05 June 2020

Steiner construction of a regular pentagon

By the Poncelet-Steiner theorem, any point of the plane that can be constructed with Euclidean tools (straightedge and compasses) can also be constructed with the straightedge alone, provided a circle and its center be given. In fact, an arc of this circle is already sufficient (see here). Straightedge constructions using a circle and its center are called Steiner constructions. What follows is a Steiner construction of a regular pentagon.

A basic fact in this context is the following: given a bisected segment, the parallel line through any given point can be constructed with the sole straightedge. (Problem 1, here) The solution consists of 7 lines, including the given segment and the parallel obtained. In our construction below, this basic operation is used three times, the segment being a diameter of the circle, bisected by the center. Each time the intermediate 5 lines have been omitted, leaving only the original diameter and (in blue) the final parallel. There are 25 lines displayed (a,b,...,y). If the 15 omitted lines are counted in, a grand total of 40 lines is required.

Verification by elementary analytical geometry:

The construction can be executed with less than the 40 lines mentioned above. Once the two orthogonal diameters are obtained, Pascal's theorem allows to obtain the tangents i and j with only three lines beyond the ones already present. In the figure below the letters are unrelated to those above. A notation like P12Q6 means that this point is P1=P2=Q6. The points P1,...,P6 form a first Pascal hexagon, with collinear intersections on the line p, and Q1,...,Q6 a second hexagon with collinear intersections on the line q. The tangents in blue require three extra lines, shown in red.

The preliminary construction leading to the two orthogonal diameters and the orthogonal tangents is as follows, if all the lines are displayed:

and this is how the whole looks with all of its lines fully displayed (preliminary construction in blue, sides of the pentagon in red):

There are 32 lines required. (Geogebra file here.)