Next: Warm-up Problems

BERKELEY MATH CIRCLE, October 18 1998

INVERSION IN THE PLANE. PART II: RADICAL AXES1

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

 (1)

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 ():

where is the midpoint of , and is the orthogonal projection of onto . Then

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.

Lemma 1   Let and be two nonconcentric circles circles, with , and let be the midpoint of . Let lie on the segment , so that

Then the radical axis of and is the line , perpendicular to and passing through .

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.

Lemma 2   Let and be two nonintersecting circles. Prove that there exists an inversion sending the two circles into concentric ones.

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)

Next: Warm-up Problems
Zvezdelina Stankova 2001-08-31