New Constants Associated to Regular Tetrahedron and Cube

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1 New Constants Associated to Regular Tetrahedron and Cube Mamuka Meskhishvili Abstract We consider the points on the sphere with center at the centroid of the regular tetrahedron and the cube. We prove that the sum of the fourth power distances from these points to the vertices of the regular tetrahedron is constant and in case of the cube the sums of the fourth and the sixth power distances are constant too. Keywords. Regular tetrahedron, cube, Platonic solid, sphere, centroid, constant sum of the square distances, constant sum of the fourth power distances, constant sum of the sixth power distances AMS Classification. 51M04, 51N20, 51N35.

2 1 Introduction Consider any regular polygon (n-gon) and a point X on any circle with center at the centroid of the n-gon. Denote the vertices of the n-gon as P i. Then it follows S (2) = n d 2 (X, P i ), where S (2) is constant, i.e. the sum of the square distances d 2 (X, P i ) remains unchanged, when a point X moves on any circle with center at the centroid 1]. The case of the second power (square) distances admits a 3D-generalization. Furthermore, an analogous result is obtained for high dimensional space 2], but nothing is known if the powers of distances are more than 2. We consider the sums of even powers of distances S (2m) = n d 2m (X, P i ), when m > 1, for the regular tetrahedron (n = 4) and the cube (n = 8). General theorem on regular polyhedrons is known only for the second power sums. Consider a regular polyhedron with n vertices and an arbitrary point X on any sphere with center at the centroid O of the polyhedron. The sum of the square distances from X to the vertices of the polyhedron is constant and equals S (2) = n(l 2 + R 2 ), (1) where L is a radius of the sphere and R is a circumradius of the regular polyhedron. This result is easily obtained by using the properties of a scalar vector product: 2

3 S (2) = = n n d 2 2 (X, P i ) = XP i = n XO 2 + 2XO n ( n 2 n 2 OP i + OP i, XO + OP i ) 2 which gives us (1) because of XO 2 = L 2, n OP i = 0 and OP i 2 = R 2. The same equality holds if the n points are all at the same distance L from their centroid, even if they fo not form a regular polyhedron: that is the case, for instance, of a semiregular polyhedron. This method cannot be used for high powers, when m > 1, because from the definition of the scalar product only the second power of the vector is defined. Instead of vectors, the coordinate method can be used, which gives huge algebraic expressions and it is difficult to predict which of their surprising simplifications will occur. From my consideration, inefficiency of the scalar vector product and complex algebraic expressions are the reasons why the results of this article were not known till present. In the article we will prove, that results about regular triangle and square 3] admit 3D-generalization. In particular, the sum of the fourth power distances, from an arbitrary point on any sphere with center at the centroid of the regular tetrahedron to the vertices of the tetrahedron, is constant and in case of the cube the sums of the fourth and the sixth powers are constant too. 3

4 2 Regular Tetrahedron Introduce the Cartesian coordinate system so that the coordinates of the vertices of the regular tetrahedron are: P 1 (4t, 0, 0), ( ) P 2 2t, 2 3 t, 0, ( ) P 3 2t, 2 3 t, 0, ( ) P 4 0, 0, 4 2 t. The origin of the coordinate system is chosen to coincide with the centroid of the triangle P 1 P 2 P 3 and x and z axes pass through the vertices P 1 and P 4, respectively. Then the centroid of the tetrahedron is: O ( 0, 0, 2 t ). The sphere equation with center at the centroid and any radius L: x 2 + y 2 + ( z 2 t ) 2 = L 2. The circumradius of the tetrahedron: R = 3 2 t. The square distances from an arbitrary X(x, y, z) point to the vertices are: 4

5 d 2 (X, P 1 ) = (x 4t) 2 + y 2 + z 2, d 2 (X, P 2 ) = (x + 2t) 2 + ( y 2 3 t ) 2 + z 2, d 2 (X, P 3 ) = (x + 2t) 2 + ( y t ) 2 + z 2, d 2 (X, P 4 ) = x 2 + y 2 + ( z 4 2 t ) 2. The sum of the square distances: 4 S (2) = d 2 (X, P i ) = 4 x 2 + y 2 + ( z 2 t ) ] t 2 = 4(L 2 + R 2 ), which is (1). The sum of the fourth power distances: S (4) = ( (x 4t) 2 + y 2 + z 2) 2 + ((x + 2t) 2 + (y 2 3 t) 2 + z 2) 2 + ((x + 2t) 2 + (y t) 2 + z 2) 2 + ( x 2 + y 2 + (z 4 2 t) 2) 2. It is difficult to imagine that this algebraic expression can be expressed only in terms of L and R, but the plane case results 3] give us this hope. 5

6 Indeed: S (4) = (x 4t) 4 + y 4 + z 4 + (x + 2t) 4 + (y 2 3 t) 4 + z 4 + (x + 2t) 4 + (y t) 4 + z 4 + (z 4 2 t) 4 + x 4 + y (x 4t) 2 y 2 + (x 4t) 2 z 2 + y 2 z 2 + (x + 2t) 2 (y 2 3 t) 2 + (x + 2t) 2 z 2 + (y 2 3 t) 2 z 2 + (x + 2t) 2 (y t) 2 + (x + 2t) 2 z 2 + (y t) 2 z 2 + (z 4 2 t) 2 x 2 + (z 4 2 t) 2 y 2 + x 2 y 2] = x 4 4x 3 4t + 6x 2 (4t) 2 4x(4t) 3 + (4t) 4 + y 4 + z x 4 + 4x 3 (2t) + 6x 2 (2t) 2 + 4x(2t) 3 + (2t) 4 + z 4 4z t + 6z 2 (4 2 t) 2 + y 4 + 6y 2 (2 3 t) 2 + (2 3 t) 4 + z 4] 4z(4 2 t) 3 + (4 2 t) 4 + x 4 + y (x 4t) 2 (y 2 + z 2 ) + y 2 z 2 + (x + 2t) 2 2(y t 2 ) + 2z 2 (x + 2t) 2 + 2z 2 (y t 2 ) + (z 4 2 t) 2 (x 2 + y 2 ) + x 2 y 2] = x 4 + y 4 + z 4 16x 3 t + 96x 2 t 2 256xt t x 4 + y 4 + z 4 + 8x 3 t + 24x 2 t 2 6

7 + 32xt t x 2 t t 4] + x 4 + y 4 + z z 3 t +192z 2 t zt t x 2 y 2 + y 2 z 2 + (y 2 + z 2 )(x t 2 8xt) + (x 2 + y 2 ) ( z zt + 32t 2) ( )] + 2 (y 2 +12t 2 )(x +2t) 2 + z 2 (x +2t) 2 + z 2 (y 2 +12t 2 ) = 4 x 4 + y 4 + z x 2 t 2 48xt t y 2 t z 3 t + 48z 2 t zt 3] + 2 2x 2 y 2 + 2y 2 z 2 + 2x 2 z y 2 t 2 8xty z 2 t 2 8xtz ztx t 2 x zty 2 ( + 2 y 2 x 2 + 4xty 2 + 4y 2 t 2 + z 2 x 2 + 4z 2 xt + 4z 2 t x 2 t xt t 4 + y 2 z z 2 t 2)] = = = 4 x 4 + y 4 + z 4 + 2x 2 y 2 + 2x 2 z 2 + 2y 2 z x 2 t y 2 t z 2 t zt t z 3 t 4 2 zx 2 t 4 ] 2 zy 2 t = 4 (x 2 + y 2 + ( z 2 t ) ) t 2( x 2 + y 2 + ( z 2 t ) 2 ) + 324t 4]. 7

8 In L and R notations: ( S (4) = 4 L 4 + R L2 R 2). Theorem 2.1. The sum of the fourth power distances S (4), from an arbitrary point on any sphere with center at the centroid of a regular tetrahedron to the vertices of the tetrahedron, is constant S (4) = 4 (L 2 + R 2 ) L2 R 2], (2) where L is the radius of the sphere and R is the circumradius of the tetrahedron. For given tetrahedron R is fixed and from the expression S (4) the equivalency follows: S (4) = const L = const, so is true vice versa: Theorem 2.2. The set of points, from which the sum of the fourth power distances S (4) to the vertices of a regular tetrahedron is constant, is: - the sphere with center at the centroid of the tetrahedron, if S (4) > 4R 4 ; - the point the centroid of the tetrahedron, if S (4) = 4R 4 ; - the empty set, if S (4) < 4R 4 ; where R is the circumradius of the tetrahedron. As for the sums of high powers (more than 4), they will no longer be constant. 8

9 3 Cube Introduce the Cartesian coordinate system with the origin at the centroid of the cube and with axes parallel to the sides of the cube. Denote the sides by 2a. Then the coordinates of the vertices are: P 1 (a, a, a), P 2 (a, a, a), P 3 (a, a, a), P 4 (a, a, a), P 5 ( a, a, a), P 6 ( a, a, a), P 7 ( a, a, a), P 8 ( a, a, a). The sphere equation with center at the centroid and any radius L: x 2 + y 2 + z 2 = L 2. The circumradius of the cube R = a 3. The square distances from an arbitrary X(x, y, z) point to the vertices are: d 2 (X, P 1 ) = (x a) 2 + (y + a) 2 + (z a) 2, d 2 (X, P 2 ) = (x a) 2 + (y + a) 2 + (z + a) 2, d 2 (X, P 3 ) = (x a) 2 + (y a) 2 + (z a) 2, d 2 (X, P 4 ) = (x a) 2 + (y a) 2 + (z + a) 2, d 2 (X, P 5 ) = (x + a) 2 + (y a) 2 + (z a) 2, d 2 (X, P 6 ) = (x + a) 2 + (y a) 2 + (z + a) 2, d 2 (X, P 7 ) = (x + a) 2 + (y + a) 2 + (z a) 2, 9

10 d 2 (X, P 8 ) = (x + a) 2 + (y + a) 2 + (z + a) 2. The sum of the square distances: S (2) = 8 d 2 (X, P i ) = 8 x 2 + y 2 + z 2 + 3a 2] = 8(L 2 + R 2 ) which is (1). The sum of the fourth power distances: S (4) = 8 d 4 (X, P i ) = 4 (x + a) 4 + (x a) 4 + (y + a) 4 + (y a) 4 + (z + a) 4 + (z a) 4] + 4 (x + a) 2 (y + a) 2 + (x + a) 2 (y a) 2 + (x + a) 2 (z + a) 2 + (x + a) 2 (z a) 2 + (x a) 2 (y + a) 2 + (x a) 2 (y a) 2 + (x a) 2 (z + a) 2 + (x a) 2 (z a) 2 + (y + a) 2 (z + a) 2 + (y + a) 2 (z a) 2 + (y a) 2 (z + a) 2 + (y a) 2 (z a) 2] = 8 x 4 + y 4 + z 4 + 6x 2 a 2 + 6y 2 a 2 + 6z 2 a 2 + 3a 4] + 4 2(x + a) 2 (y 2 + a 2 ) + 2(x + a) 2 (z 2 + a 2 ) 10

11 + 2(x a) 2 (y 2 + a 2 ) + 2(x a) 2 (z 2 + a 2 ) ] + 2(y + a) 2 (z 2 + a 2 ) + 2(y a) 2 (z 2 + a 2 ) = 8 x 4 + y 4 + z 4 + 6x 2 a 2 + 6y 2 a 2 + 6z 2 a 2 + 3a 4 + (x + a) 2 (y 2 + z 2 + 2a 2 ) + (x a) 2 (y 2 + z 2 + 2a 2 ) ] + 2(z 2 + a 2 )(y 2 + a 2 ) = 8 x 4 + y 4 + z 4 + 6x 2 a 2 + 6y 2 a 2 + 6z 2 a 2 + 3a 4 ] + 2(y 2 + z 2 + 2a 2 )(x 2 + a 2 ) + 2(z 2 + a 2 )(y 2 + a 2 ) = 8 y 4 + y 4 + z 4 + 2x 2 y 2 + 2x 2 z 2 + 2z 2 y x 2 a y 2 a z 2 a 2 + 9a 4] = 8 (x 2 + y 2 + z 2 ) a 2 (x 2 + y 2 + z 2 ) + 9a 4] = 8 L 4 + R L2 R 2]. Theorem 3.1. The sum of the fourth power distances S (4), from an arbitrary point on any sphere with center at the centroid of a cube to the vertices of the cube, is constant S (4) = 8 (L 2 + R 2 ) L2 R 2], (3) where L is the radius of the sphere and R is the circumradius of the cube. From the equivalency 11

12 S (4) = const L = const, is true vice versa: Theorem 3.2. The set of points, from which the sum of the fourth power distances S (4) to the vertices of a cube is constant, is: - the sphere with center at the centroid of the cube, if S (4) > 8R 4 ; - the point the centroid of the cube, if S (4) = 8R 4 ; - the empty set, if S (4) < 8R 4 ; where R is the circumradius of the cube. Consider the sum of the sixth power distances: S (6) = 8 d 6 (X, P i ). To simplify this expression we use the following identity: (u + v + w) 3 = u 3 + v 3 + w 3 + 3(u + v)(u + w)(v + w). S (6) = (x + a) 6 + (y + a) 6 + (z + a) 6 ((x a) 2 + (y + a) 2)( (x + a) 2 + (z + a) 2) ( (y + a) 2 + (z + a) 2)] + (x + a) 6 + (y + a) 6 + (z a) 6 12

13 ((x a) 2 + (y + a) 2)( (x + a) 2 + (z a) 2) ( (y + a) 2 + (z a) 2)] + (x + a) 6 + (y a) 6 + (z + a) 6 ((x a) 2 + (y a) 2)( (x + a) 2 + (z + a) 2) ( (y a) 2 + (z + a) 2)] + (x + a) 6 + (y a) 6 + (z a) 6 ((x a) 2 + (y a) 2)( (x + a) 2 + (z a) 2) ( (y a) 2 + (z a) 2)] + (x a) 6 + (y + a) 6 + (z + a) 6 ((x + 3 a) 2 + (y + a) 2)( (x a) 2 + (z + a) 2) ( (y + a) 2 + (z + a) 2)] + (x a) 6 + (y + a) 6 + (z a) 6 ((x + 3 a) 2 + (y + a) 2)( (x a) 2 + (z a) 2) ( (y + a) 2 + (z a) 2)] + (x a) 6 + (y a) 6 + (z + a) 6 ((x + 3 a) 2 + (y a) 2)( (x a) 2 + (z + a) 2) ( (y a) 2 + (z + a) 2)] 13

14 + (x a) 6 + (y a) 6 + (z a) 6 ((x + 3 a) 2 + (y a) 2)( (x a) 2 + (z a) 2) ( (y a) 2 + (z a) 2)] = 4 (x + a) 6 + (x a) 6 + (y + a) 6 ((x a) 2 + (y + a) 2) + (y a) 6 + (z + a) 6 + (z a) 6] ( ((x + a) 2 + (z + a) 2)( (y + a) 2 + (z + a) 2) + ( (x + a) 2 + (z a) 2)( (y + a) 2 + (z a) 2)) + ( (x + a) 2 + (y a) 2) ( ((x + a) 2 + (z + a) 2)( (y a) 2 + (z + a) 2) + ( (x + a) 2 + (z a) 2)( (y a) 2 + (z a) 2)) + ( (x a) 2 + (y + a) 2) ( ((x a) 2 + (z + a) 2)( (y + a) 2 + (z + a) 2) + ( (x a) 2 + (z a) 2)( (y + a) 2 + (z a) 2)) + ( (x a) 2 + (y a) 2) ( ((x a) 2 + (z + a) 2)( (y a) 2 + (z + a) 2) 14

15 + ( (x a) 2 + (z a) 2)( (y a) 2 + (z a) 2))]. The second addend equals: ((x 3 + a) 2 + (y + a) 2)( (x + a) 2 (y + a) 2 + (x + a) 2 (z + a) 2 + (y + a) 2 (z + a) 2 + (z + a) 4 + (x + a) 2 (y + a) 2 + (x + a) 2 (z a) 2 + (y + a) 2 (z a) 2 + (z a) 4) + ( (x + a) 2 + (y a) 2)( (x + a) 2 (y a) 2 + (x + a) 2 (z + a) 2 + (y a) 2 (z + a) 2 + (z + a) 4 + (x + a) 2 (y a) 2 + (x + a) 2 (z a) 2 + (y a) 2 (z a) 2 + (z a) 4) + ( (x a) 2 + (y + a) 2)( (x a) 2 (y + a) 2 + (x a) 2 (z + a) 2 + (y + a) 2 (z + a) 2 + (z + a) 4 + (x a) 2 (y + a) 2 + (x a) 2 (z a) 2 + (y + a) 2 (z a) 2 + (z a) 4) + ( (x a) 2 + (y a) 2)( (x a) 2 (y a) 2 + (x a) 2 (z + a) 2 + (y a) 2 (z + a) 2 + (z + a) 4 + (x a) 2 (y a) 2 + (x a) 2 (z a) 2 + (y a) 2 (z a) 2 + (z a) 4)] ((x = 3 + a) 2 + (y + a) 2) ( 2(x + a) 2 (y + a) 2 + 2(x + a) 2 (z 2 + a 2 ) 15

16 + 2(y + a) 2 (z 2 + a 2 ) + ( (z a) 4 + (z + a) 4)) + ( (x + a) 2 + (y a) 2) ( 2(x + a) 2 (y a) 2 + 2(x + a) 2 (z 2 + a 2 ) + 2(y a) 2 (z 2 + a 2 ) + ( (z + a) 4 + (z a) 4)) + ( (x a) 2 + (y + a) 2) ( 2(x a) 2 (y + a) 2 + 2(x a) 2 (z 2 + a 2 ) + 2(y + a) 2 (z 2 + a 2 ) + ( (z + a) 4 + (z a) 4)) + ( (x a) 2 + (y a) 2) ( 2(x a) 2 (y a) 2 + 2(x a) 2 (z 2 + a 2 ) + 2(y a) 2 (z 2 + a 2 ) + ( (z + a) 4 + (z a) 4)). By using (z + a) 4 + (z a) 4 = (z 2 + a 2 + 2az) 2 + (z 2 + a 2 2az) 2 = ( (z 2 + a 2 ) 2 + 4a 2 z 2) 2 the second addend is: 6 2(x + a) 4 (y 2 + a 2 ) + 2(x + a) 4 (z 2 + a 2 ) + 2(x + a) 2 (y 2 + a 2 )(z 2 + a 2 ) + 2(z 2 + a 2 ) 2 (x + a) 2 16

17 + 4a 2 z 2 (x + a) 2 + (x + a) 2( (y + a) 4 + (y a) 4) + 2(x+a) 2 (y 2 +a 2 )(z 2 +a 2 ) + (z 2 +a 2 ) ( (y+a) 4 + (y a) 4) + 2(y 2 + a 2 )(z 2 + a 2 ) 2 + 8a 2 z 2 (y 2 + a 2 ) +2(x a) 4 (y 2 + a 2 ) + 2(z 2 + a 2 )(x a) 4 + 2(x a) 2 (y 2 + a 2 )(z 2 + a 2 ) + 2(x a) 2 (z 2 + a 2 ) 2 + 4a 2 z 2 (x a) 2 + (x a) 2( (y + a) 4 + (y a) 4) + 2(x a) 2 (y 2 +a 2 )(z 2 +a 2 ) + (z 2 +a 2 ) ( (y+a) 4 + (y a) 4) ] + 2(y 2 + a 2 )(z 2 + a 2 ) 2 + 8a 2 z 2 (y 2 + a 2 ) = 6 2(y 2 + a 2 ) ( (x + a) 4 + (x a) 4) + 2(z 2 + a 2 ) ( (x + a) 4 (x a) 4) + 4(x 2 + a 2 )(y 2 + a 2 )(z 2 + a 2 ) + 4(x 2 + a 2 )(z 2 + a 2 ) 2 + 8a 2 z 2 (x 2 + a 2 ) + 2 2(x 2 + a 2 )(y 2 + a 2 ) a 2 y 2 (x 2 + a 2 ) + 4(x 2 + a 2 )(y 2 + a 2 )(z 2 + a 2 ) + 2 2(z 2 + a 2 ) ( (y 2 + a 2 ) 2 + 4a 2 y 2 ) ] + 4(y 2 + a 2 )(z 2 + a 2 ) a 2 z 2 (y 2 + a 2 ) = 12 2(y 2 + a 2 )(x 2 + a 2 ) 2 + 8(y 2 + a 2 )a 2 x 2 + 2(z 2 + a 2 )(x 2 + a 2 ) 2 + 8a 2 x 2 (z 2 + a 2 ) + 4(x 2 + a 2 )(y 2 + a 2 )(z 2 + a 2 ) 17

18 + 2(x 2 + a 2 )(z 2 + a 2 ) 2 + 8a 2 z 2 (x 2 + a 2 ) + 2(x 2 + a 2 )(y 2 + a 2 ) 2 + 8a 2 y 2 (x 2 + a 2 ) + 2(z 2 + a 2 )(y 2 + a 2 ) 2 + 8a 2 y 2 (z 2 + a 2 ) ] + 2(y 2 + a 2 )(z 2 + a 2 ) 2 + 8a 2 z 2 (y 2 + a 2 ) ( S (6) = 24 14a 2 (x 2 y 2 + x 2 z 2 + y 2 z 2 ) + 16a 4 (x 2 + y 2 + z 2 ) + 2a 2 (x 4 + y 4 + z 4 ) + (y 2 x 4 +z 2 x 4 +x 2 z 4 +x 2 y 4 +z 2 y 4 +y 2 z 4 )+8a 6 +2x 2 y 2 z 2) ( + 8 x 6 + y 6 + z a 2 (x 4 + y 4 + z 4 ) + 15a 4 (x 2 + y 2 + z 2 ) + 3a 6) = 24(y 2 x 4 + z 2 x 4 + x 2 z 4 + x 2 y 4 + z 2 y 4 + y 2 z 4 ) + ( )a 4 (x 2 + y 2 + z 2 ) + ( )a 2 (x 4 + y 4 + z 4 ) + 8(x 6 + y 6 + z 6 ) + 48x 2 y 2 z 2 + ( )a 6 = 8(x 6 +y 6 +z 6 )+24(y 2 x 4 +x 4 z 2 +z 4 x 2 +x 2 y 4 +y 4 z 2 +y 2 z 4 ) + 48x 2 y 2 z a 2 (x 4 + y 4 + z 4 ) + 336a 2 (x 2 y 2 +x 2 z 2 +y 2 z 2 )+504a 4 (x 2 +y 2 +z 2 )+216a 6. Now get back to L and R notations: 18

19 S (6) = 8L a 2 L a 4 L a 6 = 8 L 6 + 7R 2 L 4 + 7R 4 L 2 + R 6] = 8(L 2 + R 2 )(L 4 + R 4 + 6L 2 R 2 ). Theorem 3.3. The sum of the sixth power distances S (6), from an arbitrary point on any sphere with center at the centroid of a cube to the vertices of the cube, is constant ] S (6) = 8 (L 2 + R 2 ) 3 + 4L 2 R 2 (L 2 + R 2 ), (4) where L is the radius of the sphere and R is the circumradius of the cube. From the equivalency S (6) = const L = const is true vice versa: Theorem 3.4. The set of points, from which the sum of the sixth power distances S (6) to the vertices of a cube is constant, is: - the sphere with center at the centroid of the cube, if S (6) > 8R 6 ; - the point the centroid of the cube, if S (6) = 8R 6 ; - the empty set, if S (6) < 8R 6 ; where R is the circumradius of the cube. As for the sums of high powers (more than 6), they will no longer be constant. 19

20 4 Summary Summarize (1) and obtained (2) (4) results. Consider a regular tetrahedron and a cube. Denote the cercumradius by R and the centroid by O, respectively. If an arbitrary point X moves on any sphere with center at O and the radius L, the sums S (2m) of power distances d 2m (X, P i ) from this point to the vertices P i are constant for the following powers and equal: Regular Tetrahedron: S (2) = S (4) = 4 d 2 (X, P i ) = 4(L 2 + R 2 ), 4 d 4 (X, P i ) = 4 (L 2 + R 2 ) L2 R 2]. Cube: S (2) = S (4) = S (6) = 8 d 2 (X, P i ) = 8(L 2 + R 2 ), 8 8 d 4 (X, P i ) = 8 (L 2 + R 2 ) L2 R 2], ] d 6 (X, P i ) = 8 (L 2 + R 2 ) 3 + 4L 2 R 2 (L 2 + R 2 ). We have perfectly investigated two of Platonic solids the regular tetrahedron and the cube. Consideration of other Platonic solids the octahedron (n = 6), the icosahedron (n = 12) and the dodecahedron (n = 20) is the aim of the future studies. 20

21 References 1] J. Steiner, Gesammelte Werke. Herausgegeben von K. Weierstrass. Bd. II. (German) G. Reimer, Berlin, (1881,1882). 2] T. M. Apostol, M. A. Mnatsakanian, Sums of squares of distances in m-space. Amer. Math. Monthly 110 (2003), no. 6, ] M. Meskhishvili, New sense of a circle. arxiv: math.gm], 2019; Author s address: Georgian-American High School, 18 Chkondideli Str., Tbilisi 0180, Georgia. 21

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