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If the four points making up a Quadrilateral are joined pairwise by six distinct lines, a figure known as a complete quadrangle results. Note that a complete quadrilateral is defined differently from a Complete Quadrangle.
The midpoints of the sides of any complete quadrangle and the three diagonal points all lie on a Conic known as the Nine-Point Conic. If it is an Orthocentric Quadrilateral, the Conic reduces to a Circle. The Orthocenters of the four Triangles of a complete quadrangle are Collinear on the Radical Line of the Circles on the diameters of a Quadrilateral.
See also Complete Quadrangle, Ptolemy's Theorem
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 230-231, 1969.
Demir, H. ``The Compleat [sic] Cyclic Quadrilateral.'' Amer. Math. Monthly 79, 777-778, 1972.
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA:
Houghton Mifflin, pp. 61-62, 1929.
Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 101-104, 1990.