See also
Bivalent Range,
Euler Line,
Gergonne Line,
Harmonic Conjugate,
Soddy Line Explore with Wolfram|Alpha
References
Casey, J. "Theory of Harmonic Section." §6.3 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 87-94, 1888.Durell, C. V. "Harmonic Ranges and Pencils." Ch. 6 in Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 65-67, 1928.Graustein, W. C. "Harmonic Division." Ch. 4 in Introduction to Higher Geometry. New York: Macmillan, pp. 50-64, 1930.Hardy, G. H. A Course of Pure Mathematics, 10th ed. Cambridge, England: Cambridge University Press, pp. 99 and 106, 1967.Lachlan, R. "Harmonic Properties." §288-290 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 177 and 267-268, 1893.Referenced on Wolfram|Alpha
Harmonic Range Cite this as:
Weisstein, Eric W. "Harmonic Range." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HarmonicRange.html
Subject classifications
be divided internally at 
and externally at 
in the same ratio, so that 
is said to be divided harmonically at 
and 
and the points 
are said to form a harmonic range (Durell 1928, p. 65). 
and 
divide 
harmonically, then 
and 
divide 
harmonically. 
is the midpoint of 
, then 
