The Euclidean right triangle: remarks on the regular pentagon and the golden ratio

Abstract

The article contains an original construction of the Euclidean triangle and its proof, from which a simple graphical technique for the construction of the regular pentagon is derived.

Appendix: Complete (and new) construction of the regular pentagon, given its side AB

Given the side AB, from the endpoint B draw the line BD, perpendicular to it (Fig. 14). Having found the point D such that BD has the same length as AB, join C, the midpoint of the side AB, to D. With centre in C and radius CB, draw an arc of circle that intersects CD in the point E.

Fig. 14

Construction of the regular pentagon

The distance AE is the radius of the circumscribed circle of the regular pentagon having side AB. Thus, the arc centred in A with radius AE intersects the line perpendicular to AB and passing through C in the point O, centre of this circle (the point O may be found, in a more direct way, by intersecting the last-mentioned arc with another one, with the same radius and centre in B). Having drawn the circle, the vertices of the regular pentagon are determined, one by the line CO (vertex G), and the other two by the circle arc centred, respectively in A and B and radius AB (vertices F and H).

Translated from the Italian by Daniele A. Gewurz.