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

Received: September 12, 2019 Revised: January 25, 2020 Accepted: February 03, 2020 Published Online: March 30, 2020 An injection defined from Euclidean n-space E n n ( ) 2≤ < ∞ 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 < ∞).


Introduction
It is well known that some geometric transformations can be characterized by the properties of they preserve. For instance, collinearity preserving bijections of Euclidean n-space E n n � ( ) 2 ≤ <∞ characterize the affine transformations and this theorem is known as the fundamental theorem of affine geometry. The Möbius transformations of the extended complex plane can be characterized by as transformations preserving quadruples of concylic points. In Minkowski space the Alexandrov's theorem which describes Lorentz transformations as the transformations of Minkowski space preserving the speed of light. In Euclidean space E n n � ( ) 2 ≤ <∞ the Beckman-Quarles theorem which identifies as motions those functions from E n to itself preserving pairs of points of a given fixed distance apart. More precisely Beckman-Quarles theorem (Beckman and Quarles 1953) states that a function from E n to itself which preserves the relation | | x y Q − = for a fixed � Q ∈ +  must be an Euclidean motion where | | x y − denotes the distance between x y E n , ∈ . This theorem plays a major role in our result. G. Martin (unpublished) characterized the equiaffine transformations (affine and area preserving) of E 2 via the injections which preserves triangles with area 1 as follows, see (Lester 1985). Theorem 1.1: An injection from Euclidean plane to itself which preserves triangles with area 1 must be equiaffine, see (Lester 1985).
J. Lester generalized this theorem to the Euclidean space E n as follows: Theorem 1.2: Let f be an injection from Euclidean space � E n n 2 ≤ <∞ ( )to itself which preserves triangles with area 1 must be a Euclidean motion, see (Lester 1986). J. Lester also obtained the following results using triangles of perimeter 1 instead of triangles of area 1. Theorem 1.3: Let f be an injection from Euclidean plane � E 2 to itself which preserves triangles of perimeter 1 must be a Euclidean motion, see (Lester 1985). Theorem 1.4: Let f be an injection from Euclidean space E n n ( ) 2 ≤ <∞ to itself which preserves triangles of perimeter 1 must be a Euclidean motion, see (Lester 1986).

A New Proof of the Lester's Perimeter Theorem in Euclidean Space
is a triangle of perimeter 1 then must be a point on n-dimensional rotated ellipsoid with equation

is a triangle of perimeter
Then the locus of all points X E n ∈ is an n-dimensional rotated ellipsoid drilled by two points. These two points are clearly the vertices of the ellipsoid. Lemma 2.2. Let f be an injection from Euclidean space E n n ( ) 2 < <∞ to itself which preserves triangles of perimeter 1. Then f preserves the right angles. Proof: Let l 1 and l 2 be two distinct lines in E n which meets perpendicularly. Denote the common point of these lines by F 1 . Now take a point on l 2 , say F 2 , such that Now following the same way in the proof of Lemma 2.1, one can easily construct the n-dimensional rotated ellipsoid Ω with focal points F 1 and F 2 . Clearly l 1 and Ω meets at two points, say A and B. Now draw the Euclidean line passing through F 2 and parallel to L 1 . Obviously, this line and Ω meets at two points and denote them by C and D.
It is clear that either AC BD  or AD BC  . Without loss of generality we may assume AC BD  . Clearly one can easily see that AF F C 1 2 is a rectangle which consists of four triangles AF F F F C F CA 1 2 1 2 2 , , and CAF 1 . The perimeter of these triangles is 1. Clearly, by hypothesis, the perimeter of the triangles A F F F F C F C A ' ' ', ' ' ', ' is a rectangle, see (Lester 1985