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| In [[geometry]], '''harmonic division''' of a [[line segment]] ''AB'' means identifying two [[Point (geometry)|point]]s ''C'' and ''D'' such that ''AB'' is divided internally and externally in the same [[ratio]]
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| :<math>
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| \frac{CA}{CB} = \frac{DA}{DB}.
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| </math>
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| 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.
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| [[Image:Harmonic division.png|frame|center|Harmonic division of ''AB'' by points ''C'' and ''D'']]
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| 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
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| :<math>
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| \frac{BC}{BD} = \frac{AC}{AD}
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| </math> | |
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| which equals one-third in the example above. (Note that the two ratios are not equal!)
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| Harmonic division of a line segment is a special case of [[Apollonian circles|Apollonius' definition of the circle]]. It is also related to the [[cross-ratio]].
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| ==See also== | |
| * [[Harmonic series (music)]]
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| * [[Projective harmonic conjugate]]
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| ==References==
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| * [[C. Stanley Ogilvy]] (1990) ''Excursions in Geometry'', Dover. ISBN 0-486-26530-7.
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| [[Category:Euclidean plane geometry]]
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Latest revision as of 16:58, 22 December 2014
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