Harmonic Division

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


frac{CA}{CB} = frac{DA}{DB}.

In the example shown below, the ratio is two. 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 1.5 inches.

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


frac{BC}{BD} = frac{AC}{AD}

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. It is also related to the cross-ratio.

Famous quotes containing the words division and/or harmonic:

    For a small child there is no division between playing and learning; between the things he or she does “just for fun” and things that are “educational.” The child learns while living and any part of living that is enjoyable is also play.
    Penelope Leach (20th century)

    For decades child development experts have erroneously directed parents to sing with one voice, a unison chorus of values, politics, disciplinary and loving styles. But duets have greater harmonic possibilities and are more interesting to listen to, so long as cacophony or dissonance remains at acceptable levels.
    Kyle D. Pruett (20th century)