GENERALIZATIONS OF CEVA S THEOREM AND APPLICATIONS

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1 GENERLIZTIONS OF CEV S THEOREM ND PPLICTIONS Floret Smaradache Uversty of New Mexco 200 College Road Gallup, NM 87301, US E-mal: smarad@um.edu I these paragraphs oe presets three geeralzatos of the famous theorem of Ceva, whch states: If a tragle BC oe draws the cocurret straght les 1, BB 1, CC 1, the 1 B 1 C B 1C B 1 C 1 C 1 B 1. Theorem 1: Let us have the polygo , a pot M ts plae, ad a crcular permutato p Oe otes M j the tersectos of the le M wth the les + s + s+1,..., + s+t 1 + s+t (for all ad j, j { + s,..., + s + t 1}). M for all respectve dces, ad f 2s+ t, oe has: If j, + s+ t 1 Mj j ( 1) ( s ad t are atural o-zero umbers). M ( j), j 1, + s j p alytcal proof: Let M be a pot the pla of the tragle BC, such that t satsfes the codtos of the theorem. Oe chooses a Cartesa system of axes, such that the two parallels wth the axes whch pass through M do ot pass by ay pot (ths s possble). Oe cosders M (a,b), where a ad b are real varables, ad (X,Y ) where X ad Y are kow, { 1,2,...,}. The former choces esure us the followg relatos: X a 0 ad Y b 0 for all { 1,2,...,}. The equato of the le M (1 ) s: x a X a y b Y b 0. Oe otes that d(x, y; X,Y ) 0. Oe has Mj j δ ( j, M ) d( X j, Yj; X, Y ) D( j, ) M δ (, M) d( X, Y ; X, Y) D( p( j), ) j p( j) p( j) p( j) p( j) 1

2 where δ(,st ) s the dstace from to the le ST, ad where oe otes wth D(a,b) for d(x a,y a ; X b,y b ). Let s calculate the product, where we wll use the followg coveto: a + b wll mea p(p(...p (a)...)), ad a b wll mea p 1 (p 1 (...p (a)...)) b tmes M j j j + s M j j +1 + s+t 1 D( j,) D( j + 1,) + s+t 1 j + s D ( + s, ) D ( + s+ 1, ) D ( + s+ t 1, ) D ( + s+ 1,) D ( + s+ 2,) D ( + s+ t,) 1 D( + s,) D( + s,) D( + s + t,) D( s,) b tmes The tal product s equal to: D( + s,) D( s,) D(1 + s,1) D(2 + s,2) D(2s,s) D(1 s,1) D(2 s,2) D(,s) D(2s + 1, s + 1) D(1, s + 1) D(2s + 2, s + 2) D(2,s + 2) D(2s + t,s + t) D(t,s + t) D(2s + t + 1, s + t + 1) D(t + 1, s + t + 1) D(2s + t + 2, s + t + 2) D(2s + t + s,s + t + s) D(t + 2, s + t + 2) D(t + s,s + t + s) D(1 + s,1) D(1,1 + s) D(2 + s,2) D(2,2 + s) D( + s,) D(, + s) 1 1 D(2s + t,s + t) D(s,) D(s + t,2s + t) D(,s) P( + s) P() ( 1) because: Xr a Yr b Dr (, p) X p a Yp b ( Xr a)( Yr b) Pr (), D( p, r) X p a Yp b ( X p a)( Yp b) P( p) X a Y b r r the last equalty resultg from what oe otes: (X t a)(y t b) P(t). From (1) t results that P(t) 0 for all t from { 1, 2,..., }. The proof s completed. Commets regardg Theorem 1: 2

3 t represets the umber of les of a polygo whch are tersected by a le 0 M ; f oe otes the sdes +1 of the polygo, by a, the s + 1 represets the order of the frst le tersected by the le 1 M (that s a s+1 the frst le tersected by 1 M ). Example: If s 5 ad t 3, the theorem says that : - the le 1 M tersects the sdes 6 7, 7 8, the le 2 M tersects the sdes 7 8, 8 9, the le 3 M tersects the sdes 8 9, 9 10, 10 11, etc. Observato: The restrctve codto of the theorem s ecessary for the M j j exstece of the ratos. M j p( j) Cosequece 1.1: Let s have a polygo k +1 ad a pot M ts pla. For all from { 1,2,...,2k + 1}, oe otes M the tersecto of the le p() wth the le whch passes through M ad by the vertex whch s opposed to ths le. If M M {, p() } the oe has: 1. 1 M p() The demostrato results mmedately from the theorem, sce oe has s k ad t 1, that s 2k + 1. The recprocal of ths cosequece s ot true. From where t results mmedately that the recprocal of the theorem s ot true ether. Couterexample: Let us cosder a polygo of 5 sdes. Oe plottes the les 1 M 3, 2 M 4 ad 3 M 5 whch tersect M. Let us have K M 3 3 M 4 4 M 5 5 M 3 4 M 4 5 M 5 1 The oe plots the le 4 M 1 such that t does ot pass through M ad such that t forms the rato: (2) M 1 1 M 1 2 1/K or 2 / K. (Oe chooses oe of these values, for whch 4 M 1 does ot pass through M ). t the ed oe traces 5 M 2 whch forms the rato M or 1 M fucto of (2). Therefore the product: 5 M -1 wthout havg the respectve les cocurret. 1 M p() 3

4 Cosequece 1.2: Uder the codtos of the theorem, f for all ad j, j {, p 1 ()}, oe otes M j M j p( j ) ad M j j, p( j ) M j j ( 1)., j 1 M j p( j ) j {, p 1 ()} Effectvely oe has s 1, t 2, ad therefore 2s + t. { } the oe has: Cosequece 1.3: For 3, t comes s 1 ad t 1, therefore oe obtas (as a partcular case ) the theorem of Ceva. pplcato of the Geeralzatos of Ceva s Theorem s preseted below. Theorem 2: Let us cosder a polygo serted a crcle. Let s ad t be two o zero atural umbers such that 2s + t. By each vertex passes a le d whch tersects the les + s + s+1,..., + s+t 1 + s+t at the pots M,+ s,..., M + s+t 1 respectvely ad the crcle at the pot M. The oe has: M M + s+ t 1 j j + s 1 j + s Mj j+ 1 1 M + s+ t. Proof: Let be fxed. 1) The case where the pot M,+ s s sde the crcle. There are tragles M,+ s + s ad M M,+ s + s+1 whch are smlar, sce the agles M,+ s + s ad M,+ s + s+1 M equal. It results from t that: o oe sde, ad M,+ s + s ad + s+1 M,+ s M are (1) M,+ s M,+ s + s+1 + s M + s+1 4

5 +s+1 MMM M,+s M +s I a smlar maer, oe shows that the tragles M,+ s + s+1 ad M,+ s + s M are smlar, from whch: (2) M,+ s + s+1. Dvdg (1) by (2) we obta: M,+ s + s M + s (3) M,+ s + s M + s + s. M,+ s + s+1 M + s+1 + s+1 2) The case where M,+ s s exteror to the crcle s smlar to the frst, because the tragles (otatos as 1) are smlar also ths ew case. There are the same terpretatos ad the same ratos; therefore oe has also the relato (3). +s+1 +s M M,+s Let us calculate the product: 5

6 M j j j + s M j j +1 + s+t 1 M j j j + s M j +1 j +1 + s+t 1 M + s M + s+1 M M + s+1 M + s+ 2 M + s+t + s+t 1 + s + s+1 + s+1 + s+2 + s+t 1 + s+t Therefore the tal product s equal to: M + s + s 1 M + s+t + s+t sce: 1 M + s M + s+t M + s M + s+t + s + s+t 1 + s + s+t 1 1+ s 2 2+ s s 2s 1 1+ s+t 2 2+ s+t s+1 1 s+2 2s+2 s+2 2 s+t s+t t s+t +1 1 s+t +1 t +1 s+t s+t +2 t +2 s s+t 1 (by takg to accout the fact that 2s + t ). Cosequece 2.1: If there s a polygo 1 2,..., 2s 1 scrbed a crcle, ad from each vertex oe traces a le d whch tersects the opposte sde + s 1 + s M ad the crcle M the: M + s 1 M + s 1 1 M + s 1 M + s I fact for t 1, oe has odd ad s If oe makes s 1 ths cosequece, oe fds the mathematcal ote from [1], pages pplcato: If the theorem, the les d are cocurret, oe obtas: M + s ( 1). 1 M + s+t 6

7 Referece: [1] Da Barbla - Io Barbu Pag edte, Edtura lbatros, Bucharest, 1981 (Edţe îgrjtă de Gerda Barbla, V. Protopopescu, Vorel Gh. Vodă). 7