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. 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
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 harmonic and/or division:
“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)
“For in the division of the nations of the whole earth he set a ruler over every people; but Israel is the Lord’s portion: whom, being his firstborn, he nourisheth with discipline, and giving him the light of his love doth not forsake him. Therefore all their works are as the sun before him, and his eyes are continually upon their ways.”
—Apocrypha. Ecclesiasticus 17:17-9.