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harmonic division (Definition)
  • If the point $ X$ is on the line segment $ AB$ and $ XA\!:\!XB = p\!:\!q$, then $ X$ divides $ AB$ internally in the ratio $ p\!:\!q$.
  • If the point $ Y$ is on the extension of line segment $ AB$ and $ YA\!:\!YB = p\!:\!q$, then $ Y$ divides $ AB$ externally in the ratio $ p\!:\!q$.
  • If $ p\!:\!q$ is the same in both cases, then the points $ X$ and $ Y$ divide $ AB$ harmonically in the ratio $ p\!:\!q$.

Theorem 1. The bisectors of an angle of a triangle and its linear pair divide the opposite side of the triangle harmonically in the ratio of the adjacent sides.

Theorem 2. If the points $ X$ and $ Y$ divide the line segment $ AB$ harmonically in the ratio $ p\!:\!q$, then the circle with diameter the segment $ XY$ (the so-called Apollonius' circle) is the locus of such points whose distances from $ A$ and $ B$ have the ratio $ p\!:\!q$.

The latter theorem may be proved by using analytic geometry.



"harmonic division" is owned by pahio.
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See Also: bisectors theorem, Apollonius' circle

Also defines:  harmonically, divide harmonically
Keywords:  division ratio
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Cross-references: analytic geometry, distances, locus, Apollonius circle, segment, diameter, circle, adjacent sides, opposite side, linear pair, triangle, angle, bisectors, extension, ratio, line segment, point
There are 2 references to this entry.

This is version 4 of harmonic division, born on 2007-10-08, modified 2007-10-20.
Object id is 9986, canonical name is HarmonicDivision.
Accessed 770 times total.

Classification:
AMS MSC: 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
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