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the inversion theorem
If {P,Q} are inverse with respect to an i-line L and i is an inversive transformation, then
(1) i(L) is an i-line, and
(2) {i(P),i(Q)} are inverse with respect to i(L).
Proof
Since I(2) is genereated by inversions, it is enough to consider a single inversion.
By earlier remarks, we may choose axes so that this inversion is i(z) = z* or i(z) = 1/z*.
For (1), the Example actaully found equations for the images. They are i-lines.
For (2), suppose that L = C(a,α,b).
Then z,w are inverse with respect to L if and only if awz* -αz* - α*w + b = 0.
(a) i(z) = z*. Then i(L) = C(a,α*,b).
Conjugating the above, we get aw*z - α*z - αw* + b = 0.
This is precisely the condition that z*, w* are inverse with respect to C(a,α*,b),
i.e. that i(z),i(w) are inverse with respect to i(L).
(b) i(z) = 1/z*. Then i(L) = C(b,α,a).
Dividing the above by wz*, conjugating and rearranging, we get the condition
b(1/w*)(1/z) - α(1/z) - α*(1/w*) + a = 0.
This is precisely the condition that 1/z* and 1/w* are inverse with respect to C(b,α,a),
i.e. that i(z), i(w) are inverse with respect to i(L).
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