In geometry, harmonic division of a line segment AB means identifying two points C and D such that AB is divided internally and externally in the same ratio

In the example shown below, the ratio is two. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume A ratio is an expression which compares quantities relative to each other Specifically, the distance AC is one inch, the distance CB is half an inch, the distance AD is three inches, and the distance BD is 1. Inches redirects here To see the Les Savy Fav album see Inches. 5 inches.

Harmonic division of AB by points C and D

Harmonic division of a line segment is reciprocal; if points C and D divide the line segment AB harmonically, the points A and B also divide the line segment CD harmonically. In that case, the ratio is given by

which equals one-third in the example above. (Note that the two ratios are not equal!)

Harmonic division of a line segment is a special case of Apollonius' definition of the circle. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the It is also related to the cross-ratio. In Mathematics, the cross-ratio of a set of four distinct points on the Complex plane is given by (z_1z_2z_3z_4 = \frac{(z_1-z_3(z_2-z_4}{(z_1-z_4(z_2-z_3}

References

• C. Stanley Ogilvy (1990) Excursions in Geometry, Dover. ISBN 0-486-26530-7.