[§ 21. Harmonic properties of the complete quadrilateral and quadrangle.
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| Fig. 7. | Fig. 8. |
A figure formed by four lines in a plane is called a complete quadrilateral, or, shorter, a four-side. The four sides meet in six points, named the “vertices,” which may be joined by three lines (other than the sides), named the “diagonals” or “harmonic lines.” The diagonals enclose the “harmonic triangle of the quadrilateral.” In fig. 7, A′B′C′, B′AC, C′AB, CBA′ are the sides, A, A′, B, B′, C, C′ the vertices, AA′, BB′, CC′ the harmonic lines, and αβγ the harmonic triangle of the quadrilateral. A figure formed by four coplanar points is named a complete quadrangle, or, shorter, a four-point. The four points may be joined by six lines, named the “sides,” which intersect in three other points, termed the “diagonal or harmonic points.” The harmonic points are the vertices of the “harmonic triangle of the complete quadrangle.” In fig. 8, AA′, BB′ are the points, AA′, BB′, A′B′, B′A, AB, BA′ are the sides, L, M, N are the diagonal points, and LMN is the harmonic triangle of the quadrangle.
The harmonic property of the complete quadrilateral is: Any diagonal or harmonic line is harmonically divided by the other two; and of a complete quadrangle: The angle at any harmonic point is divided harmonically by the joins to the other harmonic points. To prove the first theorem, we have to prove (AA′, βγ), (BB′, γα), (CC′, βα) are harmonic. Consider the cross-ratio (CC′, αβ). Then projecting from A on BB′ we have A(CC′, αβ) = A(B′B, αγ). Projecting from A′ on BB′, A′(CC′, αβ) = A′(BB′, αγ). Hence (B′B, αγ) = (BB′, αγ), i.e. the cross-ratio (BB′, αγ) equals that of its reciprocal; hence the range is harmonic.
The second theorem states that the pencils L(BA, NM), M(B′A, LN), N(BA, LM) are harmonic. Deferring the subject of harmonic pencils to the next section, it will suffice to state here that any transversal intersects an harmonic pencil in an harmonic range. Consider the pencil L(BA, NM), then it is sufficient to prove (BA′, NM′) is harmonic. This follows from the previous theorem by considering A′B as a diagonal of the quadrilateral ALB′M.]
This property of the complete quadrilateral allows the solution of the problem:
To construct the harmonic conjugate D to a point C with regard to two given points A and B.
Through A draw any two lines, and through C one cutting the former two in G and H. Join these points to B, cutting the former two lines in E and F. The point D where EF cuts AB will be the harmonic conjugate required.
This remarkable construction requires nothing but the drawing of lines, and is therefore independent of measurement. In a similar manner the harmonic conjugate of the line VA for two lines VC, VD is constructed with the aid of the property of the complete quadrangle.