Rusty Compass Construction of Equilateral Triangle: What is it?
A Mathematical Droodle

Explanation

The standard Euclidean construction of an equilateral triangle with a given side length assumes it is possible to set up the distance between compass legs to the length of the base. Some compasses are rusty, however, so that their opening can't be changed. Given a rusty compass whose opening does not match the given side length, but is more than its half, is it still possible to complete the construction? According to Dan Pedoe [Pedoe, p. xxxv-xxxvi], a student, while doodling in class, came up with a drawing unaware of its significance.

All five circles in the drawing have exactly same radius. The first two are centered at the given points A and B. By our assumption, the two circles intersect at two points. Let D be one of them, and make it the center of the third circle. Besides A and B, the latter intersects the first two circles at points E and F that again serve as the centers of additional two circles. These meet at D and one other point, C. The triangle ABC is equilateral.

Indeed, by construction, the central angle BED is 60^o. It follows that the inscribed angle BCD is half that - 30^o. The same is true of angle ACD. Because of the symmetry, triangle ABC is isosceles (AC = BC) with the apex angle of 60^o. Q.E.D.

A short history of constructions with ruler and rusty compass could be found in Chapter 7 of G. E. Martin's book. The chapter is remarkable in that it is supposed to be written by the reader. The reader is however expected to come up with the exercises that match the hints and answers at the end of the book.

References

1. G. E. Martin, Geometric Constructions, Springer, 1998
2. D. Pedoe, Circles: A Mathematical View, MAA, 1995