Quadrangle

Harmonic properties of the quadrangle

We extend the sides and draw the diagonals AC and BD. Points {I,K,F,G} build a harmonic
tetrad (or harmonic division). The same is true for the sides AD and BC. Points {H,L,F,E} build
also a harmonic tetrad.

This theorem pertains to the realm of projective geometry. With a homography (or
projectivity) the general quadrangle can be mapped onto a square and the various relations
are equivalent with (trivial) properties of the square. Notice that the tetrads of lines passing
through G (GA, GD, GK, GE) and E (EB, EA, EH, EG) are harmonic. The line-pairs (HK,
IL) and (HI, KL) concure at points M and J, respectively, lying on the line EG. The tetrads of
lines concurring at these points are also harmonic.
Every bundle of four lines through a point, passing also through the points of a harmonic tetrad is
said to be harmonic. Every line cutting the lines of a harmonic tetrad of lines cuts them at points
forming a harmonic tetrad.
This figure can be noticed on every painting that respects the laws of perspectivity (read
projective geometry). A special case is encountered in the document: Trapezium_0.html .
An alternative proof of the properties stated results by repeated application of the basic construction
procedure for harmonic points on a line (see Harmonic.html ).
For an application of the figure, in the case of a cyclic quadrilateral look at the document
CyclicProjective.html . An other use of the figure, relating to conics, is made in the document
Polar_Construction.html .
The figure resulting by applying the previous construction to the quadrangle HKLI is displayed in
Quadrangle_1.html .

Harmonic properties of the quadrangle

See Also