The Steiner Theorem for Trapezium (Jakob Steiner, 1796-1863)

The students noted that Steiner's proof was comparable to the solution of their problem (the proof of which is given below) and thus were stimulated to continue researching the use of the Steiner theorem for the trapezium, which ultimately led to an interest in general geometric constructions according to the rules of ancient Greek mathematics, and building geometric figures using a straightedge only.

In addition, the solution requires some knowledge of the following special properties of trapeziums (of which the students are usually not aware), such as the Steiner theorem for a trapezium, and which should be introduced to them during presentation of the problem.

Property B: Is a result of the Steiner theorem for the trapezium.

Constructing an application of the Steiner theorem for the trapezium (will be used below)

It is important to note that according to the Steiner theorem for the trapezium, the following two constructions with a straightedge alone are equivalent:

There are seven different solutions, depending on the location of point P (see Figure 2), plus another option using the Steiner theorem in a more straightforward manner.

From property B, which stems from the Steiner theorem for a trapezium, line [P.sub.5][X.sub.7] is perpendicular to diameter AB.

Straightforward application of the Steiner theorem: Another option is to apply the Steiner theorem for trapezium (see Figure 15).

However, for cases 5-7 (to construct a perpendicular at a point on the segment AB or its continuation), one must apply the Steiner theorem.

Then, using the Steiner theorem for trapezium [H.sub.1][K.sub.1][K.sub.2][H.sub.2], it is possible to bisect each of the chords H1K1 and H2K2 (points L and N).

Choose two points on l and two points on k and draw the trapezium [F.sub.1][F.sub.2][G.sub.2][G.sub.1], then, using the Steiner theorem, find the midpoints of the bases of the trapezium (one midpoint is sufficient) and apply the previous construction of a diameter of the circle.