Elem. Math. 63 (2008) 126-136 0013-601 8/08 030126-1 @ Swiss Mathematical Society, 2008 I Elemente er Mathematik Lothar Heinrich Lothar Heinrich promovierte im Jahr 1980 an er Technischen Universität Dresen. Von 1995-1997 war er Professor für Wahrscheinlichkeitstheorie un Statistik an er Technischen Universität Bergakaemie in Freiberg. Seit 1997 ist er Professor für Mathematik an er Universität Augsburg. Seine Forschungsinteressen liegen in er angewanten Wahrscheinlichkeitstheorie (stochastische Geometrie, große Abweichungen, Grenzwertsätze, räumliche Statistik, probabilistische Zahlentheorie). 1 Introuction an preliminaries If points are marke on each sie of a planar triangle, one on each sie (or on a sie's extension), then the three circles (each passing through a vertex an the marke points on the ajacent sies) are concurrent at a point M. This interesting fact was first prove an publishe by Augueste Miquel [3] in 1838, see also Weisstein [4] for further etails an extensions. This result is well-known in planar geometry as Miquel's theorem, an M is calle the Miquel point. However, much less known (even amongst geometers) is the following multiimensional generalization of Miquel's theorem: If one point is marke on each of the ( + I) /2 eges of a -simplex S(xo,Xl,...,xa) : {*o+ipi (xi-xo),lu,. l, t"i> 0,i - L,..., (l) l a t i:i i:7 Der Satz von Miquel ist ein klassisches Resultat er Elementargeometrie: Wir auf jeer Seite eines gegebenen Dreiecks oer eren Verlängerung ein Funkt beliebig festgelegt, un wir urch jeweils eine Ecke un ie beien markierten Punk;te auf en Nachbarseiten ein Kreis gezeichnet, so schneien sich iese rei Kreise in einem Punk;t. In er vorliegenen Arbeit wir mit einfachen Hilfsmitteln er linearen Algebra erstmals ein vollstitniger Beweis er Verallgemeinerung es Miquelschen Satzes auf -imensionale Simplizes angegeben: Wir auf en ( + l) /ZKanten je ein Punkt beliebig markiert, ann haben ie * 1 Kugeln, wovon jee urch je einen Eckpunkt un ie markierten Punkte auf en in iesen Eckpunkt einlaufenen Kanten festgelegt
r27 with positive volume an a sphere S; is rawn through each vertex x; an the points marke on the eges which meet in x;, then these + l spheres S;, i - 0, 1,...,, all meet in a point M which will also be calle Miquel's point in this note. By the best of the author's knowlege the only known proof of this result seems to be that given by Konnully t2l. Konnully's proof is base on the fact that there exists a common orthogonal sphere with respect to the so-calle Miquel spheres So, Sr S7 an it is shown that the raius of this sphere equals zero. However, such a sphere oes not always exist even in the planar case, namely if the unique point which has the same circle power w.r.t. three pairwise non-concentric circles lies in the interior of each of these circles. The aim of the present note is to provie a rigorous analytical proof which requires only simple facts from analytic geometry an linear algebra. As a by-prouctwe obtain a family of upper bouns of Gram's eterminant (incluing Haamar's inequality) which seems to be of interest in its own rieht. This auxiliarv result in Section 3 is vali in all Eucliean vector spaces. It shoul be mentione that an analogous construction with points marke on the ( - I)- faces (instea of on the eges) of the simplex oes in general not yiel a common point that belongs to all spheres. Let the points of the Eucliean space R be represente by column vectors x _ (u,..., x)t having the Eucliean nonn llxll - /E;S, where the scalar prouct (.,.) is efine by (y,z) : y'z : yi zl+... * ya za for y: (yr,...,!)' an z- (2t,...,2)l Furthermore, we recall the well-known fact froq analytic geometry that the circumsphere of the -simplex (1) consists of all points x IR. satisfying the equation llxll2 llxoll2 llxrll2 x X g x l 1 1 1... llxa ll2 x 1-0, (2) where the left-han sie is for a ( * 2) x ( t 2) eterminant. Without loss of generality, we shall assume the vertex xs of the -simplex (1) to coincie withtheorigino:(0,...,0),antheverticesxi:(x1; linearly inepenent vectors, i.e. A::etXl0 (3) where X - (xr x6) enotes the quaratic matrix with columns Xl,..., X. For 0 < i < j <, let xil be a fixe point on the ege joining the vertices xi an x; being istinct from its en-points, i.e. vij : xi * l;j (x: - x;) for some )"ii 4 {0, 1}. For notationaleaseput Li; -I-)"ij forl < j anlii:l/2fori:0,i,...,.
t28 L. Heinrich In accorance with (2), the unique -sphere Ss passing through the origin o an the points X01,..., xg consists of all points x - (xr x)t e IR satisfying As(x) :- llxll2 o rfr,llxrll2 xf;ollxall2 xr 0 lorxrr Loaxu - 0. x 0 Lofiat Loa)c 1 I 1... Since 117:o Loi * 0, the latter equation is equivalento llxll2 lor llxr ll2 xt xn xl loa llxall2-0 (4) *o xr ": x Likewise, a point x : (xr x)t e IR belongs to the unique -sphere $ containing thevertexx;anthemarkepointsxgi,...,xi-t,i,xi,i+l,...,xiontheajacentegesif an onlv if fori-i,...,. llxll2 llxi ll2,r,fr, llxi ll2 llxi + r;; (xj- xi) ll2 xr xri )"oixtt xu I Lij @t j - xu) a; (x):- - 0 *O *,a, ),Oirxai xai * X;1@A1 - xai) 1 1 1 1 ffi By appealing to the well-known transformation rules for eterminants we obtain A;(x)-(),e;-1) llxll2 - x?r,llxillz (1* r.o;)llxill2 Äo; llxill2 llxi + l;;(xj - xi)ll2 x1 - )"six1; xfi 0 xu I Lij (xt j - xu) x -)"oixar *a, ö xai i Lij('xaj - xai) 0 0-1 1 : (Äo; - r) II r,; i*i llxll2-1,fr, llxi ll2 cii ro; llxi ll2 cij x1 - )"gix1i xfi 0 x1j x -'Loixai *,a, ö x'j 0 0-1 0 \/ j e {1,...,}\{l} je {1,...,}\{,}
where llxi + l;; (xj- xi) ll2 - llxi ll2 + (t + ro;) ll*ill2 )uij - lo; ll*ill2 + llxill2 -Lii llxl -x;ll2 for i, i =1,...,. The latter equality follows by using the ientity llu - vll2 - llull2 + llvll2-2 {a, v). Since cii : (1* ),0;) llxi ll2 we may simplify the previous eterminant by multiplying the secon column by Äo; an aing it to the first column. Consequently, in view of l";; {0, 1} for i + j, the equation A;(x) - 0 can be expresse as,,,,),,,,) llxll" + ^o; llxi ll- cit ci xt xn xh *o xt ": x Obviously, the set of points x lr satisfying both equations (4) an (5) coincies with the ( - l)-imensional sphere So n S;. By subtracting equation (4) from equation (5) an applying the summation law for eterminants iffering in only one row we obtain the linear equation t ^o; llxill- ' ) r r t cir - ^0l llxtll- ^ r t t ) ci - ^0llxll' t t ) xi xtl xi D; (x) :- - 0 *o xt ": x that hols for all x - (xr x)t IR belonging to the (uniquely etermine) hyperplane H; containing the ( - l)-sphere 56 n S,. Combining the equations D; (x) : 0, i -, yiels a system of linear equations whose solution (if it exists!) coincies with the point of intersection of the hyperplanes Hr,..., H. To fin this point we introuce the matrix Al : (o4)!,1:1with entries aii::cij -ro;llx;ll2:äo; llxill2+(1 -ro;) llx;ll2 _ Lii llxi-";112. (6) Letusfirstnotetheremarkablefactthat,inviewof Lij+Lii - lforl { j anlit -I/2, aij laji: llxill2 + llxjll2 - (xii tl1) llxi -x;ll2-2(x1,x;) for i, i :L,...,, which can be expresse concisely by I (o" + A1\ ) - G(x) :- ((xi, xi)!,i:t, where G(X) an et G(X) are calle Gram's matrix an Gram's eterminant, respectively, of the vectors Xl,..., X. In IR we have G(X) - X/X an thus et G(X) : L2. On the (7)
130 L. Heinrich other han, the relation (7) is meaningful in any real vector space V on which a symmetric, positive ef,nite, bilinear form (.,.) - briefly calle scalar prouct - is efine. Such vector spaces are usually calle Euclieanvector spaces. Proposition. For arbitrary linearly inepenent vectors Xt,..., X R an real numbers)";i,i, j :0, 1,...,, satisfuing)"ij*)"ii - Ian).ij f {0,I}fori, j :0, 1,...,, there exists auniquepointof intersection H1n... iha: x* : ("f,...,*ä),,whichis given by x*: XAnlxl with XÄ: (Äor llxrll2,...,lo llxall2),. (g) The first step in proving this result is to show that x - (\,..., x)t obeys the linear equations Di(x) -, 0 for i : I x1. This is left to the reaer as an exercise. To see the invertibility of the matrix An we use (7) an ecompose An as follows: Ar : Bn * G(X) with "n: (ln - n'n ) ) (e) Here the matrix Bn is skew-symmetric, i.e. Bl : -Bn. This skew-symmetry an the positive efiniteness of the Gram matrix G(X) show the quaratic form x/a1x - x/ G(X) x to be strictly positive for any x # o. This in turn implies An x # o for any x t' o, which is equivalent to etan I 0 for ali ),i1 satisfying Lij + Lii - 1. This combine with a simple continuity argument tells us that etan < 0 contraicts et G(X) ) 0, leaving.as the only possibility etal cvg(x)) > 0for anya > 0,entailingetBzr > 0bylettingcv + 0.HoweveretBn may take positive values only for even > 2 ue to the very efinition of skew-symmetry. Moreover, we shall boun etan uniformly from below by Gram's eterminant et G(X) in Section 3. Finally, note that the proposition oes not answer the question whether x* & hols for some or even ail i - 0, 1,...,. This will be the subiect of the next section. 2 The main result an its proof Theorem. Uner the conitions of the above proposition the point x* given in (8) belongs toeachof thespheres S;, i :0, 1,...,,thatis, x* is theuniquepointof intersection So f-l Sr n... f) Sa. In other wors, x* coincies with Miquel's point M of the simplex S(o, x1,..., xo) w.rt. the marke points xii: Xi * I;; (xj - xi) on its eges. Proof. After transposing an expaning the eterminant on the left-han sie of (4) along the first row we recognize that Ä9(x*) : 0 is equivalent to ll**ll2^-i "i j:1 xn lor llxr ll2 xr xr2 Lozllxzllz xz xr loa llxa ll2 x - 0. (10)
Miquel point M = (3.41,0.89, 1.54) xr = (3,6, 0) Fig. 1: Tetraheron,S(xg,x1,x2,x3)anthefourMiquelsphereswithcommonMiquel point M for ),61 : l/2, ),g2 - I/4, ),ß : 2/3, Lt2 : I/3, ),p : I/4, )'Y :2/3 Diviing by a t'. 0 the latter equation takes the form (X*, x* - zo) - components of zs - (zt za)' ut" given by 0, where the 1 zj:t xtl xt2 r.or llxr ll2 Lozllxzllz xl x2 for j:1,..., xt roa llxa ll2 x Applying Cramer's rule we see that z6 satisfies the linear equation X'zo _ xtr, i.e. (xi/llxill,zo) - Äo; llxill for i : I, sothatzs - (X,)-1 x,,". The geometric interpretation of the above relations reveals that the orthogonal projection of z6 onto the ege joining o an xi equals Xgi : Äo; xi for i : 1,...,. Furtherrnore, zg lies on the sphere Ss an, provie that x* So, the point zs/2 coincies with the centre of Ss by appealing to the converse of Thales' theorem. Hence, Ao(x*) : 0 hols if an only if (x*, x* - (x')-1xr,) : xi 1An1)'*' (*Ant - (x')-r ) *^ - xi crrx). : 0, where Cn :: (A;t)'X'XA;1 - (Ant),.
r32 L. Heinrich Fig. 2: Intersection of the tetraheron an the Miquel spheres in Fig. 1 with the plane z: 1..54 parallel to the xy-plane Using (7) we fin that cn : I t,t;tl' (nn +n'n) lnt - (Ant)' :i (ant - (Ant)')- -air, z thus proving the skew-symmetry of the matrix Cn which is necessary an sufficient for the quaiatic f*- ti Crr x1 to isappear for any real.1"s1 )"sa anflr1l X1,..., x R, as we wishe to prove. ItremainstoshowthatA;(x*) -0fori - l,...,.forthiswetransposetheeterminant on the left-han sie of (5) an expan it along the first row leaing to the system of equations (llx* ll2 * Äo; llxi ll2) A - I j:i ti xtt cit xi x't ci x v j-th column In the next step we subtract equation (9) from the latter equation an ivie the ifference by A I 0. Finally, using the abbreviation (6) we arrive at the equation lo;llxillt : I j:l,j rti with wi1 l A xn ait xl ;I a'i xaa \./ j-th column, j:i,...,,
r33 which is equivalentto A;(x*) - 0 for eachi - 1,...,. Again, accoring to Cramer's rule the vector wi : (wn,..., tt)i)t sasfies the equation X/wi - (an,..., ai)',that is, we may write wi : (Xr)-t A'rr.t with ei - (0,..., 1 0)/ enoting the i-th column vector of the x ientity matrix. Hence, the above equations can be rewritten as Äo;llxillz: wi'x* -ei'anx-l x* for i-1,...,. However, these equations follow irectly from (8) an vice versa. Thus the proof of the Theorem is complete. I Note that the point Zi - wi* zl-xiwhich the orthogonality relation can be shown to belong to the sphere S; satisfies (x*-zi,x*-xi) -0 for i-1,..., This is easily verifie by straightforwar computations using the expressions of x*, wi, an zs given in the above proof. As a result, the sphere S; has the centre (zi + x) /2 - (wi + zü/2. 3 Bouns for eterminants In the subsequent lemma we establish a lower boun for the eterminant of An that is uniform in all the varying parameters )";y, 0 ( i < j <, an even positive provie the vectors Xl,..., x are linearly inepenent. Note that the below inequality (11) is vali for any (> 2) elements of an arbitrary Eucliean vector space. Lemma. Let V be a Eucliean vector space equippe with scalar prouct (.,.). For any Xl,...,X V ananyrealnumberslij,i, j :0,I,...,, satisfiing),i1 *L1i - lfor i, j :0, 1,...,, the inequality etan Z etbr * etc(x) > etc(x) (11) hols, where Al, B71, an G(X) are efine by (6), (9), an (7), respectively. Equality is attaine in (lt) if,1.;; llxi -*;ll2 - (1-ro;) llxillz +xo,;1x;ll2 - (x1,x;) fo, <i < j <. Proof. As alreay pointe out at the en of Section 1 the eterminant of any skewsymmetric x matrixbisnon-negativeanequals zerolf iso. Thus,theseconpart of (11) is trivial an instea of the first part we prove the slightly more general inequality et(b + G(X)) > etb * et G(X) (r2) By applying the well-known principal axis theorem to the non-negative efinite Gram matrix G(X) we fin an orthogonal x maftix O (with eto - 1) such that D - O'G(X) O is a iagonal matrix with non-negative iagonal elements. The multiplication rule for eterminants enables us to replace G(X) by D an B by the skew-symmetric matrix
134 L. Heinrich O'B O without changing the inequality (12). Therefore it suffices to verify et(b + D) > etb*etd only for iagonal matrices D with non-negative iagonal elements )l1,... z la an any skew-symmetric x matrix B. The Taylor expansion of the function f (yt,..., ya) - et(b + D) - lr bn bu -bn lz bza -bta -bza io leas to -2 f (Yr,...,Y) * etb + I lh'''!i* etbi1,...,ik k:l l<it<...<ip< * Yr ' ''Y, wherethe(-k)x(-k)matrixbir,...,iuemergesfrombbyeletingtherowsil,...,ik an columns i 1,..., ik.obviously, all these matrices are skew-symmetric which, together with y1,..., la > 0, implies f(yr,...,ya) > etb *yt"'l: etb*etd. Thus the inequality (11) is prove. The proof of the lemma is complete by noting that Al : A/n an (7) imply equality in (11). n Remark. The above proof turns out that the inequality (I2) remains vali if G(X) is replace by any other non-negative efinite x matrix. Corollary. If xt,..., X e V are linearly inepenent, then etal is positive which in turn implies the existence of the inverse Alr for all real )";1 satisfying ),ii * ).1i : I for i, j :0, 1,...,. As a special case, (II) inclues thewell-knownhaamarinequality etg(x) < llxr P..... llxall2 for any xr,..., x4 e V, whichreas lals llxrll... llxall in R. The first part of the corollary follows from the well-known fact that et G(X) > 0 characterizes the linear inepenence of xr,..., x V. The secon part follows by setting 101 -... _ L1 :0an )rij : llxill2 /llxi-xjll2 sothatby(6) aji :0 for < i < j <. In other wors, Azr is an upper triangular matrix entailing etan - afi '... ' a. 4 Concluing remarks 1. Formula (8) an the above theorem reveal that the Miquel point M of the simplex S(o,x1,...,x)w.r.t.themarkepointsxij:xi*l;j(x:-xi)onthesimplex'egescan be represente as a linear combination of the eges Xl,..., X, **:I i:l uixi - Xu with weight vector a - (ut,..., ü)t - All xt (13)
135 From (6) an (8) it is seen that the weights ur,...,u4 epen only on the parameters )"ii anthesquareegelengthsllxi-x;ll2for0 < i < j < (withxo: o). For this reason the Miquel point can be efine in a meaningful way for any finite family of linearly inepenent elements of a Eucliean vector space V. By (13) the Miquel points *i,i - 0, I,...,, of those ( - l)-simplices having the vertices x:, i f i, can be easily etermine. Geometrically spoken,xi is the point of intersectionof themiquelspheres Sj, j t' i,withthehyperplanecontainingthe ( - l)-tace of the original simplex which x; oes not belong to. In this way a new - simplex S(xö,x{,...,*ä) arises an relations between it an the original one for given Ä;7's coul be of interest. 2. Since the matrix An is invertible for any )"ii the point x* is also well-efine for \ij e {0, 1}. For special choices of the,1.;y's we have interesting geometric intepretations, e.g. for:2,lettinglor + l,loz ) 0, an),p + l entailsthateachof thelimitingmiquel circles touches one sie of the triangle A x6x1x2 at a vertex an passes through the opposite vertex; the corresponing Miquel point turns into the Brocar point, see [1]. By means of the Theorem in Section 2 several generalizations of this point to higher imensions are possible. In particular, for a tetraheron S(o, xr,x2,x3) in IR3 the choice l0r : Loz : 103 : Ä13 : 1 an Ltz : \23 :0 (in the above setting) yiels x* as common point of the circumsphere 56 an 51 fl Sz fl 53, where e.g. Sr is the unique sphere through x3 touching the face triangle A ox1x2 atx1, an 52, 53 are eflne analogously. 3. The vertices of a further -simplex associate with,s(o, x1,..., xa) an the given,l.;;'s coinciewiththe mipointsxf of themiquel spheres S;, i :0, 1,...,.In Section 2 we have erive the followins formulas: *3: jtx')-t *, anxf -)r*'r-tan., **3, i : r,..., In the planar case simple geometric arguments show that the triangles A xsx1x2 an A xfixixi are similar, cf. e.g. [1]. It is natural to ask whether the simplex S(xfi, x!,..., *;) an the original simplex are similar for any > 2. This fact can be expresse analytically by an orthogonal matrix O an some scaling factor y > 0 by requiring *i -*3 1 - - lt*'l-tan., yox; for i - r,...,, where' 2 TetAn 1t7a \- ^r-l ln view of O/ - O-1 an (7) these relations are equivalento G-l(X) - 2y2 (n;t + (A'n)-t ). fnis equality hols actually only for. - 2 without aitional restrictions on the.1";;'s. 4. Onthe other han, for any > 2, the particular choice )"ij l/2, 0 < i < j <, yiels the Miquel point x* _ \.(x')-t (llxrll2,...,llxa;12;', which coincies with the circumcentre of the simplex,s(o, Xl,..., xa).
136 L. Heinrich Acknowlegements I wish to thank Prof. Dr. H. Martini (TU Chemnitz) who brought Konnully's paper l2l to my attention. I am grateful to Dr. L. Muche for computer simulations concerning Miquel's theorem. Many thanks also to Dr. T. Klein for his helpful comments on an earlier version of this paper an to stu. math. oec. G. Freuenreich who assiste me in the 3D visualization of Miquel's theorem. References [1] Donath, E.: Die merhuürigen Punkte un Linien es ebenen Dreiecks. VEB Deutscher Verlag er Wissenschaften, Berlin 1968. [2] Konnully, A.O.: Pivot theorems in n-space. Beitröge Algebra Geom. 10 (1980), 157-163. [3] Miquel, A.: M6moire e G6om6trie, J. Math. Pures Appl. e Liouville 1 (1838), 485487. [4] Weisstein, E.W.: Miquel's Theorem. http://mathworl.wolfram.com/lvliquelstheorem.html Lothar Heinrich University of Augsburg Institute of Mathematics D-86 1 35 Augsburg, Germany e-mail: he inri ch@math. uni - augsburg. e