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A quadruple of points on a straight line with [[Cross ratio|cross ratio]] equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h0465301.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h0465302.png" /> is a harmonic quadruple of points, one says that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h0465303.png" /> harmonically divides the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h0465304.png" />, or that the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h0465305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h0465306.png" /> are harmonically conjugate with respect to the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h0465307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h0465308.png" />; the pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h0465309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h04653010.png" /> are called harmonically conjugate.
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A quadruple of points on a straight line with [[Cross ratio|cross ratio]] equal to $-1$. If $(ABCD)$ is a harmonic quadruple of points, one says that the pair $AB$ harmonically divides the pair $CD$, or that the points $A$ and $B$ are harmonically conjugate with respect to the points $C$ and $D$; the pairs $AB$ and $CD$ are called harmonically conjugate.
  
 
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A harmonic quadruple can be defined without recourse to metric concepts. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h04653011.png" /> be a quadrangle (see Fig.), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h04653012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h04653013.png" /> be the intersection points of the opposite sides, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h04653014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h04653015.png" /> be the intersection points of the diagonals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h04653016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h04653017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h04653018.png" /> with the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h04653019.png" />. Then the quadruple of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046530/h04653020.png" /> is a harmonic quadruple. A quadruple of straight lines (or planes) passing through a single point (a single straight line) is called a harmonic quadruple of straight lines (planes) if a straight line intersects it in a harmonic quadruple of points.
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A harmonic quadruple can be defined without recourse to metric concepts. Let $PQRS$ be a quadrangle (see Fig.), let $A$ and $B$ be the intersection points of the opposite sides, and let $C$ and $D$ be the intersection points of the diagonals $SQ$ and $PR$ of $PQRS$ with the straight line $AB$. Then the quadruple of points $(ABCD)$ is a harmonic quadruple. A quadruple of straight lines (or planes) passing through a single point (a single straight line) is called a harmonic quadruple of straight lines (planes) if a straight line intersects it in a harmonic quadruple of points.
  
  

Latest revision as of 14:42, 29 April 2014

of points

A quadruple of points on a straight line with cross ratio equal to $-1$. If $(ABCD)$ is a harmonic quadruple of points, one says that the pair $AB$ harmonically divides the pair $CD$, or that the points $A$ and $B$ are harmonically conjugate with respect to the points $C$ and $D$; the pairs $AB$ and $CD$ are called harmonically conjugate.

Figure: h046530a

A harmonic quadruple can be defined without recourse to metric concepts. Let $PQRS$ be a quadrangle (see Fig.), let $A$ and $B$ be the intersection points of the opposite sides, and let $C$ and $D$ be the intersection points of the diagonals $SQ$ and $PR$ of $PQRS$ with the straight line $AB$. Then the quadruple of points $(ABCD)$ is a harmonic quadruple. A quadruple of straight lines (or planes) passing through a single point (a single straight line) is called a harmonic quadruple of straight lines (planes) if a straight line intersects it in a harmonic quadruple of points.


Comments

When the straight line is a complex one, but viewed as a Euclidean plane, one says harmonic quadrilateral, see [a1].

For example, use, etc. of harmonic quadruples see, for example, [a1][a3].

References

[a1] M. Berger, "Geometry" , I , Springer (1987) pp. 270
[a2] H.S.M. Coxeter, "Projective geometry" , Blaisdell (1964)
[a3] H.S.M. Coxeter, "The real projective plane" , McGraw-Hill (1949)
How to Cite This Entry:
Harmonic quadruple. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Harmonic_quadruple&oldid=17134
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article