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harmonic division
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(Definition)
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- 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.
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"harmonic division" is owned by pahio.
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Cross-references: analytic geometry, distances, locus, Apollonius circle, segment, diameter, circle, adjacent sides, opposite side, linear pair, triangle, angle, bisectors, theorem, 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 1956 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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Pending Errata and Addenda
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