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Harmonic Range


HarmonicRange

Let a straight line AB be divided internally at C and externally at D in the same ratio, so that

 (AC)/(CB)=-(AD)/(DB).

Then AB is said to be divided harmonically at C and D and the points ACBD are said to form a harmonic range (Durell 1928, p. 65).

If C and D divide AB harmonically, then A and B divide CD harmonically.

If O is the midpoint of AB, then

 OB^2=OC×OD.

Hardy (1967) uses the term harmonic system of points to refer to a harmonic range.


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.

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Harmonic Range

Cite this as:

Weisstein, Eric W. "Harmonic Range." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HarmonicRange.html

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