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geom3d

 AreConjugate
 test if a pair of points is conjugate with respect to a sphere

 Calling Sequence AreConjugate(A, B, s, cond)

Parameters

 s - a sphere A, B - two points cond - (optional) name

Description

 • 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.

Examples

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

For what values of a'' are the points (-a,2,a), (a,2,3) conjugate with respect to the sphere ${x}^{2}+{y}^{2}+{z}^{2}-6x+2y-4z+4=0$.

 > $\mathrm{point}\left(A,-a,2,a\right),\mathrm{point}\left(B,a,2,3\right):$
 > $\mathrm{sphere}\left(s,{x}^{2}+{y}^{2}+{z}^{2}-6x+2y-4z+4=0,\left[x,y,z\right]\right):$
 > $\mathrm{AreConjugate}\left(A,B,s,\mathrm{cond}\right)$
 AreConjugate:   "unable to determine if -a^2+a is zero"
 ${\mathrm{FAIL}}$ (1)
 > $\mathrm{cond}$
 ${-}{{a}}^{{2}}{+}{a}{=}{0}$ (2)

Hence, the values of a'' are

 > $\mathrm{solve}\left(\mathrm{cond}\right)$
 ${0}{,}{1}$ (3)

check:

 > $\mathrm{assume}\left(\mathrm{cond}\right)$
 > $\mathrm{AreConjugate}\left(A,B,s\right)$
 ${\mathrm{true}}$ (4)