Mathematical Reflections, Issue 5, INEQUALITIES ON RATIOS OF RADII OF TANGENT CIRCLES. Y.N. Aliyev
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1 themtil efletions, Issue 5, 015 INEQULITIES ON TIOS OF DII OF TNGENT ILES YN liev
2 stt Some inequlities involving tios of dii of intenll tngent iles whih inteset the given line in fied points e studied onsideing speil nd degenete ses of the onstution, some supising inequlities e otined nlogous esults fo etenll tngent iles e lso disussed Intodution If,,, nd D e fied points on line in the given ode then the lous of points not on line D fo whih = D is ile whose dimete lies on the line D Figue 0 This ile, whih is sometimes lled pollonius ile, is lso inteesting fo the othe eson Note tht the iumsied iles of the tingles D nd e tngent if lies on the pollonius ile If we fi the iumsied ile of D nd move point long this ile then the tio / D deeses So in etin sense pollonius ile is the lous of points fo whih the tio / D is miml On the othe hnd it would e inteesting to find miml nd miniml vlues of the tio / D if is on the pollonius ile We hve not sueeded in solving the lst polem ompletel Fo moe infomtion on the histo of the question see [1] ut ou investigtions led to some inteesting inequlities whih we olleted in the pesent ppe D Figue 0 1 Theoem Let,,, nd D e points on line k in this ode, nd e point not on k suh tht = D Then
3 sin sin D D D Figue 1 Poof We shll fist pove tht iumsied iles of tingles D nd with dii nd espetivel, e tngent t point Let O 1 nd O e iumentes of tingles D nd Dop pependiul H to line D It is es to show tht H= O 1 nd H= O sutting we otin = D + O 1 O Note tht = D It follows tht points, O 1 nd O e olline O O 1 H D Figue
4 Theefoe these iles e tngent t point We see tht > sine theoem sin sin D D D Theoem Let,,, nd D e points on line k in this ode, nd e point not on k suh tht = D Then D sin D D sin D D Poof Suppose fist tht Then ommon tngent line of iles t point inteset line D t point E Denote =, D =, =, E = nd E = It follows tht nd O O 1 E D Figue 3 sutting o dividing we otin Putting this in one of the pevious equlities gives Denote =μ Dop the pependiuls O 1 K nd O L to line D We otin EH os,
5 KD K, L L, os HL, os HK onsequentl, os os Sine, ftion / deeses s ngle μ ineses fom 0 to π Theefoe D E O O 1 H L K Figue 4 It emins onl to simplif the epessions on oth sides of this doule inequlit
6 multipling the numeto with its onjugte nd then dividing with the sme onjugte we otin Finll, sin sin D
7 The se is nlogous Fo the se one must pss to the limit in the lst doule inequlit tending Note The following polem is open: Pove tht D poving this inequlit the following hin of inequlities will e ompleted [1]: sin D sin D 3 ooll Let nd < e dii of two iles whih e tngent t point hod D of gete ile intesets the othe ile t points nd Then D D D D O D O 1 Figue 5 Note In the nottions of pevious polems this inequlit n lso e witten s
8 4 ooll [3] Let nd < e dii of two iles whih e tngent t point hod of gete ile is tngent to the othe ile t point Then Note This follows fom pevious polem simpl put =0 It is inteesting tht the ove inequlit gives uppe ound s tio of ithmeti men nd geometi men of two segments nd Lowe ound is of ouse 1 O O 1 Figue 6 5 ooll hod of ile w 1 psses though midpoint of hod DE of the sme ile w 1 ile w is tngent to line t point nd ile w 1 t point Line intesets w 1 t point F Then F DE
9 E w 1 w D F Figue 7 6 ooll [] Two iles w 1 nd w inteseting t points nd e tngent to ile w intenll t points nd N, espetivel Line intesets ile w t points nd D Let 1 nd e dii of iles w 1 nd w, espetivel Then D D 1 1 w 1 D w w N Figue 8 Poof Let dius of ile w e the inequlit in Eeise 3,
10 , D 1 D D D ultipling we otin the equied inequlit Note epling 1 with nd vie ves we n lso otin the uppe ound fo 1 : 1 D D So in ft the following doule inequlit holds tue D D 1 D D The following theoem nd its onsequene n e poved using the sme method 7 Theoem ile w pssing though the points nd is etenll tngent to ile w 1 Line intesets the ile w 1 t points nd D Let 1 nd e dii of iles w 1 nd w, espetivel Then D D D 1 D D If w is nothe ile pssing though the points nd D, nd etenll tngent to the ile w then D D D D 1 D D D, D whee is the dius of ile w
11 ooll Let iles w nd w 1 of dii nd e etenll tngent Let the etension of hod of the ile w e tngent to the ile w 1 t the point Let D e tngent to the ile w Then Theoem Let iles w nd w 1 of dii nd e etenll tngent line though the ente of ile w 1 is tngent to the ile w t the point Let e tngent to ile w 1 t the point Simill, line though the ente of ile w is tngent to the ile w 1 t the point Let D e tngent to ile w t the point D Then Polems fo futhe eplotions Do the lst equlit nd the inequlit hold tue if the iles w nd w 1 e 1 noninteseting inteseting? efeenes 1 YN liev, Use of dnmi geomet softwe in tehing nd eseh, 5th Intentionl onfeene on IT, 1-14 Otoe 011, Qfqz Univesit, ku, 1-14 YN liev, Polem 11689, me th onth, 10 1, Jnu 013, 77 3 YN liev, ufge 1318, Elem th ; Solution: YN liev Deptment of themtis Qfqz Univesit, Khdln Z 0101, zeijn liev@queduz
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