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The figure determined by four lines and their six points of intersection (Johnson 1929, pp. 61-62). Note that this is different from a Complete Quadrangle. The midpoints of the diagonals of a complete quadrilateral are Collinear (Johnson 1929, pp. 152-153).
A theorem due to Steiner (Mention 1862, Johnson 1929, Steiner 1971) states that in a complete quadrilateral, the bisectors of angles are Concurrent at 16 points which are the incenters and Excenters of the four Triangles. Furthermore, these points are the intersections of two sets of four Circles each of which is a member of a conjugate coaxal system. The axes of these systems intersect at the point common to the Circumcircles of the quadrilateral.
See also Complete Quadrangle, Gauss-Bodenmiller Theorem, Polar Circle
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 230-231, 1969.
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA:
Houghton Mifflin, pp. 61-62, 149, 152-153, and 255-256, 1929.
Mention, M. J. ``Démonstration d'un Théorème de M. Steiner.'' Nouv. Ann. Math., 2nd Ser. 1, 16-20, 1862.
Mention, M. J. ``Démonstration d'un Théorème de M. Steiner.'' Nouv. Ann. Math., 2nd Ser. 1, 65-67, 1862.
Steiner, J. Gesammelte Werke, 2nd ed, Vol. 1. New York: Chelsea, p. 223, 1971.