Steiner's Quadrilateral Theorems

Complete quadrilaterals and complete quadrangles

A Complete quadrangle is the figure formed by any four points A, B, C, D and the six lines joining pairs of them.

A Complete quadrilateral is the figure formed by any four lines a, b, c, d and their six points of intersecction.
[check definitions]

Each of these objects has a diagonal triangle.

A complete quadrilateral yields four triangles by dropping each edge in turn, and there are several interesting theorems got by considering well known “triangle objects” for each of these.

Each point in a complete quadrilateral, determined by the intersection of two of the lines, has a unique opposite point, determined by the intersection of the other two lines. The line connecting these points is a diagonal. The segment joining these points is the diagonal segment, on which a diagonal circle can be constructed.

In 1828 Jacob Steiner published a short note in Gergonne’s Annales containing 10 statements about the complete quadrilateral. These 10 statements are delightfully interconnected. More or less in his own words, here is what he said [we are helped by a translation of Steiner’s note by Jeanne-Pierre Ehrman, taken from his excellent paper on this topic, which includes some of the new results below].

Suppose four lines intersect two by two in six points.
(1) These four lines, taken three at a time, form four triangles whose circumcircles pass through the same point [This is called the Miquel pole of the quadrilateral.]

(2) The centers of the four circles (and the Miquel pole) lie on the same circle.

(3) The perpendicular feet from the [Miquel] point to the four lines lie on the same line [which is the Simpson line of the Miquel point with respect to each of the four triangles].

(4) The orthocenters of the four triangles lie on the same line.

(5) The two lines are parallel.

(6) The midpoints of the diagonals of the complete quadrilateral formed by the four lines lie on the same line (the Newton line).

(7) The Newton line is perpendicular to the lines in (3) and (4).

(8) Each of the four triangles in (1) has an incircle and three excircles. Their 16 centers lie, four by four, on eight new circles.

(9) These eight new circles form two sets of four, each circle of one set being orthogonal to each circle of the other set. The centers of the circles of each set lie on the same line. These two lines are perpendicular.

(10) These last two lines intersect in the [Miquel] point defined in 1.

When told these 10 results from Steiner, the Princeton mathematician Joseph Cohen remarked “and the last step is ‘God exists.’”

The first 7 of these are illustrated in the following figure

The rest, along with some new resullts, are illustrated at the end.

More recent work has added these results
(11) The three diagonal circles are coaxal with Newton line as axis and the orthocentric line as radical axis.

(12) There is one parabola tangent to all four lines. The Miquel point is its focus and the orthocentral line its directrix.

(13) The circumcenter of the diagonal triangle is on the orthocentric line.

(14) The 4 MacBeath deltoid lines are concurrent (in the MacBeath deltoid center). Recall that the MacBeath line of a triangle is the perpendicular bisector of O—H, and that the MacBeath deltoid center is the center of the unique deltoid that touches (the sides?)

(15) The 4 circumcenters form a cyclic quadrangle Q1 and the further quadrangle Q2 found by the four orthocenters of Q1 is also cyclic, with center the MacBeath deltoid center of Q, and related to Q1 by a half turn.

(16) The Miquel point is the isogonal conjugate with respect to each of the four triangles of the infinite point on the Newton line.

(17) The intersections of the bisectors drawn from the opposite ends of a diagonal are not incenters. These four intersections create segments whose diagonal circles are members of the coaxal system (shown as dashed circles in the Figure.