 HarmonicConjugate - Maple Help

geom3d

 HarmonicConjugate
 Find the harmonic conjugate of a point with respect to two other points Calling Sequence HarmonicConjugate(A, B, C, F) Parameters

 A - name B, C, F - three points Description

 • 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. Examples

 > $\mathrm{with}\left(\mathrm{geom3d}\right):$
 > $\mathrm{point}\left(B,3,3,0\right),\mathrm{point}\left(C,7,7,0\right),\mathrm{point}\left(F,\frac{21}{11},\frac{21}{11},0\right):$
 > $\mathrm{HarmonicConjugate}\left(A,B,C,F\right)$
 ${A}$ (1)
 > $\mathrm{coordinates}\left(A\right)$
 $\left[\frac{{63}}{{17}}{,}\frac{{63}}{{17}}{,}{0}\right]$ (2)

check that the harmonic conjugate of A with respect to BC is the same as F

 > $\mathrm{HarmonicConjugate}\left(\mathrm{F1},B,C,A\right)$
 ${\mathrm{F1}}$ (3)
 > $\mathrm{AreDistinct}\left(F,\mathrm{F1}\right)$
 ${\mathrm{false}}$ (4)