Zvezdelina Stankova, UC Berkeley

**Definition 2.** The *degree* of point with respect to
a circle is defined as

This is simply the square of the tangent segment from to .
Let be the midpoint of in
, and - the
altitude from , with (cf. Fig.5-6.) Mark the sides ,
and by , and , respectively. Then

where is the midpoint of .

**Definition 3.** The *radical axis* of two circles
and is the geometric place of all points which have the same degree
with respect to and :
.

Let be one of the points on the radical axis of
and
(cf. Fig.7.) We have by ():

constantpointis constant

(Show that
the direction of
is the same regardless of which
point on the radical axis we have chosen.) Thus, the radical axis
is a subset of a line
. The converse is easy.

What happens with the radical axis when the circles are concentric? In some situations it is convenient to have the circles concentric. In the following fundamental lemma, we achieve this by applying both ideas of inversion and radical axis.

PROOF: If the radical axis intersects
in point , let
d intersect in
and . Apply inversion wrt
(cf. Fig. 8.) Then
is a line through ,
. But
,
hence
, i.e. the center of lies on . It also
lies on , hence is centered at . Similarly,
is centered at .

ARRAY(0x963c8c)

- Warm-up Problems
- Problems
- Problems From Around the World
- Final Remarks on Inversion: Alternative Definition of Inversion in Terms of Complex Numbers
- Hints and Solutions to Selected Problems in Part I and Part II
- About this document ...