Theorem 3
If K is a circle lying within D, and tεH(2), then
(1) t(K) is a circle lying within D, and
(2) the interior of K maps to the interior of t(K).
Proof
Observe that t is the restriction to D of an
inversive transformation t*,
and that K is a circle, and hence an i-line.
(1) As K lies within D, t*(K) = t(K) lies in D.
Since t* is inversive, it maps the i-line K to an i-line.
Since the image lies in D, it must be a circle.
(D cannot contain an extended line.)
(2) From the Interior-Exterior Theorem, t* maps the interior of K
either to the interior or to the exterior of t(K). Since t* maps D to D,
it must map the interior of K (which is in D) to that of t(K).
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