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geometry[AreHarmonic] - test if a pair of points is harmonic conjugate to another pair of points
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Calling Sequence
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AreHarmonic(A, B, C, F)
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Description
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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.
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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.
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The command with(geometry,AreHarmonic) allows the use of the abbreviated form of this command.
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Examples
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AreCollinear: "hint: could not determine if 3*a-63/11 is zero"
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From the above hint, we see that the condition for F to be conjugate harmonic of C is a = 21/11
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