SOME COPLANAR POINTS IN TETRAHEDRON

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1 Journl of Pure n Applie Mthemtis: Avnes n Applitions Volume 16, Numer 2, 2016, Pges Aville t DOI: SOME COPLANAR POINTS IN TETRAHEDRON S. EKMEKÇİ, Z. AKÇA *, A. ALTINTAŞ n A. BAYAR Deprtment of Mthemtis Computer Sienes Eskişehir Osmngzi University Eskişehir Turkey e-mil: zk@ogu.eu.tr Astrt In this work, we etermine the onitions for oplnrity of the verties, the inenter, the exenters, n the symmein point of tetrheron. 1. Introution n Preliminries In geometry, the ryentri oorinte system is oorinte system in whih the lotion of point of simplex ( tringle, tetrheron, et.) is speifie s the enter of mss, or ryenter, of usully unequl msses ple t its verties. Coorintes lso exten outsie the simplex, where one or more oorintes eome negtive. The system ws introue in 1827 y August Ferinn Möius [1, 2] Mthemtis Sujet Clssifition: 51Axx, 51Kxx. Keywors n phrses: tetrheron, oplnrity. * Corresponing uthor is Z. Akç. Reeive Novemer 30, Sientifi Avnes Pulishers

2 110 S. EKMEKÇİ et l. In given tringle ABC, every point P is oorintize y triple of numers ( u : v : w) in suh wy tht the system of msses u t A, v t B, n w t C will hve its lne point t P. These msses n e tken in the proportions of the res of tringle PBC, PCA, n PAB. Allowing the point P to e outsie the tringle, we use signe res of oriente tringles. The homogeneous ryentri oorintes of P with referene to ABC is triple of numers ( x : y : z) suh tht x : y : z = PBC : PCA : PAB. The entroi G, the inenter I n the symmein point K hve the homogeneous ryentri oorintes ( 1, 1, 1), (,, ), n (,, ), respetively [3]. The oorintes of verties of tringle ABC re A ( 1, 0, 0), B( 0, 1, 0), C( 0, 0, 1) n the exenters re J A (,, ), J B (,, ), JC (,, ). Some ollinerities of these points in the heptgonl tringle re isusse [2]. Bryentri oorintes my e esily extene to three imensions. The 3D simplex is tetrheron, polyheron hving four tringulr fes n four verties. Let P e point insie the tetrheron. It ivies the tetrheron into four su-tetrherons. Let the volumes of su-tetrherons e PBCD = V PACD = V, A, B PABD = VC, n PABC = VD. Using the signe volumes of the oriente su-tetrherons the homogeneous ryentri oorintes of P with the referene tetrheron ABCD is ( x, y, z, t) suh tht x : y : z : t = VA : VB : VC : VD. Respet to the tetrheron ABCD, the ryentri oorintes of the verties re A ( 1, 0, 0, 0), B( 0, 1, 0, 0), C( 0, 0, 1, 0), n D ( 0, 0, 0, 1) [6]. The inenter of tetrheron is the intersetion point of plnes tht iset the ngles etween the tetrheron fes. It is the enter of insphere whih is tngent to four fes. In tetrheron ABCD, let S, S, S, S e

3 SOME COPLANAR POINTS IN TETRAHEDRON 111 the re of the fes BCD, ACD, ABD, n ABC, respetively. The ryentri oorinte of the inenter of ABCD is I ( S, S, S, S ) n the ryentri oorintes of the exsphere enters re J A ( S, S, S, S ), J B ( S, S, S, S ), J ( S, S, S, S ), J ( S, S, S, S ) [1]. C In the referene tetrheron ABCD, two plnes α n β through AB re si to e isogonl onjugtes if they re eqully inline from the sies tht form the iherl ngle etween the plnes of the tringles ABC n ABD. α is lle the isogonl onjugte of β n vie vers. If point X of ABCD is joine to the vertex A n the vertex B, the plne through XA n XB hs n isogonl onjugte t A. Similrly, joining X to verties B n D, D n C, A n C, B n C, C n D, proue five more onjugte plnes. Let M e the mipoint of CD. The plne ontining AB n tht is isogonl to the plne of tringle ABM is lle symmein plne of tetrheron ABCD. Tking the mipoints of the six sies of the tetrheron ABCD n forming the ssoite symmein plnes, we ll the intersetion point of these symmein plnes the symmein point of the tetrheron. The entroi G of the tetrheron hs ryentri oorintes G ( 1, 1, 1, 1) n symmein point K hs oorintes K ( S S, S, S ) [4]., In this work, we etermine the oplnrity reltions mong the verties A, B, C, D, the inenter I, the exenters J J, J, J, n D A, B C D the symmein point K in the referene tetrheron y using the ryentri oorinte system. 2. Coplnrity n Bryentri Coorintes in Tetrheron Bryentri oorintes (see, e.g., (Hoking n Young [8]), Chpter 5) re prtil wy to prmeterize lines, surfes, et., for pplitions tht must ompute where vrious geometri ojets interset. In prtie, the ryentri oorinte metho reues to speifying two points ( x 0, x 1 )

4 112 S. EKMEKÇİ et l. on line, three points ( x, x x ) on plne, four points ( x, x, x x ) 0 1, , in volume, et., n prmeterizing the line segment, enlose tringulr re, n enlose tetrherl volume, et., respetively. Let (,, ), (,,, ), (,,, ), n (,, ) 0, ,, e the ryentri oorintes of 0 1 2,, with respet to the stnr ffine frme for ffine spe, respetively. It is well known tht re oplnr iff [,,, ] 0. et = Proposition 2.1. Let the referene tetrheron e ABCD. Then, 3 3,,, () Any two verties n the exenters orresponing to these verties re oplnr in ABCD. () Any vertex, the exenter relte to this vertex, the inenter n the symmein point re oplnr in ABCD. Proof. () Let the verties n the exenters of ABCD e A, B n J A, J B. These four points in terms of the ryentri oorintes re A = ( 0, 0, 1, 0), B = ( 0, 0, 0, 1), J A = ( S, S, S, S ), J B = ( S, S,, S ). S Sine [ A, B,, ] et J A J B is zero, these four points A, B, J A, J B of tetrheron re oplnr. The oplnrity of other verties n orresponing exenters re prove similrly. () Let the vertex n the exenter orresponing to this vertex of tetrheron e A n J A. Sine the eterminnt of the mtrix of the ryentri oorintes of the points A, J A, I, n K is zero, these four points A, J A, I, n K of the tetrheron re oplnr. Also, the oplnrity of other verties n orresponing exenter, I n K of the tetrheron ABCD re prove similrly. Proposition 2.2. Let the referene tetrheron e ABCD. Then ny three exenters of ABCD re oplnr neither the symmein point nor the inenter point.

5 SOME COPLANAR POINTS IN TETRAHEDRON 113 Proof. Let ny three exenters e ABCD. We ssume tht, of the tetrheron, n the symmein point K of ABCD re oplnr, then the eterminnt in terms of the ryentri oorintes of J A, J B, JC, K of the tetrheron is zero. S = S + S + S is otine from this eterminnt. But the eqution S S + S + S = ontrits the sum of the res of the three fes in tetrheron is greter thn the re of fourth fe in [5]. So, ny three exenters n the symmein point of tetrheron nnot e oplnr. Similrly, we ssume tht, n the inenter point I of ABCD re oplnr in orer to fin ontrition, then the eterminnt in terms of the ryentri oorintes of J A, J B, JC, I is zero. From this eterminnt, the reltion S S S S = 0 is otine ontriting the ft tht S, S, S, S > 0. Our lim is prove. Proposition 2.3. Let ny three exenters e entroi e G in the tetrheron ABCD. Let S S, S, S, n the, e the re of the fes BCD, ACD, ABD, n ABC of ABCD. Then the points J J, J C, n G re oplnr iff + + =. S S S S A, B Proof. Suppose tht J J, J, n G re oplnr. Sine the A, B C eterminnt in terms of the ryentri oorintes of J J, J, n G is zero, 1 S 1 S = + + is otine. S S S Conversely, it is seen tht [ J, J, J G ] A B C, A, B C et is zero y using = + +. This implies tht the points J A, J B, JC, n G S S S re oplnr.

6 114 S. EKMEKÇİ et l. Proposition 2.4. Let ny two exenters e J J, the inenter e I, A, B the symmein point e K, n the entroi e G in the tetrheron ABCD. Let S, S, S, S e the re of the fes BCD, ACD, ABD, n ABC of ABCD. Then if S = S, the points J A, J B, G, I, n K re oplnr. Proof. We ssume tht S = S. Then it is otine tht the eterminnt in terms of the ryentri oorintes of J A, J B, G, n I is zero. Hene, the points J A, J B, G, n I re oplnr. Similrly, the eterminnt in terms of the ryentri oorintes J A, J B, G, n K is zero, it is seen tht the points J A, J B, G, n K re oplnr. So, the points J A, J B, G, I, n K re oplnr. Referenes [1] A. Plln, Misellneous Results on Tetrheron. [2] Einr Hille, Anlyti Funtion Theory, Volume I, Seon Eition, Fifth Printing, Chelse Pulishing Compny, New York, 1982, ISBN , pge 33. [3] A. Altıntş, Some ollinerrities in heptgonl tringle, Forum Geometriorum 16 (2016). [4] P. Yiu, Introution to Geometry of Tringle, Flori Atlnti University, [5] J. Sek, M. Bni n H. N. Rhee, Isogonl onjugtes in tetrheron, Forum Geometriorum 16 (2016). [6] Niri M. Serkyn, Another look t the Volume of Tetrheron, Cnin Mthemtil Soiety 27 (2001), g