The locus of points $P\left(x\,y\,z\right)$ which have the same power with respect to the two given spheres s1, s2 is a plane called radical plane.

Let us introduce a third sphere s3. Now we have three radical planes that form a pencil whose axis is the straight line. This line is called the radical line of the three sphere.

Now add a fourth sphere s4, and we have four radical lines. These four lines are clearly concurrent at the radical center.

$\mathrm{with}\left(\mathrm{geom3d}\right)\:$

$\mathrm{sphere}\left(\mathrm{s1}\,x^2+y^2+z^2=1\,\left[x\,y\,z\right]\right)\:$

$\mathrm{sphere}\left(\mathrm{s2}\,\left[\mathrm{point}\left(B\,\left[5\,5\,5\right]\right)\,2\right]\right)\:$

$\mathrm{RadicalPlane}\left(p\,\mathrm{s1}\,\mathrm{s2}\right)$

$p$

$\mathrm{Equation}\left(p\right)$

$-72+10x+10y+10z=0$

$\mathrm{NormalVector}\left(p\right)$

$\left[10\,10\,10\right]$

$\mathrm{randpoint}\left(P\,p\right)$

$P$

$\mathrm{powerps}\left(P\,\mathrm{s1}\right)-\mathrm{powerps}\left(P\,\mathrm{s2}\right)$

$0$

$\mathrm{draw}\left(\left[p\,\mathrm{s1}\,\mathrm{s2}\right]\,\mathrm{style}=\mathrm{patchnogrid}\,\mathrm{orientation}=\left[-26\,96\right]\,\mathrm{lightmodel}=\mathrm{light1}\,\mathrm{title}=\mathrm{`Radical\; plane\; of\; two\; given\; spheres`}\right)$

$\mathrm{sphere}\left(\mathrm{s3}\,\left[\mathrm{point}\left(A\,1\,2\,3\right)\,3\right]\right)$

$\mathrm{s3}$

$\mathrm{RadicalLine}\left(l\,\mathrm{s1}\,\mathrm{s2}\,\mathrm{s3}\right)$

$l$

$\mathrm{detail}\left(l\right)$

$\begin{array}{ll}\text{name of the object}& l\\ \text{form of the object}& \mathrm{line3d}\\ \text{equation of the line}& \left[x=\frac{57}{5}+20\mathrm{\_t}\,y=-\frac{21}{5}-40\mathrm{\_t}\,z=20\mathrm{\_t}\right]\end{array}$

$\mathrm{sphere}\left(\mathrm{s4}\,\left[\mathrm{point}\left(A\,-3\,7\,1\right)\,3\right]\right)$

$\mathrm{s4}$

$\mathrm{RadicalCenter}\left(P\,\mathrm{s1}\,\mathrm{s2}\,\mathrm{s3}\,\mathrm{s4}\right)$

$P$

$\mathrm{form}\left(P\right)$

$\mathrm{point3d}$

$\mathrm{coordinates}\left(P\right)$

$\left[\frac{249}{100}\,\frac{471}{100}\,0\right]$