Two points A and B are said to be conjugate with respect to the sphere s if the points of intersection P and Q divide A and B harmonically.

The routine returns true if A and B are conjugate with respect to s; false if they are not; and FAIL if it is unable to reach a conclusion.

If FAIL is returned, and the optional argument cond is given, the condition that makes the points conjugate is assigned to this argument.

The command with(geom3d,AreConjugate) allows the use of the abbreviated form of this command.

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

$\mathrm{point}\left(A\,-a\,2\,a\right),\mathrm{point}\left(B\,a\,2\,3\right)\:$

$\mathrm{sphere}\left(s\,x^2+y^2+z^2-6x+2y-4z+4=0\,\left[x\,y\,z\right]\right)\:$

$\mathrm{AreConjugate}\left(A\,B\,s\,\mathrm{cond}\right)$

AreConjugate:   "unable to determine if -a^2+a is zero"

$\mathrm{FAIL}$

$\mathrm{cond}$

$-a^2+a=0$

$\mathrm{solve}\left(\mathrm{cond}\right)$

$0,1$

$\mathrm{assume}\left(\mathrm{cond}\right)$

$\mathrm{AreConjugate}\left(A\,B\,s\right)$

$\mathrm{true}$