harmonic division
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If the point is on the line segment
and , then divides internally in the ratio .
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If the point is on the extension of line segment and , then divides externally in the ratio .
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If is the same in both cases, then the points and divide harmonically in the ratio .
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 and divide the line segment harmonically in the ratio , then the circle with diameter the segment (the so-called Apollonius’ circle) is the locus of such points whose distances
from and have the ratio .
The latter theorem may be proved by using analytic geometry.
Title | harmonic division |
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Canonical name | HarmonicDivision |
Date of creation | 2013-03-22 17:34:29 |
Last modified on | 2013-03-22 17:34:29 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 7 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 51N20 |
Classification | msc 51M04 |
Related topic | BisectorsTheorem |
Related topic | ApolloniusCircle |
Defines | harmonically |
Defines | divide harmonically |