Projective Harmonic Conjugate - From Complete Quadrangle

From Complete Quadrangle

Another approach to the harmonic conjugate is through the concept of a complete quadrangle such as KLMN in the above diagram. Based on four points, the complete quadrangle has pairs of opposite sides and diagonals. In the expression of projective harmonic conjugates by H. S. M. Coxeter, the diagonals are considered a pair of opposite sides:

D is the harmonic conjugate of C with respect to A and B, which means that there is a quadrangle IJKL such that one pair of opposite sides intersect at A, and a second pair at B, while the third pair meet AB at C and D.

It was Karl von Staudt that first used the harmonic conjugate as the basis for projective geometry independent of metric considerations:

...Staudt succeeded in freeing projective geometry from elementary geometry. In his Geometrie der Lage Staudt introduced a harmonic quadruple of elements independently of the concept of the cross ratio following a purely projective route, using a complete quadrangle or quadrilateral.

To see the complete quadrangle applied to obtaining the midpoint, consider the following passage from J. W. Young:

If two arbitrary lines AQ and AS are drawn through A and lines BS and BQ are drawn through B parallel to AQ and AS respectively, the lines AQ and SB meet, by definition, in a point R at infinity, while AS and QB meet by definition in a point P at infinity. The complete quadrilateral PQRS then has two diagonal points at A and B, while the remaining pair of opposite sides pass through M and the point at infinity on AB. The point M is then by construction the harmonic conjugate of the point at infinity on AB with respect to A and B. On the other hand, that M is the midpoint of the segment AB follows from the familiar proposition that the diagonals of a parallelogram (PQRS) bisect each other.