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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.