Eight Solved and Eight Open Problems in Elementary Geometry

Transcription

1 Eght Solved ad Eght Ope Problems Elemetary Geometry Floret Smaradache Math & Scece Departmet Uversty of New Mexco, Gallup, US I ths paper we revew eght prevous proposed ad solved problems of elemetary D geometry [], ad we exted them ether from tragle to polygos or from D to 3D-space ad make some commets about them Problem We draw the projectos M of a pot M o the sdes + of the polygo Prove that: M + + M M + + M + M Soluto It results that: From where: For all we have: + + MM M M M M + + M M M M ( M M + ) ( M M+ ) 0 Ope Problem If we cosder a 3D-space the projectos M of a pot M o the edges + of a polyhedro the what kd of relatoshp (smlarly to the above) ca we fd? But f we cosder a 3D-space the projectos M of a pot M o the faces F of a polyhedro wth k 4 faces, the what kd of relatoshp (smlarly to the above) ca we fd? Problem

2 Let s cosder a polygo (whch has at least 4 sdes) crcumscrbed to a crcle, ad D the set of ts dagoals ad the les jog the pots of cotact of two o-adjacet sdes The D cotas at least 3 cocurret les Soluto Let be the umber of sdes If 4, the the two dagoals ad the two les jog the pots of cotact of two adjacet sdes are cocurret (accordg to Newto's Theorem) The case > 4 s reduced to the prevous case: we cosder ay polygo (see the fgure) P + B j- O B B 4 j B 3 j+ - R crcumscrbed to the crcle ad we choose two vertces, j ( j) such that j j + P ad j j + R Let B, h h {,,3,4 } the cotact pots of the quadrlateral PjR wth the crcle of ceter O Because of the Newto s theorem, the les j, BB 3 ad BB 4 are cocurret Ope Problem I what codtos there are more tha three cocurret les? What s the maxmum umber of cocurret les that ca exst (ad what codtos)? 3 What about a alteratve of ths problem: to cosder stead of a crcle a ellpse, ad the a polygo ellpsoscrbed (let s vet ths word, ellpso-scrbed, meag a polygo whose all sdes are taget to a ellpse whch sde of t): how may cocurret les we ca fd amog ts dagoals ad the les coectg the pot of cotact of two o-adjacet sdes? 4 What about geeralzg ths problem a 3D-space: a sphere ad a polyhedro crcumscrbed to t? 5 Or stead of a sphere to cosder a ellpsod ad a polyhedro ellpsodo-scrbed to t? Of course, we ca go by costructo reversely: take a pot sde a crcle (smlarly for a ellpse, a sphere, or ellpsod), the draw secats passg through ths pot that tersect the

3 crcle (ellpse, sphere, ellpsod) to two pots, ad the draw tagets to the crcle (or ellpse), or taget plaes to the sphere or ellpsod) ad try to costruct a polygo (or polyhedro) from the tersectos of the taget les (or of taget plaes) f possble For example, a regular polygo (or polyhedro) has a hgher chace to have more cocurret such les I the 3D space, we may cosder, as alteratve to ths problem, the tersecto of plaes (stead of les) Problem 3 I a tragle BC let s cosder the Cevas ', BB ' ad CC ' that tersect P Calculate the mmum value of the expressos: P PB PC EP ( ) + + P' PB ' PC ' ad P PB PC FP ( ) P' PB' PC ' where ' [ BC], B ' [ C], C ' [ B] Soluto 3 We ll apply the theorem of Va ubel three tmes for the tragle BC, ad t results: P C ' B ' + P' C ' B B ' C PB B' BC ' + PB ' ' C C ' PC C' CB ' + PC ' ' B B ' If we add these three relatos ad we use the otato C ' x 0 CB ' >, B ' y 0 BC ' >, B' z 0 C ' > the we obta: EP ( ) x+ + x+ + z y y z The mmum value wll be obtaed whe x y z, therefore whe P wll be the gravtato ceter of the tragle Whe we multply the three relatos we obta

4 FP ( ) x+ x+ z+ 8 y y z Ope Problem 3 3 Istead of a tragle we may cosder a polygo ad the les,,, that tersect a pot P Calculate the mmum value of the expressos: P P P P' P' P ' EP ( ) P P P P' P' P ' FP ( ) 3 The let s geeralze the problem the 3D space, ad cosder the polyhedro ad the les,,, that tersect a pot P Smlarly, calculate the mmum of the expressos E(P) ad F(P) Problem 4 If the pots, B, C dvde the sdes BC, C respectvely B of a tragle a rapport k, determe the mmum of the followg expresso: + BB + CC Soluto 4 Suppose k > 0 because we work wth dstaces B k BC, CB k C, C k B We ll apply tree tmes Stewart s theorem the tragle BC, wth the segmets, BB, respectvely CC : where smlarly, ( ) ( ) 3 B BC k + C BC k BC BC k k ( ) ( ) k B + k C k k BC ( ) ( ) ( ) ( ) BB k BC + k B k k C CC k C + k CB k k B By addg these three equaltes we obta: ( )( ) + BB + CC k k+ B + BC + C,

5 whch takes the mmum value whe k, whch s the case whe the three les from the eoucemet are the medas of the tragle 3 The mmum s ( B BC C ) Ope Problem 4 4 If the pots,,, dvde the sdes, 3,, of a polygo a rapport k>0, determe the mmum of the expresso: ' + ' + + ' 4 Smlarly questo f the pots,,, dvde the sdes, 3,, the postve rapports k, k,, k respectvely Problem 5 I the tragle BC we draw the les, BB, CC such that B + BC + C B + BC + C I what codtos these three Cevas are cocurret? Partal Soluto 5 They are cocurret for example whe, B, C are the legs of the medas of the tragle BC Or, as Prof Io Pătrașcu remarked, whe they are the legs of the heghts a acute agle tragle BC More geeral The relato from the problem ca be wrtte also as: a( B C ) + b( BC C ) + c( C CB ) 0, where a, b, c are the sdes of the tragle We ll deote the three above terms as α, β, ad respectve γ, such that α + β + γ 0 α α a( B C ) B C C a where α a C a a a a a a + α a C a a a α C C a α a α C The

6 Smlarly: BC B b b B C a a + β ad β + α α I coformty wth Ceva s theorem, the three les from the problem are cocurret f ad oly f: B BC C ( a + α )( b + β)( c + γ) ( a α)( b β)( c γ) C B C B C CB c c + γ γ Usolved Problem 5 Geeralze ths problem for a polygo Problem 6 I a tragle we draw the Cevas, BB, CC that tersect P Prove that P PB PC B BC C P PB PC B B C C Soluto 6 I the tragle BC we apply the Ceva s theorem: C B CB B C BC () I the tragle B, cut by the trasversal CC, we ll apply the Meelaus theorem: C BC P P C BC () I the tragle BBC, cut by the trasversal, we apply aga the Meelaus theorem: C B P C B B C BP BP B C (3) We apply oe more tme the Meelaus theorem the tragle CC cut by the trasversal BB : B CPCB B CPCB (4) We dvde each relato (), (3), ad (4) by relato (), ad we obta:

7 P BC B (5) P B BC PB C CB (6) PB CB C PC B C (7) PC C B Multplyg (5) by (6) ad by (7), we have: P PB PC B BC C B BC C P PB PC B BC C B BC C but the last fracto s equal to coformty to Ceva s theorem Usolved Problem 6 Geeralze ths problem for a polygo? Problem 7 Gve a tragle BC whose agles are all acute (acute tragle), we cosder ' BC, ' ' the tragle formed by the legs of ts alttudes I whch codtos the expresso: ' B' B' C' + B' C' C' ' + C' ' ' B' s maxmum? b-y z B C c-z y C B x a-x We ote Soluto 7 We have It results that ΔBC~ ΔBC ' ' '~ ΔBC ' ~ Δ BC ' ' () B' x, CB' y, C' z

8 ' C a x, B' b y, C' B c z BC B ' ' C B' C '; BC B ' C ' ' B ' C '; BC BC ' ' B ' C ' From these equaltes t results the relato () C ' ' x ΔBC ' '~ ΔBC ' ' () a x ' B' C ' ' c z ΔBC ' ' ~ ΔBC ' ' (3) z B' C' BC ' ' b y ΔBC ' '~ ΔBC ' ' (4) y B ' ' From (), (3) ad (4) we observe that the sum of the products from the problem s equal to: a b c x( a x) + y( b y) + z( c z) ( a + b + c ) x y z 4 a b c whch wll reach ts maxmum as log as x, y, z, that s whe the alttudes legs are the mddle of the sdes, therefore whe the Δ BC s equlateral The maxmum of the expresso s ( ) 4 a + b + c Usolved Problem 7 Geeralze ths problem to polygos Let be a polygo ad P a pot sde t From P we draw perpedculars o each sde + of the polygo ad we ote by the tersecto betwee the perpedcular ad the sde + podare polygo s formed What propertes does ths podare polygoe have? Problem 8 Gve the dstct pots,, o the crcumferece of a crcle wth the ceter O ad of ray R o 80 Prove that there exst two pots, j such that O + Oj R cos Soluto 8 Because O + O + + O + O 360 o 3

9 ad {,,, }, O 0 o + >, t result that t exst at least oe agle (otherwse t follows that S > 360 o ) O j 360 o - O j M O + Oj OM O + Oj OM The quadrlateral OMjs a rhombus Whe α s smaller, OM 360 o s greater s α o α 80 results that: OM R cos R cos, t Ope Problem 8: Is t possble to fd a smlar relatoshp a ellpse? (Of course, stead of the crcle s radus R oe should cosder the ellpse s axes a ad b) Refereces: [] F Smaradache, Problèmes avec et sas problèmes!, Sompress, Fés, Morocoo, 983 [] Cătăl Barbu, Teorema lu Smaradache, hs book Teoreme fudametale d geometra trughulu, Chapter II (Teoreme fudametale d geometra trughulu), Secto II57, p 338, Edtura Uque, Bacău, 008