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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.