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of points

A quadruple of points on a straight line with cross ratio equal to . If is a harmonic quadruple of points, one says that the pair harmonically divides the pair , or that the points and are harmonically conjugate with respect to the points and ; the pairs and are called harmonically conjugate.

Figure: h046530a

A harmonic quadruple can be defined without recourse to metric concepts. Let be a quadrangle (see Fig.), let and be the intersection points of the opposite sides, and let and be the intersection points of the diagonals and of with the straight line . Then the quadruple of points 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.