308 CHAPTER 8 GEOMETRY FOR AMERICANS
co, draw the two tangents to co. The two points of tangency determine
a chord whose midpoint is X'.The following facts should be evident from the figure above.
Fact 8.5.13 Let T denote inversion with respect to a circle co with center O.
(a) If A is on co, then T(A) = A. Thus, T(co) = co.
(b) T takes points inside co into points outside co, and vice versa.
(c) As a point X moves toward 0, the image T(X) moves out "towards infinity."
Conversely, as X moves further and further away from 0, its image T(X) moves
closer and closer towards O.
(d) For any point X, T(OX) = OX. (Note that OX is a ray, not a vector, in this
context.)
(e) Let ybe a circle concentric with co, with radius t. Then T(y) will also be a circle
concentric with co, with radius r^2 It.
Inversion encourages us to include the "point at infinity" since that is the limit
of the image of a point as it approaches the inversion center. This concept can be
made rigorous (and is a standard idea in projective geometry), but we will keep things
informal for now. Therefore, we can amend statement (d) above to read
Iff is a line passing through 0, then T(f) = f.
Try to visualize this. Imagine a point X traveling on a line f that passes through
o in a north-south direction. Simultaneously, imagine T(X). Start on the northern
intersection of f with circle co, and move south toward the center o. The image T(X)
will move north on f, towards (northern) "infinity." When we pass through 0 and begin
moving towards the southern boundary of the circle, the image swoops north towards
the circle boundary from the southern "infinite" location on f, reaching the boundary
just as X does. Then, as X continues on its path, towards "southern infinity," the image
T(X) crawls north, toward O. This "point at infinity" is not north or south, but all
directions at once. We denote it by 00 and thus T(oo) = 0 and T(O) = 00.
Now that we have 00, we can generalize the notion of circles to include lines.
This is not insane : A line can be thought of as a "circle" that passes through oo! Its
radius is infinite, which explains the zero curvature (flatness). Every ordinary circle is
determined by three points. Likewise, we can think of a line as just a circle specified
by three points: two on the line and one at 00. So now we will refer to the set of
ordinary circles and ordinary lines as "circles." Using this definition, we can also say
that two lines are tangent if and only if they coincide or are parallel. In the latter case,
they are "circles" whose point of tangency is 00.
We have seen that, in some cases, inversion takes circles to circles and lines to
lines. It turns out that this is always the case!Fact 8.5.14 Fundamental Properties of Inversion. Let X' denote the image of the
point X by inversion with respect to a circle co with center 0 and radius r.
(a) For any points X, Y, the triangles OXY and OX'Y' are oppositely similar. In other
words, .6.0XY rv .6. OY 'X'.