How many triangles? (English) Zbl 0920.51010

Grünbaum, Branko

Geombinatorics 8, No. 1, 154-159 (1998).

A result known as Robert’s Theorem (1889) states that \(n\) lines in general position in the plane determine at least \(n-2\) triangular regions. However, his arguments were not convincing (as mentioned in B. Grünbaum, Regional Conf. Series in Math. No. 10 Am. Math. Soc. (1972; Zbl 0249.50011)). R. W. Shannon gave a proof in 1979 [Geom. Dedicata 8, 179-187 (1979; Zbl 0423.51013)] which is not elementary. In this note a proof given by A. Ya. Belov [Russ. Math. Surv. 47, No. 3, 167-168 (1992); translation from Usp. Mat. Nauk 47, No. 3(285), 151-152 (1992; Zbl 0850.51006)] is neatly reproduced.

Reviewer: T.Thrivikraman (Cochin)

MSC:

51N20

Euclidean analytic geometry

Keywords:

triangles; homogeneous linear equations

Citations:

Zbl 0249.50011; Zbl 0423.51013; Zbl 0850.51006