Complete_Quadrangle

Complete Quadrangle

This is the figure formed by four points A, B, C, D, no three of which are collinear. These points are the [vertices] of the complete quadrangle. The six lines formed by pairs of points are called [sides]. The intersection points E, F, G of the sides other than the four vertices are called [diagonal points] of the complete quadrilateral. The corresponding triangle is called [diagonal] triangle of the complete quadrilateral. There are several facts resulting immediately from the properties of a quadruple of harmonic conjugate points on a line, discussed in Harmonic.html :
1) The sides of the [diagonal] triangle are divided harmonically by the sides of the quadrangle. For example, in the figure below L, M are harmonic conjugate to G, F.
2) The sides of the complete quadrangle are also divided harmonically by the sides of the [diagonal triangle]. For example, G, I are conjugate to C, A.
3) The intersection points of the sides of the quadrangle and the sides of the [diagonal triangle] are, by three collinear. For example H, L, J are collinear. By the way, the line of H, L, J is the [trilinear polar] of the point A with respect to the triangle EFG.

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