Let be a circle of radius r and center O. For any point
the inverse P' of P with respect to the circle
is the
unique point
so that
. In this case
is called the circle of inversion.
These are Euclidean geometry theorems about circles, so we may use the facts that we know about Euclidean geometry.
Proposition 15.1:
For and
as above,
.
then
, and vice
versa.
Proposition 15.2: Let and let TU be the chord through P
perpendicular to
. Then P'=P(TU), the pole of TU,
i.e., the intersection of the tangents to
at T and U.
Proof: Suppose the tangent to at T cuts
at the
point P'. The right triangle
is similar to right triangle
OTP' (since they have
in common and the angle sum is
). Hence corresponding sides are proportional. Since
,
we get that
which shows that P' is the inverse to P. Reflect across the line
, and we see that the tangent to
at U also passes
through P', so that P' is the pole of TU.
Proposition 15.3: If P is outside of , let Q be the midpoint of OP.
Let
be the circle of radius QP centered at Q. Then
.
and
are tangent to
.
.
Proof: By the circular continuity principle, and
do
intersect in two points T and U. Since
and
are
inscribed in semicircles of
, they are right angles. Therefore
and
are tangent to
. If TU intersects
OP in a point P', then P is the inverse of P', by the previous
proposition. Thus, P' is the inverse of P in
.
The next proposition shows how to construct the Poincaré line joining two
ideal points--the line of enclosure.
Its proof shows that in the previous figure is indeed a right
angle, as we needed.
Proposition 15.4: Let T and U be points on that are not contained
on a the same diameter, and let P be the pole of TU. Then
n
,
,
, and
with center P and radius PT cuts
orthogonally at T and U.
Proof: By definition of pole, and
are right
angles, so by the Hypotenuse-Leg criterion,
. Therefore,
and
. The base angles
and
of the isosceles triangle
are
thus congruent, and the angle bisector
is perpendicular to the
base TU. The circle
is then well-defined because
and
intersects
orthogonally by the hypothesis that
and
are tangent to
.
Let P be a point in the plane and a circle with center O. The
power of P with respect to
is defined to be
Proposition 15.5:
Assume and assume that two lines through P intersect
in points
and
. Then
.
at T, then
.
Proposition 15.6: Let and
, the center of
. Let
be a circle through P.
is orthogonal to
if and only
if
passes through P', the inverse of P with respect to
.
Proposition 15.7: Let have radius r,
have radius t and let P be
the center of
.
is orthogonal to
if and only if
, where the power is computed with respect to
.
Let O be a point and k>0. The dilation with center O and ratio k
is a mapping of the Euclidean plane that fixes O and maps every
point to a unique
such that
. Call this
map
.
Proposition 15.8: Let be a circle with center
and radius s.
maps
to a circle
with center
and
radius
. If
, the tangent to
at
is parallel to
the tangent to
at Q.
Proof: Choose rectangular coordinates so that O is the origin. Then
the dilation is given by the mapping . The image of the
line have equation ax+by=c is the line having equation
, so the
image is parallel to the original line. In particular,
is
parallel to
, and their perpendiculars at Q and
,
respectively, are also parallel. If
has equation
, then
has equation
.
Proposition 15.9: Let be the circle of radius r centered at O and
let
be the circle of radius s centered at C. Assume O lies
outside
; let p=Pw(O) with respect to
and let
.
The image
of
under inversion in
is the circle of
radius
whose center is
. If
and
P' is the inverse of P with respect to
, then the tangent t' to
at P' is the reflection across the perpendicular bisector of PP'
of the tangent to
at P.
Corollary is orthogonal to
if and only if
is mapped to
itself by inversion in
.
Proposition 15.10: Let be a line so that
. The image of
under
inversion by
is a punctured circle with missing point O. The
diameter through O of the completed circle
is perpendicular to
.
Proposition 15.11: Let be a circle passing through the center O of
. The image of
minus O under inversion in
is a line
so that
and
is parallel to the tangent to
at O.
Proposition 15.12: A directed angle of intersection of two circles is preserved in magnitude by an inversion. The same applies to the angle of intersection of a circle and a line or the intersection of two lines.
Proposition 15.13: Let be orthogonal to
. Inversion in
maps
onto
and the interior of
onto itself. Inversion in
preserves incidence, betweenness and congruence in the sense of the
Poincaré disk model.