harmonic division

•
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 socalled 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^{}.
Title  harmonic division 

Canonical name  HarmonicDivision 
Date of creation  20130322 17:34:29 
Last modified on  20130322 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 