Properties of Inversion
Inversion across C

fixes the circle C,

interchanges the inside and outside (5K) of
the circle C,

is a contraction (5K) on sets outside
the circle C,

preserves circles S orthogonal (5K) to the
inverting circle C,

is an involution (5K)
(that is, if (x', y')
is the inverse of (x, y) across C, then
(x, y)
is the inverse of (x', y') across C),

takes a circle not passing through
(a, b) to a
circle (5K)
not passing through
(a, b), and

takes a circle passing through
(a, b) to a
line (5K).
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