Chitkara Open Access Journals - Blog

A New Proof of the Lester’s Perimeter Theorem in Euclidean Space

Author(s):  Oğuzhan Demirel, Leyla Aslan, Damla Topal,  Afyon Kocatepe University 

Keywords:  Euclidean isometry, Euclidean motion, Beckman-Quarles theorem


An injection defined from Euclidean n-space E^n(2<=n<∞) to itself which preserves the triangles of perimeter 1 is an Euclidean motion. J. Lester presented two different proofs for this theorem in Euclidean plane (Lester 1985) and Euclidean space (Lester 1986). In this study we present a general proof which works both in Euclidean plane (n=2) and Euclidean space (2<n< ∞).



Beckman F. S. and Quarles D. A.: On isometries of Euclidean spaces. Proc. Amer. Math. Soc. 4, 810-815 (1953).

Lester, J. A.: Euclidean plane point transformations preserving unit area or unit perimeter. Arch. Math. (Basel ) 45, 561-564 (1985).

Lester, J. A.: Martin’s the or em for Euclidean-space and a generalization to the perimeter case. J. Geom. 27, 29-35 (1986).