The routine returns true if C and F are harmonic conjugates of each other with respect to A and B; false if C and F are not harmonic conjugates; and FAIL if it cannot determine whether C and F are harmonic conjugates.

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

The command with(geometry,AreHarmonic) allows the use of the abbreviated form of this command.

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

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{false}$

$\mathrm{point}\left(F\,\frac{21}{11}\,a\right)$

$F$

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

AreCollinear:   "hint: could not determine if 3*a-63/11 is zero"

Error, (in geometry:-CrossRatio) unable to determine if 3*a-63/11 is zero

$a≔\frac{21}{11}\:$

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

$\mathrm{true}$