The Complete Quadrilateral

The mathematical object that goes by the name of complete quadrilateral is neither complete nor quadrilateral, at least not in the sense in which the word "quadrilateral" appears in, say, Brahmagupta's theorem about cyclic quadrilaterals. Simply put, it's not a 4-sided polygon. (The latter is sometimes called a simple quadrilateral.) Seldom mentioned in elementary courses, it plays an important role in projective geometry, where it is used for a ruler only construction of harmonic conjugates and as a corner stone of the projective coordinate system [Möbius, p. 91, Kline, p. 127-128, Courant, p. 179].

A complete quadrilateral is a configuration of four straight lines in general position and six points at which the lines intersect. This is the (projective) configuration that Hilbert and Cohn-Vossen used to denote as (6₂4₃) [Hilbert, p. 96] meaning a system of 6 points and 4 lines, 2 lines through each point, 3 points per line. A triangle in this notations appears as (3₂3₂) and has the property that the straight lines joining the points of the configuration are necessarily the lines of the configuration. This is obviously not the case with the configuration (6₂4₃). There are three lines joining points of the configuration that do not count among the four configuration lines. (These are known as the diagonals of the complete quadrilateral.) This is why the configuration may be called incomplete. Adjoining the three lines to the configuration does not save the situation because such an operation adds new points of intersection. Adding them leaves room for new lines and so on. In fact, all but trivial configurations are incomplete.

A typical quadrilateral (4-sided polygon) is represented by the symbol (4₂4₂) and is therefore a typical quadrangle (4-angled polygon). However, the complete quadrangle is the configuration dual to (6₂4₃), i.e., (4₃6₂), which is the configuration of 4 points and 6 straight lines, 2 points on a line, 3 lines through a point.

In plane geometry, there are quite a few curious theorems associated with the complete quadrilateral [Wells, p. 34-35], some of which are illustrated by the applet below.

1. First of all, we have the Theorem of Complete Quadrilateral: the midpoints of the three diagonals are collinear.

2. Next we consider the four triangles formed by the four lines (omitting one of them at a time.) The orthocenters of the triangles are collinear and the line (Ortholine in the applet) is perpendicular to the line (Midline in the applet) of the three mid-diagonals.

3. Also, the ortholine serves as the common radical axis of the three circles constructed on the diagonals as diameters, such that whenever the circles intersect, all three of them intersect in two points on the ortholine.

4. The circumcircles of the four triangles meet in a point, the Miquel point of the complete quadrilateral.

5. The perpendiculars from the 9-point centers of the four triangles to the respective lines omitted from the 4-line in order to obtain the triangles, meet in a point. The common point lies on the ortholine.

6. If the Euler line of one of the four triangles is parallel to the respective omitted line, the same holds true of the remaining three triangles. (The proof depends on a property of Siamese triangle.)

(In the applet, the four lines are each defined by two draggable points. Dragging one of the points rotates the line around the other. The line may be also translated by dragging it anywhere away from the points. The four triangles are also shown in translated positions to avoid cluttering the diagram. Try moving -- not dragging -- the cursor over one of the translated triangles.)

Complete quadrilateral has further properties:

7. Any diagonal of a complete quadrilateral is harmonically cut by other two diagonals.

References

1. J. Aubrey, Brief Lives, Penguin Books, 2000
2. R. Courant and H. Robbins, What is Mathematics?, Oxford University Press, 1996
3. J. Fauvel, J. Gray (eds), The History of Mathematics. A Reader, The Open University, 1987
4. J. Fauvel et al (eds), Möbius and His Band, Oxford University Press, 1993
5. D. Hilbert and S. Cohn-Vossen, Geometry and Imagination, Chelsea Publishing Co, NY 1990.
6. C. Kimberling, Triangle Centers and Central Triangles, Utilitas Mathematica Publishing Inc., 1998
7. M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972
8. F. Morley, Orthocentric Properties of the Plane n-Line, Trans Amer Math Soc, 4 (1903) 1-12.
9. D. Wells, Curious and Interesting Geometry, Penguin Books, 1991

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