8.5 TRANSFORMATIONS 307
of BX as X ranges freely about Cl. In other words, the centers are the image of Cl
under the homothety with center B and ratio 1/2. But this is just another circle, one
with radius 1/2, located halfway between the circles.
As in Example 4.2.14, we can now see that the locus of points M forms an annular
region with center halfway between the circles, with inner radius 1 and outer radius 2 .•
There is a third solution, similar to the one above, that uses vectors. Find it!
Inversion
The inversion transformation, discovered in the 19 th century, drastically alters lengths
and shapes. Yet it is almost magically useful, because it provides a way to interchange
lines with circles.
An inversion with center a and radius r takes the point X i= a to a point X' on ray
OX so that OX. OX' = r^2. Since a and r uniquely determine a circle co, equivalently
we speak of inversion with respect to the circle co. Inversion is sort of a homothety
with-variable-ratio, where the ratio varies inversely with respect to the distance to the
center. Hence the name. Now let's see how it works, with a diagram that also suggests
an easy Euclidean construction for inversion.
X'
Consider circle co with center a and radius r. Place point X inside the circle. We
wish to find its inverse X I. Draw chord BC through X and perpendicular to ray OX,
and then draw a line perpendicular to OB through B. This line meets ray OX at the
point X'. By similar triangles, we easily see that
OB OX
OX' OB
Thus OX. OX' = OB^2 = r^2 , and indeed X' is the image of X under the inversion about
co. Conversely, X is the inverse of X'. (It is clear by the definition that the inverse of
the inverse of any point P is P.) The algorithm for inversion, then is the following:
If X lies inside co, draw the chord through X that is perpendicular to
OX. This chord intersects CO in two points : the intersection of th e tan
gents at these two points is X'. Conversely, if X lies outside the circle