The routine returns A which is the harmonic conjugate of F with respect to B and C.

If A, B, C, F are four collinear points such that the cross-ratio(AF, BC) = -1 (so that B and C divide AF one internally and the other externally in the same numerical ratio), the segment AF is said to be divided harmonically by B and C. The points A and F are called harmonic conjugates of each other with respect to B and C, and the four points A, F, B, C are said to constitute a harmonic range.

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

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

$\mathrm{point}\left(B\,3\,3\,0\right),\mathrm{point}\left(C\,7\,7\,0\right),\mathrm{point}\left(F\,\frac{21}{11}\,\frac{21}{11}\,0\right)\:$

$\mathrm{HarmonicConjugate}\left(A\,B\,C\,F\right)$

$A$

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

$\left[\frac{63}{17}\,\frac{63}{17}\,0\right]$

$\mathrm{HarmonicConjugate}\left(\mathrm{F1}\,B\,C\,A\right)$

$\mathrm{F1}$

$\mathrm{AreDistinct}\left(F\,\mathrm{F1}\right)$

$\mathrm{false}$