Radical Center

radcntr.html

Exploration

Prove that the radical axes of three circles taken in pairs intersect in a common point. This point is called the radical center of the three circles.

Approach

Create the three radical axes, intersect them in pairs and show that the coordinates of the points of intersection are equal.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Create three general circles.

Clear[h1,k1,r1,h2,k2,r2,h3,k3,r3];
C1=Circle2D[{h1,k1},r1];
C2=Circle2D[{h2,k2},r2];
C3=Circle2D[{h3,k3},r3];

Construct the radical axis lines in pairs.

L12=Line2D[C1,C2];
L13=Line2D[C1,C3];
L23=Line2D[C2,C3];

Intersect the lines in pairs to find the intersection points.

p1=Point2D[L12,L13];
p2=Point2D[L12,L23];
p3=Point2D[L13,L23];

Show that the coordinates of the intersection points are equal.

{XCoordinate2D[p1]-XCoordinate2D[p2],
XCoordinate2D[p1]-XCoordinate2D[p3],
YCoordinate2D[p1]-YCoordinate2D[p2],
YCoordinate2D[p1]-YCoordinate2D[p3]} //FullSimplify

Discussion

This is the plot of a numerical example.

Sketch2D[{C1,C2,C3,L12,L13,L23,p1,p2,p3} /.
   {h1->-2, k1->0,  r1->1,
    h2->3,  k2->3,  r2->1.5,
    h3->5,  k3->-2, r3->2},
   CurveLength2D->14]

Graphics saved as "radcnt01.eps".