Časopis pro pěstování matematiky

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1 Časopis pro pěstování ateatiky Miroslav Fiedler Isodynaic systes in Euclidean spaces and an n-diensional analogue of a theore by Popeiu Časopis pro pěstování ateatiky, Vol. 102 (1977), No. 4, Persistent URL: Ters of use: Institute of Matheatics AS CR, 1977 Institute of Matheatics of the Acadey of Sciences of the Czech Republic provides access to digitized docuents strictly for personal use. Each copy of any part of this docuent ust contain these Ters of use. This paper has been digitized, optiized for electronic delivery and staped with digital signature within the project DML-CZ: The Czech Digital Matheatics Library

2 Časopis pro pěstováni ateatiky, roč. 102 (1977). Praha ISODYNAMIC SYSTEMS IN EUCLIDEAN SPACES AND AN n-dimensional ANALOGUE OF A THEOREM BY POMPEIU MIROSLAV FIEDLER, Praha (Received April 29, 1977) INTRODUCTION Isodynaic tetrahedrons and ore generally, isodynaic n-siplexes have been studied in [1], [2]. We shall investigate here isodynaic systes of points in Euclidean spaces, i.e. unordered systes of points A l9..., A such that for soe positive nubers c l9..., c w, (1) Q(A i9 A k ) = c t c k for all i 9 k = 1,...,, i4= k. By Q we ean throughout the whole paper the Euclidean distance. In particular, we shall be interested in axial isodynaic systes in an n-diensional Euclidean space and their properties. PRELIMINARIES Under an (n l)-sphere we understand here and in the sequel a hypersphere in an n-diensional Euclidean space; a generalized (n l)-sphere is either an (n 1)- sphere or an (n l)-diensional linear space. We say that, in an n-space, a generalized (n l)-sphere K t bisects the (n l)-sphere K 2 with centre A 2 and radius r 2 iff either r\ = Q 2 (A U A 2 ) + r\ in the case K 1 is an (n l)-sphere with centre A t and radius r l9 or if K t contains A 2 in the case K t is a hyperplane. If K t and K 2 are two (n l)-spheres in E n with centres A t and radii r t (i = 1, 2) then we call (K l9 K 2 )- haronic (n l)-sphere the set {X; Q(X, A 1 )JQ(X, A 2 ) = r 1 \r 2 }. It is thus the (generalized) sphere of Apollonius of the points A 1 and A 2 with the ratio r x \r 2. As usual, the power of a point X in E n with respect to an (n l)-sphere K (in E n ) with centre A and radius r is Q 2 (X 9 A) r 2. If K is a hyperplane, we shall agree that any point of K has any real nuber as its power with respect to K, and no point outside K has a defined power with respect to K. 370

3 Two points X 9 Y in E are inverse with respect to a generalized (n l)-sphere.k iff either they are syetric with respect to K if K is a hyperplane or, in the other case, if they lie on one ray starting in the centre A of K and Q(X 9 A). Q(Y 9 A) = r 2, r being the radius of K. RESULTS It will be useful to assign to an isodynaic syste satisfying (1), a new set of nubers t l t by U = c 2, i = 1,...,. We shall call these nubers t { radii of the isodynaic syste corresponding to the points A t. The following theore is easy to prove. Theore 1. Let A l9..., A 9 _ 3, be points in a Euclidean space which for an isodynaic syste with the corresponding radii t l9..., t 9 i.e. (2) Q 2 (A i9 A k ) = t t t k, i * k, i, k = 1,...,. Then the radii t t are uniquely deterined by (2). Another trivial observation is forulated in the following Theore 2. Any subsyste of an isodynaic syste of points is as well. isodynaic Theore 3. A syste {A l9..., four points is isodynaic. A } of points is isodynaic iff any subsyste with Proof. The "only if" part following fro Th. 2, assue that any subsyste with four points is isodynaic. To prove that the given syste is isodynaic, we shall use induction with respect to. For g 4, the assertion is clearly true. Suppose that > 5 and the assertion holds for any syste with 1 points. Thus A l A ^1 is isodynaic with (uniquely deterined) radii t l9..., t - v Let 1 1 l9 1 29? 3 be radii of the isodynaic subsyste {A 9 A l9 A 29 A 3 }. Since the radii of {A l9 A 29 A 3 } are uniquely deterined by Th 1, we have 1 t = t { i = 1, 2, 3. Siilarly, if k > 3, let \ 9 i l9 \ l9 \ k be radii of the subsyste {A 9 A l9 A l9 A k }. Then \ =?,?i = t l9 \ 2 = t l9 \ k = t k so that Q\A i9 A k ) = Ut k (i*k) is satisfied for all i, k = 1,..., if t = 1. This copletes the proof. Reark. It is easily seen that a quadruple {A l9 A l9 A 3, A^} of points is isodynaic iff these points are utually distinct and Q(A 19 A 2 ) Q(A 39 A A ) = Q(A 19 A 3 ) Q(A 29 A*) = Q(A U A 4 ) Q(A 19 A 3 ). 371

4 To investigate existence of isodynaic systes, we recall the following theore essentially due to MENGER [3] which is a point analogue of the well known theore that the Grapi atrix of a vector syste is positive seidefinite and conversely. Theore 4. Let be a positive integer. The 2 real nubers e u = e ji9 i 9 j = = 1,...,, are squares of distances of soe points A l9..., A in a Euclidean space: Q 2 (A i9 Aj) = e tj, iff en = 0, i = 1,...,, and for any real -tuple (x l9..., x ) satisfying (3) I>. = 0, i=l the inequality (4) I eijxpcj = 0 *,; = - holds. If this is the case, the points A l9..., A are linearly independent iff the only - tuple satisfying (3) for which equality in (4) is attained, is the zero -tuple. More generally, all linear dependence relations aong the points A l9..., A are exactly those relations satisfying for which ty,ai = o i=l l.vi = 0, i=l E eijyiyj = 0. i,i-=l Now we are able to state the existence theore on isodynaic systes. Theore 5- Let _ 2, let t l9..., t be positive nubers. A necessary and sufficient condition that there exist in a Euclidean n-diensional (and not (n 1). diensional) space points A l A the utual distances Q(A i9 Aj) of which satisfy (5) Q 2 (A i9 Aj) = t t tj (i 4= j 9 i 9 j = 1,..., ) is : either (i) n -= 1 and i / 1 \ 2 (6) 0"-i)I i <(L7), 372

5 or w (ii) n = 2 and 1 / 1 V («--) 4-(if). *=i r t v*=t tj In the second case, the only relation aong the points A u..., A is ( 1 \ _1 1 / 1 \ _1 1 I M I 7*-(I 4) :U = - k=i r k / k=i * ft \ k=1 *k/ *=- -k Proof. LetfirstA l9..., A satisfy (5). We shall show that then 1 / 1 \ (9) (-l)lf =Ylf). 2 By Th. 4, k=i t k \k=i **/ I ******** -S 0, l = i<k^ whenever x l9..., x satisfy x f = 0. Especially, the nubers y u..., y where i=l i * i i (10) yi = 1 I V ^ I 1 ' i-l.---, ^k=i t k t t k=i r k satisfy >> fc = 0; therefore, (U) 2 V. fajw.rgo. l = i<k^ The left hand side is equal to M-H-fJ)- The first factor is by the Schwarz inequality nonnegative, and positive if not all the f f's are equal. If the ^'s are equal, (9) is satisfied. If not, thefirstfactor is positive and (9) is satisfied by (11). Observe that (7) iplies that for e u = Q(A (9 AJ) = tfo (i * j) and e H = 0, t,j-i 373

6 The nubers (10) are easily seen not to be all equal to zero. By Th. 4, (7) iplies that the points A l9..., A are linearly dependent, i.e. (12) - n = - 2, and oreover, (8) holds. This eans that if A l9..., A are linearly independent then (6) is satisfied. Let us show now that conversely, (9) iplies that there exist, in a Euclidean space, points A l9..., A satisfying (5) and even that (6) iplies that they are linearly independent. We shall use Th. 4 again. Let x i9..., x be real nubers satisfying x f = 0. Assue first (6). Then and we can write for e н = 0, e ik = e ki = tfa (i Ф к): 1 E Єijxpcj = - (( - 1) Ç tix t ) 2 - ( ЂЫ - Q> i*,) 2 )) = І,J = I 7^(-^(^! ( 2 ф, ; - ( I V ' )! ) + [ Уx У УfiXi ] 1 < f ( Yx; Y УípcЛ - V L U L ) ) Һ 1 -)^ *«7 - (ҷ I Jj - (l ) T ) 2 ) («!»?*? - (I^) 2 )) = ^'((sa--^- 4- by the Schwarz inequality. By Th. 4, this iplies the existence of linearly independent points A l9..., A in a Euclidean space which satisfy (5). If only (9) is assued, a siilar chain of inequalities as above yields ]T e^x^- ^ 0 and by Th. 4, points i4 t,..., A satisfying (5) also exist but are not necessarily linearly independent. 374

7 It reains to show that if (7) is fulfilled then n = 2. By (12), it suffices to disprove that n < 2. Suppose n < 2. Then soe n -f- 1 points, say A l A n+l of the points A l9..., A are linearly independent and the points A i , A n+3 also satisfy (5). Consequently, for each fe, 1 ^ k g n + 3, the relation corresponding to (7) holds, i.e.» + 3 /n + 3 * \2 (13) («+ l)zf "(if). since the points A i9...,a k - i9 A k+1,..., A n+3 are linearly dependent. Also n + 3 /n + 3 i \2 (14) («+-)Zf = (l-) by the sae reason. However, (13) can be rewritten in the for* (1 5 ) i (f - ft="z K. i^kj^n+3 vr, tj i=i rf (14) in the for /l 1 \ 2 n+3 t (i6) Z (f-f)=zf- Subtracting (15) fro (16), we obtain Z (i-m =1, fc = l,...,n + 3. '=- Vi '*/ '* Therefore, by suing up these equalities, 2 I ('--iy-ei. l j<j «+3 \f fy/ k=l t\ a contradiction with (16). The proof is coplete. This theore enables us to call coplete such an isodynaic syste which consists of ^ 3 points and is contained in an ( 2)-diensional Euclidean space. Theore 6. (i) In a Euclidean n-diensional space, n J> 1, the axiu nuber of points in an isodynaic syste is n -F 2. (ii) A linearly independent isodynaic syste with ^ 3 points is contained in exactly two coplete isodynaic systes in the sae space, with the only exception that the points A i A for vertices of a regular ( l)-swtp/ex; in this case, there is only one coplete isodynaic syste in the sae space in which the given syste is contained. The additional point is the center of the siplex. 375

8 (iii) For any J> 3, there exist coplete isodynaic systes with points. (iv) Any coplete isodynaic syste j with ^ 3 points in w _ 2 is contained in a coplete isodynaic syste S 2 with + 1 points in E 1 {containing _ 2 ). 2 is deterined in E 1 uniquely up to congruence leaving all points of fs _ 2 invariant. The radius of the ( + l)-th point is <...-f-si.fr \ li=-i t t J where t l9..., t are the radii of the points of j. (v) Any isodynaic syste which contains a coplete isodynaic subsyste is coplete. (vi) A coplete isodynaic syste contains a inial coplete isodynaic subsyste, i.e. a coplete isodynaic subsyste which is contained in every coplete isodynaic subsyste of. This inial subsyste contains exactly those points of whose coefficient in the (up to a factor unique) relation aong the points in is different fro zero. (vii) If {A l9..., A n } is a coplete isodynaic syste and {A l9..., A k ) its inial coplete isodynaic subsyste then A k+l9..., A are vertices of a regular siplex. (viii) Any three different points in a line for a coplete isodynaic syste. Proof, (i) is a consequence of Th. 5. To prove (ii), let {A l9..., A ) ( ^ 3) be a linearly independent isodynaic syste so that (5) and (6) holds. Assue this syste to be contained in a coplete isodynaic syste {A l9..., A +1 } (by (i), not ore than + 1 points exist). By Th. 1, the corresponding + 1 radii are unique and the first coincide with t i9 i = 1,...,. Let t +1 be the ( 4- l)-th. Then, an analogous relation to (7) holds: fih-rt' 1 -)' so that * = i t k \*-i tj ( ~ 1 )^ -2- X 7 + -j- - =0. t +i t +1 *-i ti i-i r f V'-- U) The discriinant of this quadratic equation for \\t +1 is easily coputed to be positive t>y (6). If the f j's are not all equal, the absolute eber of the equation is positive by the Schwarz inequality and the two positive roots yield two distinct coplete isodynaic systes. If all the fj's are equal, t t == t 9 i == 1,...,, i.e. if the given syste is the set of the vertices of a regular ( l)-siplex (with all edges having the sae length), one root of the equation is zero and there is only one positive root '«+!= 1. 2

9 (iii) follows e.g. fro the preceding case of the vertices and center of the regular siplex. To prove (iv), assue - consists of the points A l9...,a with radii t l t so that <2 1 / 1 \ («--) Wlf). 2 i=i í, \i=i íj Assue 2 arises fro x by adding a point with radius f +1 (the radii t l9..., f coincide). Then I V'-- t 2 t+l/ V'"- ti t+l/ fro which, the discriinant of the quadratic equation for ljt +1 being zero, 1 ' i 1 t+i - 1 i = l t Since the converse is also true, 2 exists by Th. 5. The distances Q(A i9 A +l ) are thus uniquely deterined which copletes the proof of (iv). (v) follows fro the fact that the assuption iplies the points of the syste are linearly dependent so that case (ii) of Th. 5 occurs. To prove (vi), we shall also use the fact that an isodynaic syste is coplete iff its points are linearly dependent. Thus, if the essentially unique relation aong the points of has non-zero coefficients corresponding to points Aj for j e J, the subsyste {Aj} jb j is coplete and every coplete subsyste contains this subsyste. Before proving (vii), we shall prove the following lea: Lea. Let k, n be integers, 2 ^ k < n. Let x l9..., that Then (fc-i)i* 2 = d>.) 2. i=-i i=i (»-i)l* 2 = (l>.) 2 i=-l i=l iff 1 * V **+l ~... X в - la X І k 1 І--I Proof. Fro the equality x n be real nubers such *+i i *+i * *+i i * + 1 * i * h \ / i * \2 i^-mi^) 2 = ix?-^(exo 2 + ^(x t+1 --J-i:^ i«i k Í«I s Í=-I k ~ 1 f--i k \ k - 1 i -1 / 377

10 it follows that with equality iff Thus, *+i i k+i Z*i -hi**) 2 *Z*? -~-(Zx t y, i=l k i=l i=l fc 1 *==! 1 * **+i = 7 :Z x i k li-=i î^-- J -:(Ž^) 2 è-^і;x,?- r L т (ix i ) г, i=l n 1 i = l i=l fc 1 i=l with equality of the first and last eber iff I * ^ fc+i 1 * x*+i = - - Z x *' x *+2 =7 Z x - = ~ 7 Z fe 1 i=l fe i=l fe 1 i = l fc J «-l 1 x «= Z x i = r : Z *t - n 2 i=i fe 1 i=i The lea then follows. To prove (vii), use the lea for n = 9 x t = ljt i9 i = 1,...,. The assertion (viii) being trivial, the proof is coplete. Reark. The two (or one) additional points in (iii) of Th. 6 are the isodynaic centres [2] of the corresponding ( l)-siplex. In the following ain theore about coplete isodynaic systes several characterizations are given. Theore 7. Let A l9..., A n+2 be different points in a Euclidean n-space E n. Then the following conditions are equivalent: 1 A l A n+2 is a coplete isodynaic syste in E n9 i.e. there exist positive nubers t l9..., t n+2 such that Q 2 (A i9 A k ) = tit k for all i 9 fe == 1,..., n + 2, i 4= fe ; 2 there exists a syste of n + 3 real (n l)-spheres K 09 K l K n+2 such that 21 Ki has centre in A t for i = 1,..., n + 2 and bisects K for each pair i 9 j (i =t= j) 9 i 9 j = 1,..., n + 2, the (K i9 Kj)-haronic (n - 1)- sphere K tj contains all points A k for i 4= fe 4= j; 3 there exists a syste of I ) generalized (n l)-spfteres K y (=X;0- f, j = 1,..., n + 2, i # j, sucft that 378

11 31 A { and A s are inverse with respect to K ij9 32 K t j contains all points A k for i + k + j; 33 there exists a point having the sae negative power with respect to all (n i)-spheres K tj. 4 there exists a point R in E n and a point B 0 + R in a Euclidean (n + 1)- space containing E n, on the line perpendicular to E n in R such that the second intersection points B ( (i = 1,..., n + 2) 0f the lines A t B 0 with the n-sphere K = = {X; Q(X, R) = Q(B 0, R)} for vertices of a regular (n + i)-siplex. 5 there exists, in a Euclidean (n + i)-space E n+1 containing E n, a regular (n + i)-siplex I such that A l9...,a n+2 correspond to the vertices of I in an inversion in E n+1. 6 there exists, in a Euclidean (n + i)-space n+1 a regular (n + i)-siplex with vertices B l9...,b n+1 and a point X (different fro all the points B f ) on its circuscribed n-sphere such that, for soefc > 0, U.4)- fc (B i9 X) (Bj,X) for all i 9 j = 1,..., n + 2, i + j. 7 there exists, in a Euclidean (n + i)-space n+1, a regular (n + i)-siplex with vertices B i9..., B n+1 and a point X (different fro all the points B t ) such that, for soefc > 0 fc e{ai ' Aj) = $B^JW^) for all i,j = 1,..., n + 2, i # j. Proof. We shall prove the iplications 1 => 2 => 3 => 4 => 5 => 6 => 7 => 1. Assue 1. By (iv) of Th. 6, the syste [A l9..., A n+2 } is contained in a coplete isodynaic syste, with the additional point A n+3, of an (n + l)-diensional space E n+1 containing E n. Define for i = 1,..., n + 2, K { = {X e E n ; Q 2 (X, A t ) = = Q 2 (A n+3, A f )}. If R is the orthogonal projection of the point A n+3 on E n and r = = Q(R, A n+3 ) then K 0 = {X e E n ; Q(X, R) = r} satisfies Q 2 (A i9 R) = Q 2 (A i9 A n+3 ) - r 2 which eans that K t bisects K 0. Moreover, let i 4= j. The (K U K 2 ) haronic (n l)-sphere K tj is easily checked to contain the points A k for allfc,i + fc + j. Thus 1 => 2. To prove that 2 iplies 3, it suffices to show that the (K i9 Kj) haronic spheres K tj satisfy 31, 32, follows fro the haronic property of A i9 Aj and the intersection points of the line A t Af with K ij9 32 is iediate. To prove 33, take R as the centre of K 0 in 2. Since K 0 is bisected by K t and K j9 it is bisected by K (J (belonging to the pencil deterined by K t and Kj) as well. Thus R has the sae negative power with respect to all K t fs which are nonlinear. According to our agreeent, this is also true if soe but not all of the K*/s are linear. However, all the K t fs carot be linear since in this case the utual distances of n + 2 points A x in E n would be equal. 379

12 Assue 3. Let n+1 be any Euclidean (n + l)-space containing E n. Let B 0 be a point on the line perpendicular to E n passing through R 9 such that Q 2 (B 09 R) = = p, p being the power of JR with respect to all K tj 9 s. Let, 7 (i 4= I, i,i = 1,......, n + 2)*be the generalized n-sphere in F M+1 with the sae centre and radius as K ( j if K t j is an (n l)-sphere; if K (j is linear, let R (j be that n-diensional linear space in E n+i which contains K (j and is orthogonal to E n. It follows that R {j contains the point B 0 for all i, j = 1,..., n + 2, i 4= j. Let X be the n-sphere with centre in R and radius Q(R 9 B 0 ) 9 let JB f (i = 1,..., n + 2) be the second intersection point of the line A t B with K. Denote by the n-sphere with centre B 0 which bisects K. Using the well known properties of inversion, it follows that E n corresponds to K in the inversion «/ with respect to ; A t corresponds to B t in «/, R {j corresponds to a hyperplane ify, i, j = 1,..., n + 2, i 4= j. Since ^y is orthogonal to E n9 H tj is orthogonal to K and thus contains R 9 as well as all the points B k for i 4= k 4= j. A t and Aj being inverse with respect to R ij9 B t and Bj are syetric with respect to H tj (since any sphere containing both B { and i?, is orthogonal to H ij9 this being true for their transfors in J). Consequently, Q(B {9 B k ) = Q(B J9 B k ) for all i, j, fc, i 4= j 4= k 4= i. It follows that the points B i9 i = 1,..., n + 2, for vertices of a regular (n + 1)- siplex. The proof of 3 => 4 is coplete. The iplication 4 => 5 is iediate since B t and A t correspond to each other in the inversion deterined by the n-sphere & having the centre B 0 and bisecting K. Assue 5. Denote by J the inversion, by B t (i = 1,..., n + 2) the points in E n+l corresponding to A t in J so that B { are vertices of a regular (n + l)-siplex I. Let X be the centre of the inversion J. Thus X 4= B t for all i = 1,..., n + 2. If C is the circuscribed n-sphere of I, C corresponds to E n in J and thus contains X. We have then for i 4= j 9 i, j = 1,..., n + 2 (17) e(-4 f, X) <?(,, X) = o(^, K) Q(B J9 X) so that the triangles A t AjX and B^K are siilar to each other. Thus as well as By ultiplication, e(a i9 Aj)JQ(A i9 X) = Q(B i9 BJ)IQ(B J9 X) Q(A i9 AJ)IQ(A J9 X) = Q(B i9 Bj)JQ(B i9 X). Q 2 (A i9 A,) - e 2^, ;) Q(A i9 X) Q(AJ 9 X) ( Q (B i9 X) Q(B J9 X))- 1 = - a 2 Q 2 (B i9 Bj)l( Q 2 (B i9 X) Q 2 (B J9 X)) by (17), if the coon value is denoted by a. Since Q 2 (B i9 Bj) is constant for all pairs Ui% i * if 6 follows (where = n+1, $ = Q is taken). The iplications 6 => 7 as well as 7 => 1 being trivial, the proof is coplete. A well known theore fro plane geoetry, soeties called Popeiu's theore, states: 380

13 If A t A 2 A 3 is an equilateral triangle and X another point of the plane thcnxa i9 XA 29 XA 3 for lengths of sides of a triangle iff X does not belong to the circuscribed circle of A i A 2 A 3. We shall generalize now this theore as follows: Theore 8. Let A i9.., A n + i be vertices of a regular n-siplex 1 in E n. If X is a point in E n then there exists an n-siplex with vertices B i9..., B n+1 such that edges B t B k (i 4= fc, i, k = 1,..., n + 1) have lengths proportional to (Q(A 19 X).. Q(A U, X))~ X iffx does not belong to the circuscribed (n i)sphere of I. Proof. Assue first that X belongs to the circuscribed (n l)-sphere of I. If X = A t for fcoe i 9 the n-siplex clearly does not exist. If X 4= A t for all i = = 1,..., n + 1, the equivalence of 7 and 1 in Th. 7 shows that the realization of the points B t leads to a coplete isodynaic syste which is linearly dependent. Assue now that X does not belong to the circuscribed (n l)-sphere of I. Let J be any inversion with centre X. If B t are points which correspond to the points A t in J 9 we have siilarly as in the proof of 5 => 6 in Th. 7, Q(B i9 B k ) = k( Q (A i9 X) Q(A k9 X))' 1. Moreover, the points B t do not belong to a hyperplane since this would correspond in J to the circuscribed sphere of 1 and this would contain the centre of inversion X 9 a contradiction. The proof is coplete. References [1] N. A. Court: Sur le tétraedre isodynaique. Mathesis 49 (1935), [2] S. R. Mandan: Isodynaic and isogonic siplexes. Annali di Mateatica pura ed appl. Ѕer. IV., 53(1961), [3] K. Menger: Untersuchungen über allgeeine Metrik. Math. Annalen 100 (1928), Authoŕs address: Praha 1, Žitná 25 (Mateatický ústav ČЅAV). 381