AreHarmonic - Maple Help

geometry

 AreHarmonic
 test if a pair of points is harmonic conjugate to another pair of points

 Calling Sequence AreHarmonic(A, B, C, F)

Parameters

 A, B, C, F - four points

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{geometry}\right):$
 > $\mathrm{point}\left(A,0,0\right),\mathrm{point}\left(B,3,3\right),\mathrm{point}\left(C,7,7\right),\mathrm{point}\left(F,\frac{21}{11},\frac{21}{11}\right):$
 > $\mathrm{AreHarmonic}\left(A,B,C,F\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{AreHarmonic}\left(B,A,C,F\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{AreHarmonic}\left(A,C,B,F\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{point}\left(F,\frac{21}{11},a\right)$
 ${F}$ (4)
 > $\mathrm{AreHarmonic}\left(A,B,C,F\right)$
 AreCollinear:   "hint: could not determine if 3*a-63/11 is zero"

From the above hint, we see that the condition for F to be conjugate harmonic of C is a = 21/11

 > $a≔\frac{21}{11}:$
 > $\mathrm{AreHarmonic}\left(A,B,C,F\right)$
 ${\mathrm{true}}$ (5)