harmonic division
-
•
If the point is on the line segment and , then divides internally in the ratio .
-
•
If the point is on the extension of line segment and , then divides externally in the ratio .
-
•
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 |
---|---|
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 |