See also
Bivalent Range,
Euler Line,
Gergonne Line,
Harmonic
Conjugate,
Soddy Line
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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 Web 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
