harmonic division

  • If the point X is on the line segmentMathworldPlanetmath 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 bisectorsMathworldPlanetmath of an angle of a triangle and its linear pair divide the opposite side of the triangle harmonically in the ratio of the adjacent sidesMathworldPlanetmathPlanetmath.

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

The latter theorem may be proved by using analytic geometryMathworldPlanetmath.

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