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).