Given a point Z and an inversion circle
with center at point Q and radius
, consider the following construction to obtain the inverse Z' of Z in
. Draw the circle centered at Z passing through Q; let it intersect
at points A and B. Draw two circles
and
centered at A and B both passing through Q. Their second point of intersection is the inverse of Z in
. We justify this by the following reasoning: if we invert
and
in
we obtain (red dotted) lines passing through Z; these lines intersect at
and Z, hence their inverses must intersect at the inverses of
and Z, namely Q and Z'. This Demonstration lets you drag the point Z (red); the construction works for all points Z at a distance from Q greater than
.
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