§ 22. Harmonic Pencils.—The theory of cross-ratios may be extended from points in a row to lines in a flat pencil and to planes in an axial pencil. We have seen (§ 13) that if the lines which join four points A, B, C, D to any point S be cut by any other line in A′, B′, C′, D′, then (AB, CD) = (A′B′, C′D′). In other words, four lines in a flat pencil are cut by every other line in four points whose cross-ratio is constant.
Definition.—By the cross-ratio of four rays in a flat pencil is meant the cross-ratio of the four points in which the rays are cut by any line. If a, b, c, d be the lines, then this cross-ratio is denoted by (ab, cd).
Definition.—By the cross-ratio of four planes in an axial pencil is understood the cross-ratio of the four points in which any line cuts the planes, or, what is the same thing, the cross-ratio of the four rays in which any plane cuts the four planes.
In order that this definition may have a meaning, it has to be proved that all lines cut the pencil in points which have the same cross-ratio. This is seen at once for two intersecting lines, as their plane cuts the axial pencil in a flat pencil, which is itself cut by the two lines. The cross-ratio of the four points on one line is therefore equal to that on the other, and equal to that of the four rays in the flat pencil.
If two non-intersecting lines p and q cut the four planes in A, B, C, D and A′, B′, C′, D′, draw a line r to meet both p and q, and let this line cut the planes in A″, B″, C″, D″. Then (AB, CD) = (A′B′, C′D′), for each is equal to (A″B″, C″D″).
§ 23. We may now also extend the notion of harmonic elements, viz.
Definition.—Four rays in a flat pencil and four planes in an axial pencil are said to be harmonic if their cross-ratio equals -1, that is, if they are cut by a line in four harmonic points.
If we understand by a “median line” of a triangle a line which joins a vertex to the middle point of the opposite side, and by a “median line” of a parallelogram a line joining middle points of opposite sides, we get as special cases of the last theorem:
The diagonals and median lines of a parallelogram form an harmonic pencil; and
At a vertex of any triangle, the two sides, the median line, and the line parallel to the base form an harmonic pencil.