Harmonic division

In geometry, harmonic division of a line segment "AB" means identifying two points "C" and "D" such that "AB" is divided internally and externally in the same ratio

:$frac\{CA\}\{CB\}\; =\; frac\{DA\}\{DB\}.$

In the example shown below, the ratio is two. Specifically, the distance "AC" is one inch, the distance "CB" is half an inch, the distance "AD" is three inches, and the distance "BD" is 1.5 inches.

Harmonic division of a line segment is "reciprocal"; if points C and D divide the line segment AB harmonically, the points A and B also divide the line segment CD harmonically. In that case, the ratio is given by

:$frac\{BC\}\{BD\}\; =\; frac\{AC\}\{AD\}$

which equals one-third in the example above. (Note that the two ratios are not equal!)

Harmonic division of a line segment is a special case of Apollonius' definition of the circle. It is also related to the cross-ratio.

References

* C. Stanley Ogilvy (1990) "Excursions in Geometry", Dover. ISBN 0-486-26530-7.