Harmonic division

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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 AB by points C and D
Harmonic division of AB by points C and D

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.

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