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The figure determined by four lines, no three of which are concurrent, and their six points of intersection (Johnson 1929, pp. 61-62). Note that this figure is different from a complete quadrangle. A complete quadrilateral has three diagonals (compared to two for an ordinary quadrilateral). The midpoints of the diagonals of a complete quadrilateral are collinear on a line (Johnson 1929, pp. 152-153).

A theorem due to Steiner (Mention 1862ab, 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.

Newton proved that, if a conic section is inscribed in a complete quadrilateral, then its center lies on (Wells 1991). In addition, the orthocenters of the four triangles formed by a complete quadrilateral lie on a line which is perpendicular to . Plücker proved that the circles having the three diagonals as diameters have two common points which lie on the line joining the four triangles' orthocenters (Wells 1991).

Complete Quadrangle, Gauss-Bodenmiller Theorem, Midpoint, Orthocenter, Polar Circle, Quadrilateral

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## References

Carnot, L. N. M. De la corrélation des figures de géométrie. Paris: l'Imprimerie de Crapelet, p. 122, 1801.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 230-231, 1969.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 81, 1928.Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 25, 1930.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, 1862a.Mention, M. J. "Démonstration d'un Théorème de M. Steiner." Nouv. Ann. Math., 2nd Ser. 1, 65-67, 1862b.Steiner, J. Gesammelte Werke, 2nd ed., Vol. 1. New York: Chelsea, p. 223, 1971.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 35, 1991.

## Cite this as:

Weisstein, Eric W. "Complete Quadrilateral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CompleteQuadrilateral.html