c ⎠ c ⎠ c c ⎠ We leave it as an exercise to show that this expression also leads directly to theorem D. However, here is a clear case where geometric reasoning saves work over algebra, as in the previous proofs. Theorem E is proved in a manner similar to theorem D. Figure 10.10 depicts the inverse circle shown in figure 10.6 with the tangent L drawn in. As in equation (9) we can write
L
T
r'
P'
Q'
C'
Figure 10.10. The inverse circle C′ from figure 10.6 with radius r′. An elementary theorem says that L2 = TP ′ · TQ′.