themtil efletions, Issue 5, 015 INEQULITIES ON TIOS OF DII OF TNGENT ILES YN liev
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
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
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,
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
multipling the numeto with its onjugte nd then dividing with the sme onjugte we otin Finll, sin sin D
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
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
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,
, 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
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 68 013 18; Solution: 69 014 164-165 YN liev Deptment of themtis Qfqz Univesit, Khdln Z 0101, zeijn liev@queduz