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

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

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.

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