Multiplying and Dividing Curves by Points

Transcription

1 Forum Geometriorum Volume 1 (2001) FORUM GEOM ISSN Multiplying nd Dividing Curves y Points Clrk Kimerling Astrt. Pointwise produts nd quotients, defined in terms of ryentri nd triliner oordintes, re extended to produts P Γ nd quotients Γ/P, where P is point nd Γ is urve. In triliners, for exmple, if Γ 0 denotes the irumirle, then P Γ 0 is prol if nd only if P lies on the Steiner insried ellipse. Bryentri division y the tringle enter X 110 rries Γ 0 onto the Kiepert hyperol Γ ;ifp is on Γ 0, then the point P = P/X 110 is the point, other thn the Trry point, X 98, in whih the line PX 98 meets Γ, nd if Ω 1 nd Ω 2 denote the Brord points, then P Ω 1 / P Ω 2 = P Ω 1 / P Ω 2 ; tht is, P nd P lie on the sme Apollonin irle with respet to Ω 1 nd Ω Introdution Pul Yiu [7] gives mgnifient onstrution for produt P Q of points in the plne of tringle ABC. If P = α 1 : β 1 : γ 1 nd Q = α 2 : β 2 : γ 2 (1) re representtions in homogeneous ryentri oordintes, then the Yiu produt is given y P Q = α 1 α 2 : β 1 β 2 : γ 1 γ 2 (2) whenever {α 1 α 2,β 1 β 2,γ 1 γ 2 } {0}. Cyril Prry [3] onstruts n nlogous produt using triliner oordintes. In view of the ppliility of oth the Yiu nd Prry produts, the nottion in equtions (1) nd (2) will represent generl homogeneous oordintes, s in [6, Chpter 1], unless otherwise noted. We lso define the quotient P/Q := α 1 β 2 γ 2 : β 1 γ 2 α 2 : γ 1 α 2 β 2 whenever Q/ {A, B, C}. Speiliztion of oordintes will e ommunited y phrses suh s those indited here: { ryentri triliner } multiplition produt division quotient. Pulition Dte: July 12, Communiting Editor: Floor vn Lmoen.

2 100 C. Kimerling If S is set of points, then P S := {P Q : Q S}. In prtiulr, if S is urve Γ, then P Γ nd Γ/P re urves, exept for degenerte ses, suh s when P {A, B, C}. In ll tht follows, suppose P = p : q : r is point not on sideline of tringle ABC, so tht pqr 0, nd onsequently, U/P = u p : v q : w r for ll U = u : v : w. Exmple 1. If Γ is line lα + mβ + nγ =0, then P Γ is the line (l/p)α + (m/q)β +(n/r)γ =0nd Γ/P is the line plα + qmβ + rnγ =0. Given the line QR of points Q nd R, it is esy to hek tht P QR is the line of P Q nd P R. In prtiulr, P ABC = ABC, nd if T is evin tringle, then P T is evin tringle. 2. Conis nd Cuis Eh oni Γ in the plne of tringle ABC is given y n eqution of the form uα 2 + vβ 2 + wγ 2 +2fβγ +2gγα +2hαβ =0. (3) Tht P Γ is the oni (u/p 2 )α 2 +(v/q 2 )β 2 +(w/r 2 )γ 2 +2(f/qr)βγ +2(g/rp)γα +2(h/pq)αβ =0 (4) is ler, sine α : β : γ stifies (3) if nd only if pα : qβ : rγ stisfies (4). In the se of irumoni Γ given in generl form y f α + g β + h =0, (5) γ the produt P Γ is the irumoni pf α + qg β + rh γ =0. Thus, if X is the point suh tht X Γ is given irumoni u α + v β + w γ =0, then X = u f : v g : w h. Exmple 2. In triliners, the irumoni Γ in (5) is the isogonl trnsform of the line L given y fα+ gβ + hγ =0. The isogonl trnsform of P L is Γ/P. Exmple 3. Let U = u : v : w. The oni W (U) given in [1, p. 238] y u 2 α 2 + v 2 β 2 + w 2 γ 2 2vwβγ 2wuγα 2uvαβ =0 is insried in tringle ABC. The oni P W (U) given y (u/p) 2 α 2 +(v/q) 2 β 2 +(w/r) 2 γ 2 2(vw/qr)βγ 2(wu/rp)γα 2(uv/pq)αβ =0 is the insried oni W (U/P). In triliners, we strt with Γ = inirle, given y u = u(,, ) =( + ),v = u(,,),w = u(,, ), nd find 1 1 The onis in Exmple 3 re disussed in [1, p.238] s exmples of type denoted y W (Xi), inluding inirle = W (X 55), Steiner insried ellipse = W (X 6), Kiepert prol = W (X 512), nd Yff prol = W (X 647). A list of X i inluding triliners, ryentris, nd remrks is given in [2].

3 Multiplying nd dividing urves y points 101 Coni Triliner produt Bryentri produt Steiner insried ellipse X 9 Γ X 8 Γ Kiepert prol X 643 Γ X 645 Γ Yff prol X 644 Γ X 646 Γ Exmple 4. Here we omine notions from Exmples 1-3. The irumirle, Γ 0, my e regrded s speil irumoni, nd every irumoni hs the form P Γ 0. We sk for the lous of point P for whih the irumoni P Γ 0 is prol. As suh oni is the isogonl trnsform of line tngent to Γ 0, we egin with this sttement of the prolem: find P = p : q : r (triliners) for whih the line L given y p α + q β + r γ =0meets Γ 0, given y α + β + γ =0in extly one point. Eliminting γ leds to α r p q ± (p + q r) = 2 4pq. β 2p We write the disriminnt s Φ(p, q, r) = 2 p q r 2 2qr 2rp 2pq. In view of Exmple 3 nd [5, p.81], we onlude tht if W (X 6 ) denotes the Steiner insried ellipse, with triliner eqution Φ(α, β, γ) =0, then P Γ 0 is hyperol prol ellipse ording s P lies inside W (X 6 ) on W (X 6 ) outside W (X 6 ). (6) Returning to the se tht L is tngent to Γ 0, it is esy to hek tht the point of tngeny is (X 1 /P ) X 6. (See Exmple 7 for Cev onjugy, denoted y.) If the method used to otin sttement (6) is pplied to ryentri multiplition, then similr onlusion is rehed, in whih the role of W (X 6 ) is repled y the insried oni whose ryentri eqution is α 2 + β 2 + γ 2 2βγ 2γα 2αβ =0, tht is, the ellipse W (X 2 ). Exmple 5. Suppose points P nd Q re given in triliners: P = p : q : r, nd U = u : v : w. We shll find the lous of point X = α : β : γ suh tht P X lies on the line UX. This on-lying is equivlent to the determinnt eqution expressile s irumoni: u(q r) α u v w α β γ pα qβ rγ + v(r p) β + =0, w(p q) γ =0. (7) One my strt with the line X 1 P, form its isogonl trnsform Γ, nd then reognize (7) s U Γ. For exmple, in triliners, eqution (7) represents the hyperols of

4 102 C. Kimerling Kiepert, Jerek, nd Feuerh ording s (P, U) =(X 31,X 75 ), (X 6,X 48 ), nd (X 1,X 3 ); or, in ryentris, ording s (P, U) =(X 6,X 76 ), (X 1,X 3 ), nd (X 2,X 63 ). Exmple 6. Agin in triliners, let Γ e the self-isogonl ui Z(U) given in [1, p. 240] y uα(β 2 γ 2 )+vβ(γ 2 α 2 )+wγ(α 2 β 2 )=0. This is the lous of points X suh tht X, X 1 /X, nd U re olliner; the point U is lled the pivot of Z(U). The quotient Γ/P is the ui upα(q 2 β 2 r 2 γ 2 )+vqβ(r 2 γ 2 p 2 α 2 )+wrγ(p 2 α 2 q 2 β 2 )=0. Although Γ/P is not generlly self-isogonl, it is self-onjugte under the P 2 - isoonjugy defined (e.g., [4]) y X X 1 /(X P 2 ). Exmple 7. Let X P denote the X-Cev onjugte of P, defined in [1, p.57] for X = x : y : z nd P = p : q : r y X P = p( p x + q y + r z ):q( q y + r z + p x ):r( r z + p x + q y ). Assume tht X P. It is esy to hek tht the lous of point X for whih X P lies on the line XP is given y α p (β2 q 2 γ2 r 2 )+β q (γ2 r 2 α2 p 2 )+γ r (α2 p 2 β2 )=0. (8) q2 In triliners, eqution (8) represents the produt P Γ where Γ is the ui Z(X 1 ). The lous of X for whih P X lies on XP is lso the ui (8). 3. Brord Points nd Apollonin Cirles Here we disuss some speil properties of the tringle enters X 98 (the Trry point) nd X 110 (the fous of the Kiepert prol). X 98 is the point, other thn A, B, C, tht lies on oth the irumirle nd the Kiepert hyperol. Let ω e the Brord ngle, given y ot ω =ota +otb +otc. In triliners, X 98 = se(a + ω) :se(b + ω) :se(c + ω), X 110 = 2 2 : 2 2 : 2 2. Theorem. Bryentri division y X 110 rries the irumirle Γ 0 onto the Kiepert hyperol Γ. For every point P on Γ 0, the line joining P to the Trry point X 98 (viz., the tngent t X 98 if P = X 98 ) intersets Γ gin t P = P/X 110. Furthermore, P/X 110 lies on the Apollonin irle of P with respet to the two Brord points Ω 1 nd Ω 2 ; tht is P Ω 1 P Ω 2 = P Ω 1 P Ω 2. (9)

5 Multiplying nd dividing urves y points 103 X 98 B P P Ω 2 K O Ω 1 C A X 110 Figure 1 Proof. In ryentris, Γ 0 nd Γ re given y 2 α + 2 β + 2 γ =0 nd α β nd, lso in ryentris, γ =0, X 110 = 2 2 : 2 2 : 2 2 so tht Γ =Γ 0 /X 110. For the reminder of the proof, we use triliners. A prmetri representtion for Γ 0 is given y P = P (t) =(1 t) :t : t(t 1), (10) for <t<, nd the ryentri produt P/X 110 is given in triliners y (1 t) 2 2 : t 2 2 : t(t 1) 2 2. Tht this point lies on line PX 98 is equivlent to the following esily verified identity:

6 104 C. Kimerling (1 t) 2 2 t 2 2 t(t 1) 2 2 (1 t) t t(t 1) se(a + ω) se(b + ω) se(c + ω) =0. We turn now to formul [1, p.31] for the distne etween two points expressed in normlized 2 triliners (α, β, γ) nd (α,β,γ ): 1 [ os A(α α 2σ ) 2 + os B(β β ) 2 + os C(γ γ ) 2 ], (11) where σ denotes the re of tringle ABC. Let D = 2 t 2 ( )t + 2, S = Normlized triliners for (10) nd the two Brord points follow: P = ((1 t)h, th, t(t 1)h), where h = 2σ D, nd ( h1 Ω 1 =, h 1, h ) ( 1 h1, Ω 2 =, h 1, h ) 1, where nd h 1 = 2σ S. Arevite os A, os B, os C, nd 1 t s,,, nd t respetively, nd write ( t E = ) h h 1 2 ( ) th + h1 2 ( tt + ) h h 1 2, (12) ( t F = ) h h 1 2 ( ) th + h1 2 ( tt + ) h h 1 2. (13) Eqution (11) then gives P Ω 1 2 P Ω 2 2 = E F. (14) In (12) nd (13), reple os A y ( )/2, nd similrly for os B nd os C, otining from (14) the following: P Ω 1 2 P Ω 2 2 = t2 2 t( )+ 2 t 2 2 t( ) Sometimes triliner oordintes re lled norml oordintes. We prefer triliners, so tht we n sy normlized triliners, not normlized normls. One might sy tht the ltter doule usge of norml n e voided y sying tul norml distnes, ut this would e unsuitle for normliztion of points t infinity. Another reson for retining triliner nd qudriplnr not repling oth with norml is tht these two terms distinguish etween lines nd plnes s the ojets with respet to whih norml distnes re defined. In disussing points reltive to tetrhedron, for exmple, one ould hve oth triliners nd qudriplnrs in the sme sentene.

7 Multiplying nd dividing urves y points 105 Note tht if the numertor in the lst frtion is written s f(t,,, ), then the denomintor is t 2 f( 1 t,,,). Similrly, P Ω 1 2 g(t,,, ) P = Ω 2 2 t 4 g( 1 t,,,), where g(t,,, ) =t 4 e 4 + t 3 e 3 + t 2 e 2 + te 1 + e 0, nd e 4 = 4 2 ( 2 2 ) 2, e 3 = 2 ( 2 2 )( ), e 2 = 2 2 ( 2 2 ) ( 2 2 ) ( ) ( ), e 1 = 2 ( 2 2 )( ), e 0 = 4 2 ( 2 2 ) 2. One my now verify diretly, using omputer lger system, or mnully with plenty of pper, tht t 2 f(t,,, )g( 1 t,,,)=f(1,,,)g(t,,, ), t whih is equivlent to the required eqution (9). Referenes [1] C. Kimerling, Tringle Centers nd Centrl Tringles, Congressus Numerntium, 129 (1998) [2] C. Kimerling, Enylopedi of Tringle Centers, k6/enylopedi/. [3] C. Kimerling nd C. Prry, Produts, squre roots, nd lyers in tringle geometry, Mthemtis nd Informtis Qurterly, 10 (2000) [4] C. Kimerling, Conjugies in the plne of tringle, Aequtiones Mthemtie, 61 (2001) forthoming. [5] S. L. Loney, The Elements of Coordinte Geometry, Prt II: Triliner Coordintes, Et., Mmilln, London, [6] E. A. Mxwell, The Methods of Plne Projetive Geometry Bsed on the Use of Generl Homogeneous Coordintes, Cmridge University Press, [7] P. Yiu, The uses of homogeneous ryentri oordintes in plne euliden geometry, Int. J. Mth. Edu. Si. Tehnol., 31 (2000) Clrk Kimerling: Deprtment of Mthemtis, University of Evnsville, 1800 Linoln Avenue, Evnsville, Indin 47722, USA E-mil ddress: k6@evnsville.edu