Nine Solved and Nine Open Problems in Elementary Geometry
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1 Ne Solved ad Ne Ope Problems Elemetary Geometry Floret Smaradache Math & Scece Departmet Uversty of New Mexco, Gallup, US I ths paper we revew e prevous proposed ad solved problems of elemetary D geometry [4] ad [6], ad we exted them ether from tragles to polygos or polyhedros,or from crcles to spheres (from D-space 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 F( P) 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 C' B >, 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 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 EP ( ) ' ' ' P P P P P P P' P' P ' F( P) 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 the same rato k > 0, 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 : B BC ( k ) + C BC k 3 BC BC ( k ) k where ( k) B + k C ( k) k BC smlarly, 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 If the pots,,, dvde the sdes, 3,, of a polygo the same rato k>0, determe the mmum of the expresso: ' + ' '. 4.. Smlarly questo f the pots,,, dvde the sdes, 3,, the postve ratos k, k,, k respectvely Geeralze ths problem for polyhedros. 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. Cocluso : If we ote the legths of the sdes of the tragle Δ BC by B c, BC a, C b, ad the legths of the sdes of ts orthc tragle Δ `B`C` by `B` c`, B`C` a`, C`` b`, the we proved that: 4(a`b` + b`c` + c`a`) a + b + c. Usolved Problems Geeralze ths problem to polygos. Let m be a polygo ad P a pot sde t. From P, whch s called a pedal pot, we draw perpedculars o each sde + of the polygo ad we ote by the tersecto betwee the perpedcular ad the sde +. Let s exted the defto of pedal tragle to a pedal polygo a straght way:.e. the polygo formed by the orthogoal projectos of a pedal pot o the sdes of the polygo. The pedal polygo m s formed. What propertes does ths pedal polygo have? 7.. Geeralze ths problem to polyhedros. Let be a polyhedro ad P a pot sde t. From P we draw perpedculars o each edge j of the polyhedro ad we ote by j the tersecto betwee the perpedcular ad the sde j. Let s ame the Ths s called the Smaradache s Orthc Theorem (see [], [3]).
9 ew formed polyhedro a edge pedal polyhedro,. What propertes does ths edge pedal polyhedro have? 7.3. Geeralze ths problem to polyhedros a dfferet way. Let be a polyhedro ad P a pot sde t. From P we draw perpedculars o each polyhedro face F ad we ote by the tersecto betwee the perpedcular ad the sde F. Let s call the ew formed polyhedro a face pedal polyhedro, whch s p, where p s the umber of polyhedro s faces. What propertes does ths face pedal polyhedro have? ad of ray R. Problem 8. Gve the dstct pots,..., o the crcumferece of a crcle wth the ceter O Prove that there exst two pots, such that j 80 O + Oj R cos o Soluto 8. Because O + O O + O 360 o ad {,,..., } 3, 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
10 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.) Problem 9: Through oe of the tersectg pots of two crcles we draw a le that tersects a secod tme the crcles the pots M ad M respectvely. The the geometrc locus of the pot M whch dvdes the segmet MM a rato k (.e. M M k MM ) s the crcle of ceter O (where O s the pot that dvdes the segmet of le that coects the two crcle ceters O ad respectvely O to the rato k,.e. OO k OO) ad radus O, wthout the pots ad B. Proof Let OE MM adof MM. Let O OO such that OO k OO ad M MM, where MM k MM. G F M M E O O O B Fg.. We costruct OG MM ad we make the otatos: ME E x ad F FM y. The, G GM, because k ( ) x + G EG E x + y x ky k + k + ad k x+ ky x+ ky GM MM M G ( x + y) x. k + k + k + Therefore we also have OM O. The geometrc locus s a crcle of ceter O ad radus O, wthout the pots ad B (the red crcle Fg. - called Smaradache s Crcle). Coversely.
11 , the le M tersects the two crcles M ad M respectvely. We cosder the projectos of the pots O, O, O o the le MM E, FG, respectvely. Because OO k OO t results that EG k GF. Makg the otatos: ME E x ad F FM y we obta that If M ( GO, O) \{, B} MM M+ M M+ G x+ ( EG E) k k k x + ( x+ y) x ( x+ y) MM. k + k + k + For k we fd the Problem IV from [5]. Ope Problem The same problem f stead of two crcles oe cosders two ellpses, or oe ellpse ad oe crcle. 9..The same problem 3D, cosderg stead of two crcles two spheres (ther surfaces) whose tersecto s a crcle C. Drawg a le passg through the crcumferece of C, t wll tersect the two sphercal surfaces other two pots M ad respectvely M. Cojecture: The geometrc locus of the pot M whch dvdes the segmet MM a rato k (.e. M M k MM ) cludes the sphercal surface of ceter O (where O s the pot that dvdes the segmet of le that coects the two sphere ceters O ad respectvely O to the rato k,.e. OO k OO) ad radus O, wthout the tersecto crcle C. partal proof s ths: f the le MM whch tersect the two spheres s the same plae as the le O O the the 3D problem s reduce to a D problem ad the locus s a crcle of radus O ad ceter O defed as the orgal problem, where the pot belogs to the crcumferece of C (except two pots). If we cosder all such cases (ftely may actually), we get a sphere of radus O (from whch we exclude the tersecto crcle C ) ad cetered O ( ca be ay pot o the crcumferece of tersecto crcle C ). The locus has to be vestgated for the case whe M M ad O O are dfferet plaes. 9.3.What about f stead of two spheres we have two ellpsods, or a sphere ad a ellpsod? Refereces: [] Cătăl Barbu, Teorema lu Smaradache, hs book Teoreme fudametale d geometra trughulu, Chapter II, Secto II.57, p. 337, Edtura Uque, Bacău, 008. [] József Sádor, Geometrc Theorems, Dophate Equatos, ad rthmetc Fuctos, R Press, pp. 9-0, Rehoboth 00.
12 [3] F. Smaradache, Ne Solved ad Ne Ope Problems Elemetary Geometry, arxv.org at [4] F. Smaradache, Problèmes avec et sas problèmes!, pp. 49 & 54-60, Sompress, Fés, Morocco, 983. [5] The dmsso Test at the Polytechc Isttute, Problem IV, 987, Romaa. [6] Floret Smaradache, Proposed Problems of Mathematcs (Vol. II), Uversty of Kshev Press, Kshev, Problem 58, pp , 997.
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