tetrad (or

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.html

CyclicProjective.html

Polar_Construction.html

Quadrangle_1.html

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