The Metric Characterization ofthegeneralized FermatPoints
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1 Global Journal of Science Fronier Research Mahemaics and Decision Sciences Volume Issue 0 Version 0 Year 0 Type : Double Blind Peer Reviewed Inernaional Research Journal Publisher: Global Journals Inc (US Online ISSN: & Prin ISSN: The Meric Characerizaion of he Generalized Ferma Poins By Sándor N Kiss bsrac- We build on he sides of a riangle BC, ouwards and inwards, similar riangles wih one anoher We prove some relaions referring o he generalized Ferma poins We deduce hese resuls based on rigonomerical ideniies GJSFR-F Classificaion : MSC 00: J8, D4 The Meric Characerizaion ofhegeneralized FermaPoins Sricly as per he compliance and regulaions of : 0 Sándor N Kiss This is a research/review paper, disribued under he erms of he Creaive Commons ribuion- Noncommercial 0 Unpored License hp://creaivecommonsorg/licenses/by-nc/0/, permiing all non commercial use, disribuion, and reproducion in any medium, provided he original work is properly cied
![R ef The Meric Characerizaion of he Generalized Ferma Poins 0 [] H Mowaffaq, n dvanced Calculus pproach o Finding he Ferma Poin, Mah Mag 67, 9-4, 994 Sándor N Kiss bsrac- We](/docs-images/66/55279378/images/2-0.jpg "build on he sides of a riangle BC, ouwards and inwards, similar riangles wih one anoher We prove some relaions referring o he generalized Ferma poins We deduce hese resuls")
2 R ef The Meric Characerizaion of he Generalized Ferma Poins 0 [] H Mowaffaq, n dvanced Calculus pproach o Finding he Ferma Poin, Mah Mag 67, 9-4, 994 Sándor N Kiss bsrac- We build on he sides of a riangle BC, ouwards and inwards, similar riangles wih one anoher We prove some relaions referring o he generalized Ferma poins We deduce hese resuls based on rigonomerical ideniies I Inroducion If we build ouwards on he sides BC, C, B of an arbirary riangle BC hree equilaeral riangles BCX, CY, BZ hen X BY CZ ± F± BF± CF, ( where F X BY CZ is he firs (ouward Ferma poin of riangle BC (Figure minus sign being aken if he angle of riangle BC a ha verex exceeds 0 If we build inwards on he sides BC, C, B of an arbirary riangle BC hree equilaeral riangles BCX, CY, BZ hen X BY CZ ± F ± BF ± CF, ( where F X BY CZ is he second (inward Ferma poin of riangle BC (Figure minus sign being aken a each verex where he angle of riangle BC is larger han 60 In his paper we generalize hese resuls: on he sides of riangle BC we build, ouwards resp inwards, hree similar riangles We menion bu in his aricle we don discuss he Ferma-Seiner problem: Given a riangle BC, how can we find a poin P in is plane for which PPBPC is minimal? The soluion and some generalizaion of his problem was a deal for many mahemaicians (see [], [4], [5] The hisorical survey of he Ferma-Seiner problem could be found in [] Le be given a riangle BC wih he usual noaions:, B, C he angles, a, b, c he sides, R he circumradius, he area of he riangle BC, S he wice of he area of he riangle ( S Global Journal of Science Fronier Research F Volume XIII Issue X V ersion I Year 9 Sándor N Kiss: Consanin Brâncuşi Technology Lyceum, Sau Mare, Romania dsandorkiss@gmailcom 0 Global Journals Inc (US
3 0 Global Journal of Science Fronier Research F Volume XIII Issue X V ersion I Year 40 BF CE, CBD BF, CE BCD (Figure Since B C π, herefore BDC, CE, FB We work wih barycenric coordinaes wih reference o he riangle BC ccording o he Conway s formulas, he barycenric coordinaes of he poins D, E, F are: ( D a : S SC : S SB, E ( S SC : b : S S, F ( S SB : S S: c where S θ II, S cgθ Consequenly S bc cos, S ca cos B, S ab cosc The equaions of he lines D, BE, CF are: ( ( 0 B S S y S S z, B ( S S x ( S S z 0, ( S S x ( S S y 0 (( B( C :( C( :( ( B K S S S S S S S S S S S S : : S S S SB S S C We inroduce he following noaions: ( BC,,,,, λ λ 0 The Firs Generalized Ferma Poin of he Triangle We build on he sides of he riangle BC ouwards he similar riangles BCD, CE and BF in ha way, ha Swmming random by wo of he above equaions, we ge he hird equaion, he lines D, BE and CF mee each oher in a poin K D BE CF This poin K will be called he firs generalized Ferma poin of he riangle The barycenric coordinaes of he poin K are: ( sin sin B sin C cgsin cgb sin cgc sin sin sin sin ( S sin sin sin sin sin sin SB SC S (,,, BC,, λ λ ( cg cg B cg C B C sinsin sin sin sin sin sin sin sin sinsin sin sinsin sin ( as bs cs S S ( a cg b cg c cg S C C C B We will proove ha λ0 λ: ( cg cg B cg C B C ( sinsin sin sin sin sin sin sin sin sin sin B sin C cgsin cgb sin cgc sin sin sin sin 0 Global Journals Inc (US
4 ( ( ( C ( B C ( ( cos cos cos cos sin cos cos coscos sin B cos cos coscos sin cos cos cos cos sin B B C C C B cos cos cos cos sin cos cos cos cos sin ( ( ( B C sin cos sin Bcos sin Ccos coscos cos sin sin B sin C sin cos sin cos B sin cos C cos cos BcosC sin sin sin cos cos cos cos cos cos ( B C B C( cos cos cos cos cos cos cos cos cos cos cos cos 0 cos cos cos cos cos B cosc cos cos cos cos cos B cosc 0, and his is rue a The lengh of he segmens D, BE and CF In absolue barycenric coordinaes he disance beween wo poins P ( xyz,, and P ( x, y, z cossin sin sin sincos sin sin B sinsin cos sin C cos sin Bsin Csin sin cos Bsin Csin sin sin BcosCsin B C is given by ( ( ( PP x x S y y S z z S In order o calculae he lengh of he segmens D, BE and CF we inroduce he following noaions: cos cos S Ssin S S Scg Scg S sin (, sin sin sin sin sin sin S sin S S sin sin, S sin S S sinsin The sum of he coordinaes of he poin D, E resp F is, resp The lengh of he segmen D is: D ( a S ( S S S ( S S S C B B C ( a ( as S S SS SS BC B C ( a S S S S S S B C sin cos cos ( a S SS SS B SS C sin sin sin sin a sin sin Ssinsin sin Ssin SBsin cos SC cos sin sin Ssin SBsin SCsin Ssinsin sin sin 0 Global Journal of Science Fronier Research F Volume XIII Issue X V ersion I Year 4 0 Global Journals Inc (US
5 0 Global Journal of Science Fronier Research F Volume XIII Issue X V ersion I Year 4 For his reason: sin sin D ( a cg b cg c cg S, ( sin sin sin BE ( a cg b cg c cg S, (4 sin sinsin CF ( a cg b cg c cg S (5 sin Based on he formulas (, (4 and (5: ( D sin BE sin CF sin sinsin sin a cg b cg c cg S (6 b The disance of he firs generalized Ferma poin from he verices of he reference riangle In wha follows we calculae he lengs of he segmens K, BK and CK The sum of coordinaes of he poin K is: Therefore 0 ( S SB( S SC ( S SC( S S ( S S( S SB Φ ( ( ( S S S S S S S S S S S S S S S BC C B B C C B The lengh of he segmen K is: as bs cs S sin sin sin sin sin sin λ ( as bs cs S Φ S S K ( S S ( S S Φ S ( S S ( S S S ( S S ( S S S Φ S S B C C B B C ( ( ( SB SC S S S S SC SB S SB SC S Φ S ( ( ( a S S SC SB S SB SC Φ ( a cg b cg c cg S R λ R λ sinsin sin sin sin sin S S S S R λ sin S D S S sin Φ Φ sin R λ R λsin ( S bc sin ( sin ( a λ R λ Therefore: bc K sin (, R λ ca sin, (8 R λ (7 BK ( B 0 Global Journals Inc (US
6 ab CK sin ( C (9 R λ c Relaions referring o he segmens K, BK and CK Now we will demonsrae he following condiional ideniy: ( casin sin( B absin sin( C λ bc sin sin (0 Indeed bcsinsin ( ca sin sin ( B absin sin ( C sin Bsin Csin( sin cos sincos sin Csin sin ( sin Bcos sin cos B sin sin Bsin sin Ccos sin cosc ( ( sin sin B sin C cgsin cgb sin cgc sin sin sin sin ( sin cgbsin cgc sin sin sin sin λ sin sin Bsin C cg Since B C π, from inequaliies >π, B >π, C >π only one can be rue If <π, B <π, C <π, hen he K is an inside poin of he riangle BC and based on he (0 we have Dsin BE sin CF sin R λ K sin BK sin CK sin ( If for insance >π, hen Dsin BE sin CF sin R λ K sin BK sin CK sin ( III The Second Generalized Ferma Poin of he Triangle We build on he sides of he riangle BC inwards he similar riangles BCL, CM and BN in ha way, ha BN CM, CBL BN, CM BCL (Figure 4 Since B C π, herefore BLC, CM, NB ccording o he Conway s formulas, he barycenric coordinaes of he poins L, M, N are: ( : C : B, M ( S SC : b : S S, N ( S SB : S S: c L a S S S S The equaions of he lines L, BM, CN are: ( ( 0 B S S y S S z, ( S S x ( S S z 0, ( S S x ( S S y 0 Since adding whaever wo equaion, we ge he hird equaion, he lines L, BM and CN mees each oher in a poin T L BM CN This poin T will be called he second generalized Ferma poin of he riangle The barycenric coordinaes of he poin T are: (( B( C :( C( :( ( B T S S S S S S S S S S S S : : S S S SB S S C C C B 0 Global Journal of Science Fronier Research F Volume XIII Issue X V ersion I Year 4 0 Global Journals Inc (US
7 0 Global Journal of Science Fronier Research F Volume XIII Issue X V ersion I Year 44 We inroduce he following noaions: ( ( S sin sin sin SB SC Ssinsin sin µ µ,,, BC,, 0 ( ( cg cg B cg C B C sinsin sin sin sin sin sin sin sin sinsin sin sinsin sin ( as bs cs S S ( a cg b cg c cg S In his case he equaliy µ 0 µ is valid oo a The lengh of he segmens L, BM and CN The sum of he coordinaes of he poin L, M resp N are, resp The lengh of he segmen L is: L ( a S ( S S S ( S S S C B B C ( a ( as S S SS SS BC B C ( a S S S S S S B C sin cos cos ( a S SS SS B SS C sin sin sin sin a sin sin Ssinsin sin Ssin SBsin cos SC cos sin sin Ssin SBsin SCsin Ssinsin sin sin R µ R µ sinsin sin ( a cg b cg c cg S 0 sin sin sin For his reason: ( BC,,,,, µ µ sin sin B sin C cgsin cgb sin cgc sin sin sin sin sin sin L ( a cg b cg c cg S, ( sin sin sin BM ( a cg b cg c cg S, (4 sin sinsin CN ( a cg b cg c cg S (5 sin 0 Global Journals Inc (US
8 Based on he (, (4 and (5 we obain: ( L sin BM sin CN sin sinsin sin a cg b cg c cg S (6 b The disance of he second generalized Ferma poin from he verices of he reference riangle The sum of coordinaes of he poin T is: Therefore: ( S SB( S SC ( S SC( S S ( S S( S SB Γ ( ( ( S S S S S S S S S S S S S S S BC C B B C C B ( as bs cs S as bs cs S sin sin sin sin sin sin µ ( as bs cs S Γ S S The lengh of he segmen T is: T ( S S ( S S Γ S ( S S ( S S S ( S S ( S S S Γ Therefore: S S B C C B B C ( ( ( S S SB SC S S SC SB S SB SC Γ S S ( ( ( a S S SC SB S SB SC Γ S S S S R µ sin S L S S sin Γ Γ sin R µ R µ sin S bc bc sin ( sin ( sin a µ R µ R µ ( bc ca T sin, (7 BT sin B, (8 R µ R µ ab CT sin C (9 R µ 0 Global Journal of Science Fronier Research F Volume XIII Issue X V ersion I Year 45 c Relaions referring o he segmens T, BT and CT Here is valid oo he following condiional ideniy: bcsin sin ( casin sin( B absin sin( C µ (0 Since B C 0, only wo of he >, B >, C > inequaliies 0 Global Journals Inc (US
9 0 Global Journal of Science Fronier Research F Volume XIII Issue IX V ersion I Year 46 Or if <, B <, C >, hen Lsin BM sin CN sin R µ T sin BT sin CT sin ( IV In his secion we use he following relaions: The proof of (: abc ( cg cgb cgc Speciale Case ( cg cgb cgc R( a b c abc, ( ( sin sin B sin C cg cgb cgc sin sin Bsin C cos cos Bcos C, ( sin sin B sin C cg cgb cgc sin sin Bsin C cos cos Bcos C (4 (5 ( bccos cacos B abcosc R( b c a c a b a b c R R( a b c The proof of (4 and (5: sin sin Bsin C cg ± The Meric Characerizaion of he Generalized Ferma Poins can be rue a he same ime For insance if <, B >, C >, hen based on he formula (0 Lsin BM sin CN sin R µ T sin BT sin CT sin ( cos cos B cosc abc sin sin B sin C ( cgb cgc ± cos sin B sin C cos B sin C sin cosc sin sin B ± sin sin B sin C cos cos B cosc cos sin sin Bsin C sin cos cos B cosc Rabc a b c cos B cosc ± sin B sin C sin sin Bsin C If π, hen K F and T F, where F resp F is he firs resp he second Ferma s poin of he riangle BC ( X resp X4 in [] In his case ( π π π λ λ, B, C,,, sin sin B sinc cg cgb cgc 4 ( a b c S ( sin sin Bsin C cos cos Bcos C, R 4 π π π µ µ, B, C,,, sin sin B sinc ( cg cgb cgc 4 ( a b c S ( sin sin Bsin C cos cos Bcos C R 4 0 Global Journals Inc (US
10 Based on (6 we have a b c S X BY CZ λ (6 ccording o (7, (8 and (9 bc π sin, ca π F (7 BF sin, R λ B (8 R λ If < π, B < π, C < π, hen If for insance > π, hen Based on (6 we have ab π CF sin C (9 R λ X BY CZ λ F BF CF (0 X BY CZ λ F BF CF ( X BY CZ µ ccording o (7, (8 and (9 a b c S ( bc π ca π F sin, ( BF sin B, (4 R µ R µ If for insance < π, B > π, C > π, hen Or if < π, B < π, C > π, hen ab π CF sin C (5 R µ X BY CZ µ F BF CF (6 X BY CZ µ F BF CF (7 0 Global Journal of Science Fronier Research F Volume XIII Issue X V ersion I Year 47 0 Global Journals Inc (US
11 Oher arrangemens of he riangles BCD, CE and BF are possible To invesigae he valabiliies of he above meric relaions in his cases is a possible subjec for furher researches 0 Global Journal of Science Fronier Research F Volume XIII Issue X V ersion I Year 48 References Références Referencias [] C Kimberling, Triangle Ceners and Cenral Triangles, Congr Numer 9, -95, 998 [] H Mowaffaq, n dvanced Calculus pproach o Finding he Ferma Poin, Mah Mag 67, 9-4, 994 [] Shay Gueron and Ran Tessler, The Ferma-Seiner Problem, Monhly 09, 44-45, 00 [4] P G Spain, The Ferma Poin of a Triangle, Mah Mag 69, -, 996 [5] J Tong and Y S Chua, The Generalized Ferma s Poin, Mah Mag 68, 4-5, 995 The Meric Characerizaion of he Generalized Ferma Poins Figure 0 Global Journals Inc (US
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