The Inspherical Gergonne Center of a Tetrahedron

Transcription

1 Journl for Geometry nd Grphis Volume 8 (2004), No. 1, The Inspheril Gergonne Center of Tetrhedron Mowffq Hjj 1, Peter Wlker 2 1 Deprtment of Mthemtis, Yrmouk University Irbid, Jordn emil: mhjj@yu.edu.jo 2 Deprtment of Mthemtis nd Sttistis, Amerin University of Shrjh P.O. Box 26666, Shrjh, United Arb Emirtes emil: peterw@us..e Abstrt. With the trditionl definition of the Gergonne enter of tringle in mind, it is nturl to onsider, for given tetrhedron, the intersetion of the evins tht join the verties to the points where the insphere touhes the fes. However, these evins re onurrent for the limited lss of wht we ll inspheril tetrhedr. Another pproh is to note tht the evins through ny point inside tringle divide the sides into 6 segments, nd tht the Gergonne enter is hrterized by the requirement tht every two segments shring vertex re equl. Similrly, the evins through ny point inside tetrhedron divide the fes into 12 subtringles, nd one my define the Gergonne enter s the point for whih every two subtringles tht shre n edge re equl in re. This ws done by the uthors in [8], where the existene nd uniquenes of suh point is estblished. It turns out tht tetrhedr whose Gergonne enter hs the stronger property tht every two subtringles tht shre n edge re ongruent (resp. skew-ongruent) re preisely the inspheril (resp. equifil) ones. This is proved in Setion 4. In Setions 2 nd 3, we give hrteriztion of the inspheril tetrhedr nd we outline method for onstruting them. Key Words: Bryentri oordintes, Cev s theorem, Gergonne point, Fermt- Torrielli point, isoseles or equifil tetrhedron, isogonl tetrhedron. MSC 2000: 51M04, 51M20, 52B10 1. Introdution nd terminology The Gergonne enter of tringle ABC is defined to be the point of intersetion of the evins AA, BB, nd CC, where A, B, nd C re the points where the inirle touhes the sides ISSN /$ Heldermnn Verlg

2 24 M. Hjj, P. Wlker: The Inspheril Gergonne Center of Tetrhedron BC, CA, nd AB, respetively, s in Fig. 1. Suh evins re onurrent by Cev s Theorem sine A, B, C n be equivlently defined by the lgebri reltions AB = AC, BC = BA, CA = CB, where XY denotes the length of the line segment XY. It is then quite nturl to onsider the point of intersetion (if ny) of the evins AA, BB, CC, nd DD of tetrhedron ABCD, where A, B, C, nd D re the points where the insphere touhes the fes. We shll see tht suh evins re onurrent for limited lss of tetrhedr nd we shll desribe suh lss. We ll the resulting enter (when it exists) the inspheril Gergonne enter, or simply the inspheril enter to distinguish it from the Gergonne enter defined in [8], nd we shll ll tetrhedron with n inspheril enter n inspheril tetrhedron. The reltion between these two enters is disussed in Setion 4. A C C. B. B B A Figure 1: C Throughout the rest of this rtile, the term tetrhedron stnds for non-degenerte tetrhedron. An equifil or isoseless tetrhedron is tetrhedron whose fes re ongruent, or equivlently hve the sme re; see [10, Chpter 9, pges 90 97], [12, Theorem 4.4, pge 156], [2, Cor. 307, pge 108] or [7]. If A 1 A 2 A 3 is tetrhedron, then by its j-th fe we shll men the fe opposite to the j-th vertex A j. Two edges of tetrhedron will be lled opposite if they hve no vertex in ommon. Two tringles ABC nd A B C re lled ongruent if there is n isometry tht rries A, B, nd C to A, B, nd C in the sme order. Finlly, we ll two n-tuples (x 1,..., x n ) nd (y 1,..., y n ) equivlent, nd we write (x 1,..., x n ) (y 1,..., y n ) if one of them is positive multiple of the other. Thus two n- tuples (x 1,..., x n ) nd (y 1,..., y n ) re bryentri oordintes of the sme point if nd only if they re equivlent. 2. A hrteriztion of inspheril tetrhedr In Theorem 1 below, we prove tht tetrhedron is inspheril if nd only if its insphere touhes the fes t their Fermt-Torrielli points. However, to see how lrge the lss of suh tetrhedr is nd to desribe how they n be onstruted, stronger nd more tehnil sttement is needed nd is provided by Theorem 2. For ese of referene nd to be self-ontined, we strt with two firly well-known lemms. The first desribes the Fermt-Torrielli point of tringle nd the seond is threedimensionl version of Cev s Theorem. Both will be used freely.

3 M. Hjj, P. Wlker: The Inspheril Gergonne Center of Tetrhedron 25 Lemm 1. Let ABC be non-degenerte tringle. Then there exists unique point F whose distnes from the verties hve minimum sum. If the ngles of ABC re less thn 120 eh, then F is interior nd is hrterized by the equi-ngulr property AF B = BF C = CF A = 120. Otherwise, F is the vertex tht holds the lrgest ngle. In ll ses, F is lled the Fermt-Torrielli point of ABC. Proof. See [6], or [1] nd [13] for higher-dimensionl versions. Lemm 2. Let T = A 1 A 2 A 3 be tetrhedron nd let D j, 1 j 4, be point on its j-th fe. Then the evins A j D j, 1 j 4, re onurrent if nd only if there exist M 1, M 2, M 3 nd M 4 suh tht for ll rrngements (i, j, k, t) of {1, 2, 3, 4}, (M i, M j, M k ) re bryentri oordintes of D t reltive to A i A j A k. In this se, (M 1, M 2, M 3, M 4 ) re bryentri oordintes of the point of onurrene reltive to A 1 A 2 A 3. Proof. See [14]. Theorem 1. Let A 1 A 2 A 3 be tetrhedron nd let D j, 1 j 4, be the point where the insphere touhes the j-th fe. Then the evins A j D j, 1 j 4, re onurrent if nd only if D j, 1 j 4, is the Fermt-Torrielli point of the j-th fe. A 1 α d d α β γ γ β α β γ A 3 b b b α β γ d A 2 Figure 2: Proof. Following the rgument in [10, pge 93], we first note tht the line segments A 1 D 2, A 1 D 3, nd A 1 D 4 hve equl lengths sine they re tngent lines to the sme sphere from the sme point A 1. We denote their ommon length by, nd we define b,, d similrly. This is shown in Fig. 3 where our tetrhedron is flttened. It follows tht the tringles A 1 A 2 D 3

4 26 M. Hjj, P. Wlker: The Inspheril Gergonne Center of Tetrhedron nd A 1 A 2 D 4 re ongruent nd hene A 1 D 3 A 2 = A 1 D 4 A 2. We denote the mesure of this ngle by γ, nd we define α, β, α, β, γ similrly, s in Fig. 2. The equtions α + β + γ = α + β + γ = α + β + γ = α + β + γ = 360 (1) led immeditely to the onlusion α = α, β = β, γ = γ. (2) Now ssume tht the evins A j D j, 1 j 4, re onurrent. If (M 1, M 2, M 3, M 4 ) re bryentri oordintes of their point of intersetion reltive to A 1 A 2 A 3, then by Lemm 2, (M 1, M 2, M 3 ) re bryentri oordintes of D 4 reltive to A 1 A 2 A 3. Thus (b sin α, sin β, b sin γ) (M 1, M 2, M 3 ) (bd sin β, d sin α, b sin γ) (M 1, M 2, M 4 ). Therefore b sin α sin β = bd sin β d sin α ( = M ) 1 M 2 nd hene α = β. Similrly, α = γ nd therefore α = β = γ = 120. Hene D 1 is the Fermt-Torrielli point of the fe A 1 A 2 A 3. Similrly for the other fes. Thus if our evins re onurrent, then the D j s re the Fermt-Torrielli points of the fes. A 1 D 2 D 3 D 4 A 3 b D 1 A 2 Figure 3:

5 M. Hjj, P. Wlker: The Inspheril Gergonne Center of Tetrhedron 27 Conversely, suppose tht the D j s re the Fermt-Torrielli points of the fes. It is ler tht the bryentri oordintes of D 4 with respet to A 1 A 2 A 3 re ( 1 (b sin 120, sin 120, b sin 120 ), 1 b, 1 ). Similr sttements hold for the other D j s. Lemm 2 now implies tht the evins A j D j re onurrent (nd tht they onur t the point whose bryentri oordintes reltive to A 1 A 2 A 3 re (1/, 1/b, 1/, 1/d)). 3. Constrution of generl inspheril nd non-inspheril tetrhedr Theorem 1 desribes how to onstrut non-inspheril tetrhedron. Strt with ny tringle ABC nd ny point P inside ABC tht is different from the Fermt-Torrielli point of ABC. Ple smll sphere S so tht it touhes the plne of ABC t P nd drw three plnes through the sides of ABC tht re tngent to S. If D is the point where these three plnes interset, then ABCD would be non-inspheril tetrhedron. To ensure tht these plnes interset within the hlfspe of the plne of ABC whih ontins S, we tke the rdius of S suffiiently smll. For exmple, it is suffiient (but by no mens neessry) to tke the rdius of S to be less thn eh of the distnes of P from the sides of ABC. On the other hnd, to desribe how to onstrut inspheril tetrhedr, we need the following theorem, whih will lso be needed in proving Theorem 4. X A 1 X y D 2 Z D 3 z D 4 A 3 Y b x Z D 1 Y A 2 Figure 4: Theorem 2. Let A 1 A 2 A 3 be tetrhedron whose fes hve interior Fermt-Torrielli points D 1, D 2, D 3 nd D 4. If A i D j = A i D k = A i D t for ll rrngements (i, j, k, t) of {1, 2, 3, 4}, then our tetrhedron is inspheril.

6 28 M. Hjj, P. Wlker: The Inspheril Gergonne Center of Tetrhedron Proof. Let denote the ommon length of the line segments A 1 D 2, A 1 D 3, nd A 1 D 4, nd define b,, d similrly, s shown in Fig. 4. Let I be the inenter nd r the inrdius of our tetrhedron, nd let I j be the orthogonl projetion of I on the j-th fe. To prove tht I j = D j for ll j, it is suffiient by symmetry to prove tht I 4 = D 4. Equivlently, it is suffiient to prove tht the distnes from I 4 to A 1, A 2, nd A 3 re, b, nd, respetively. Agin, it is suffiient by symmetry to prove tht I 4 A 1 =, or equivlently α 2 r 2 = 2, (3) where α = IA 1. Without loss of generlity, we my ssume tht I is the origin. To omplish this, we note tht ll the elements of our tetrhedron n be omputed in terms of, b,, nd d. We find r (in terms of, b,, d) by first finding the lengths of the edges (using the Lw of Cosines), then the res F j of the fes nd the volume V of the tetrhedron, nd then using the formul V = (r/3)(f 1 +F 2 +F 3 +F 4 ). Then we find α = A 1 by solving system of 10 liner equtions in the unknowns A i A j, i j. We then hek tht (3) is indeed stisfied. We find it more onvenient to denote the lengths of the edges by x, X,... s shown in Fig. 4. Thus we hve x = b b, y = , z = 2 + b 2 + b, X = 2 + d 2 + d, Y = b 2 + d 2 + bd, Z = 2 + d 2 + d. (4) The re F j of the j-th fe is given by F j = ( 3/4)f j, where f 1 = b + d + db, f 2 = + d + d, f 3 = b + bd + d, f 4 = b + b +. (5) The volume V of the tetrhedron is given by 144V 2 = (xx + yy + zz)(x + X + y + Y + z + Z) (xyz + xy Z + XyZ + XY z) = 2(x 2 X + X 2 x + y 2 Y + Y 2 y + z 2 Z + Z 2 z). (6) Using (4), this simplifies into where Sine 16V 2 = bds, (7) S = b + b + + d + bd + d = f 1 + f 2 + f 3 + f 4. (8) 2 V = r 3 (F 1 + F 2 + F 3 + F 4 ) = r 3 it follows tht From (7) nd (9), it follows tht 3 4 (f 1 + f 2 + f 3 + f 4 ) = r (2S) = r 3 6 S, 16V 2 = 4r2 S 2. (9) 3 r 2 = 3bd 4S. (10)

7 M. Hjj, P. Wlker: The Inspheril Gergonne Center of Tetrhedron 29 It remins to ompute the distnes from the inenter I to the verties. The bryentri oordintes of I with respet to A 1 A 2 A 3 re (f 1, f 2, f 3, f 4 ). Therefore I = 0 = f 1 A 1 + f 2 A 2 + f 3 A 3 + f 4. Tking the slr produt with eh A j, nd letting A ij = A i A j, we obtin the first four liner equtions represented by (11) below. Next, every edge gives rise to n eqution in the mnner tht z = A 2 A 3 2 = (A 2 A 3 ) (A 2 A 3 ) = A 22 + A 33 2A 23. The six equtions thus obtined re the lst 6 equtions of the system (11). The system f f 2 f 3 f A f f f 3 f 4 0 A f f 1 0 f 2 0 f 4 A f f 1 0 f 2 f A A 13 = x y A 14 z A 23 X A 24 Y A 34 Z hs unique solution. Using (4) nd (5), we obtin the vlues of the A ij in terms of, b,, nd d. In prtiulr, we obtin α 2 = A 11 = 2 + 3bd 4S. This together with (10) implies tht α 2 r 2 proof. (11) = 2, s desired in (3), nd ompletes the Theorem 2 bove desribes ompletely how to onstrut generl inspheril tetrhedron. We drw three vetors OA 1, OA 2, nd OA 3 tht mke n ngle of 120 with eh other nd tht hve rbitrry lengths, b, nd (see Fig. 5). In other words, we tke tringle A 1 A 2 A 3 with Fermt-Torrielli point O. By vlley-folding long the edges of A 1 A 2 A 3, we rete three more tringles B 1 A 2 A 3, A 1 B 2 A 3, nd A 1 A 2 B 3 tht re identil with A 1 A 2 A 3. Let O 1, O 2, nd O 3 be the Fermt-Torrielli points of these tringles, respetively. For eh j, tke point C j on the line segments O j B j in suh wy tht O 1 C 1, O 2 C 2, nd O 3 C 3 hve the sme length, d sy. Now, we ut long the edges A 1 C 2, C 2 A 3, A 3 C 1, C 1 A 2, A 2 C 3, nd C 3 A 1 nd we vlley-fold long the sides of A 1 A 2 A 3 until the points C 1, C 2, nd C 3 oinide nd oupy the sme position, sy. The resulting tetrhedron A 1 A 2 A 3 stisfies the onditions of Theorem 2, nd thus is inspheril. It is quite legitimte to question the lim, impliitly mde bove, tht totlly rbitrry positive numbers, b,, d do give rise to fesible tetrhedron. After ll, very similr sitution ws met when onstruting n equifil tetrhedron. Trying to get the verties of given tringle to oinide nd oupy the sme position by vlley-folding long the three segments tht join, two by two, the mid-points of the sides, one soon disovers tht this is possible only if the given tringle is ute-ngled. Theorem 3 below ssures us tht the onstrution desribed in the previous prgrph is vlid for ll hoies of positive, b,, d. Its proof mkes use of the following lemm.

8 30 M. Hjj, P. Wlker: The Inspheril Gergonne Center of Tetrhedron Lemm 3. Let x, y, z, X, Y, Z be given positive numbers. Then there exists tetrhedron suh tht the lengths of the sides of one of its fes re x, y, z nd the lengths of the opposite sides re X, Y, Z, respetively, if nd only if the right hnd side of (6) is positive nd ll the relevnt tringle inequlities pertining to the six fes re stisfied. Proof. See [9] or [3]. Theorem 3. Let 1, 2, 3, 4 be rbitrry positive numbers. Then there exists tetrhedron A 1 A 2 A 3 whose fes hve interior Fermt-Torrielli points D 1, D 2, D 3, D 4, suh tht A i D j = A i D k = A i D t = i for ll rrngements (i, j, k, t) of {1, 2, 3, 4}. C 2 B 2 A 1 O 2 C 3 B 3 O 3 O A 3 O 1 B 1 A 2 C 1 Figure 5: Proof. Let 1, 2, 3, 4 be renmed s, b,, d nd let x, X,... be defined s in (4). In view of the onstrution exhibited in Fig. 5, nd in view of Lemm 3, we need only show tht the right hnd side of (6) is positive. This holds sine the right hnd side of (6) simplifies into the positive quntity bds given in (7). 4. Reltion to other Gergonne enters We end this note by disussing possible lterntive definitions of the Gergonne enter. Observe tht ny evins A 1 D 1, A 2 D 2, nd A 3 D 3 of tringle A 1 A 2 A 3 divide the sides into six segments A i D j, i j, nd tht these evins meet t the Gergonne enter if nd only if ny two of these segments tht shre vertex re equl, or equivlently A i D j = A i D k for every rrngement (i, j, k) of {1, 2, 3}. Similrly, ny evins A 1 D 1, A 2 D 2, A 3 D 3, nd D 4 of tetrhedron A 1 A 2 A 3 divide the fes into 12 tringles A i A j D k, i, j, k re pirwise distint. In [8], it ws proved tht there

9 M. Hjj, P. Wlker: The Inspheril Gergonne Center of Tetrhedron 31 exist unique onurrent evins suh tht the twelve tringles into whih they divide the fes hve the property tht ny two whih shre n edge hve equl re, or equivlently A i A j D s nd A i A j D t hve the sme re for every rrngement (i, j, s, t) of {1, 2, 3, 4}. (12) The bove disussion shows tht the inspheril enter is obtined by repling (12) with the stronger requirement tht A i A j D s nd A i A j D t re ongruent for every rrngement (i, j, s, t) of {1, 2, 3, 4}. (13) This shows in prtiulr tht if the inspheril enter exists, then it is the Gergonne enter itself. Thus in sense, the two pprohes to defining the Gergonne enter of tetrhedron led to the sme enter. Also, ny referene to the insphere is mde obsolete in view of (13). It is nturl to lso investigte the result of repling (13) by the similr ondition A i A j D s nd A j A i D t re ongruent for every rrngement (i, j, s, t) of {1, 2, 3, 4} (14) nd to see whether this skew-ongruene ondition, too, is equivlent to some nturl geometri requirement. The next theorem provides the nswer. Theorem 4. Let T = A 1 A 2 A 3 be tetrhedron nd let A j D j, 1 j 4, be the unique onurrent evins (gurnteed in [8]) tht stisfy (12). Then (13) holds if nd only if T is inspheril (in whih se the D j s re the Fermt-Torrielli points of the fes), nd (14) holds if nd only if T is equifil (in whih se the D j s re the entroids of the fes.) Proof. If (13) holds, then the first prgrph in the proof of Theorem 1 shows tht the D j s re the Fermt-Torrielli points of the fes. By Theorem 2, T is inspheril. Conversely, if T is inspheril, then the Fermt-Torrielli points stisfy (13) nd hene (12). By their uniqueness, the D j s re the Fermt-Torrielli points nd hene they stisfy (13). If (14) holds, then letting, b, nd α, α,... be s shown in Fig. 2, we see tht (1) holds nd leds to the onlusion (2). Thus the fes of T re ongruent nd T is equifil. Conversely, if T is equifil, then the entroids of the fes stisfy (12). By their uniqueness, the D j s re the entroids nd hene they stisfy (14). Epilogue nd knowledgments After writing this rtile, it me to our ttention tht the problem of hrterizing inspheril tetrhedr ws onsidered by Edwin Koźniewski nd Rent A. Górsk in [11], where n lterntive hrteriztion is hieved. We re grteful to E. Koźniewski for sending us opy of [11] nd for drwing our ttention to the ft tht our Theorem 1 is lredy known to Aleksey Zslvsky s nnouned in [15]. We re lso grteful to Zslvsky for writing nd telling us tht the mteril in [15] will pper (in Russin) in the Mrh 2004 issue of Mthemtiheskoe Prosveshenije nd tht the ontents re ontributions by sholrs D. Kosov (Mosow), M. Isev (Brnul) nd V. Filimonov (Ekterinburg). Lter, we found out tht Theorem 1 is more thn one entury old, s implied by Footnote 1 of [4, pge 373], where the term isogonl is used to men inspheril. The sme footnote sttes tht the three ngles subtended by the sides of fe of tetrhedron t the point where the insphere touhes tht fe re the sme for ll fes. Thus if the insphere of tetrhedron touhes fe ABC t X, then the mesures of the three ngles AXB, BXC, nd CXA

10 32 M. Hjj, P. Wlker: The Inspheril Gergonne Center of Tetrhedron re independent of tht prtiulr fe. This beutiful, yet obsure, theorem is lso stted in [5, 2nd prgrph, pge 174], nd it lerly implies tht the ondition, in Theorem 1, tht ll points of ontt of the insphere of ABCD with the fes re the Fermt-Torrielli points of the respetive fes is equivlent to the seemingly weker ondition tht one of these points is. The first-nmed uthor is supported by reserh grnt from Yrmouk University nd would like to express his thnks for tht grnt. The uthors re lso grteful to Aymn Hjj for drwing the figures nd to the editors for onverting them into PostSript. Referenes [1] S. Abu-Symeh, M. Hjj: On the Fermt-Torrielli points of tetrhedr (nd higher dimensionl simplexes). Mth. Mg. 70, (1997). [2] N. Altshiller-Court: Modern Pure Solid Geometry. Chelse Publishing Co., N.Y [3] L.M. Blumenthl: A budget of urios metri. Amer. Mth. Monthly 66, (1959). [4] B.H. Brown: A theorem on isogonl tetrhedr. Amer. Mth. Monthly 31, (1924). [5] J.L. Coolidge: Some unsolved problems in solid geometry. Amer. Mth. Monthly 30, (1923). [6] M. Hjj: An dvned lulus pproh to finding the Fermt point. Mth. Mg. 67, (1994). [7] M. Hjj: A vetor proof of theorem of Bng. Amer. Mth. Monthly 108, (2001). [8] M. Hjj, P. Wlker: The Gergonne nd Ngel enters of tetrhedron. J. Geom. 75, (2002). [9] F. Herzog: Completely tetrhedrl sextuples. Amer. Mth. Monthly 66, (1959). [10] R. Honsberger: Mthemtil Gems II. Dolini Mthemtil Expositions No. 2, Mthemtil Assoition of Ameri, Wshington, DC, [11] E. Koźniewski, R.A. Górsk: Gergonne nd Ngel points for simplies in the n- dimensionl spe. J. Geometry Grphis 4, (2000). [12] Y.S. Kupitz, H. Mrtini: The Fermt-Torrielli point nd isoseles tetrhedr. J. Geom. 49, (1994). [13] Y.S. Kupitz, H. Mrtini: Geometri spets of the generlized Fermt-Torrielli problem. Intuitive Geometry, Bolyi Soiety Mth. Studies 6, (1997). [14] S. Lndy: A generliztion of Cev s theorem to higher dimensions. Amer. Mth. Monthly 95, (1988). [15] A. Zslvsky: Comprtive Geometry in the Tringle nd the Tetrhedron. Reeived November 30, 2003; finl form July 19, 2004