Euclidean Properties
Wells (1991) describes several additional properties of complete quadrilaterals that involve metric properties of the Euclidean plane, rather than being purely projective. The midpoints of the diagonals are collinear, and (as proved by Isaac Newton) also collinear with the center of a conic that is tangent to all four lines of the quadrilateral. Any three of the lines of the quadrilateral form the sides of a triangle; the orthocenters of the four triangles formed in this way lie on a second line, perpendicular to the one through the midpoints. The circumcircles of these same four triangles meet in a point. In addition, the three circles having the diagonals as diameters belong to a common pencil of circles the axis of which is the line through the orthocenters.
Read more about this topic: Complete Quadrangle
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“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)