32. The Tangency Problem of Apollonius.


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1 . The Tngeny olem of Apollonius. Constut ll iles tngent to thee given iles. This eleted polem ws posed y Apollinius of eg ( BC), the getest mthemtiin of ntiquity fte Eulid nd Ahimedes. His mjo wok Koniká etended with stonishing ompehensiveness the peiod s slight knowledge of oni setions. His tetise De Ttionius, whih ontined the solution of the tngeny polem stted ove, hs unfotuntely een lost. Fnçois Viète, lled Viet, the getest Fenh mthemtiin of the siteenth entuy (54060), ttempted in out 600 to estoe the lost tetise of Apollonius nd solved the tngeny polem y teting eh of its speil ses individully, deiving eh suessive one fom the peeding one. In ontst to this, the solutions of Guss (Complete Woks, vol. IV, p.99), Gegonne (Annles de Mthémtiques, vol. IV), nd eesen (Methoden und Theoien) solve the genel polem. We estit ouselves to the elegnt solution of Gegonne. His solution is sed on the devie of seeking the unknown iles in pis, the thn individully. Note. The onstution is omplited enough nd (s witten y Döie) involves so mny likely unfmili onepts nd tems, tht I will eplin just enough things so tht the inteested ede n mke his o he onstutions. It s good eeise to do this with geomety softwe. Needless to sy, wht follows is not n et o even ppoimte tnsltion of Döie, ut the ides e the sme. In genel thee e eight iles tngent to thee given iles. Hee s why. When thee e just two iles, one n entiely suound the othe o not.
2 Beuse of the ntue of the polem, we know tht the tngent iles nnot e inside ny of the thee given iles. Sine tngent ile to eh of the thee given iles n e one to two type, thee e 8 possile tngent iles. Fo two iles nd, let denote the point of intesetion of thei two etenl tngents, nd " denote the point of intesetion of thei two intenl tngents.  + The onstution of the tngent iles equies The owe Cente of thee iles,,. This is the intesetion of the dil es of pis of iles. (See No..) Monge s Theoem. If thee iles,, e tken in pis,,, the points t whih the etenl tngents to eh pi e olline; similly the etenl intesetion point of one pi, nd the intesetion points of intenl tngents to the othe pis e lso olline. These lines e lled the simility es of the thee iles.
3 Note. Döie edits this theoem to d Alemet; most ooks ll it Monge s Theoem, pehps fo his simple poof. Note. If ),*,+ e eh o nd )*+, then Monge s Theoem n e stted in the fom ),* nd + e olline. I will denote this line y )*+ with the suppessed, so fo emple, the line fo though ",," is " " Simility es We lso need some esults out invesion in ile. Definition. Let e ile with ente nd dius. Two points nd U on y fom e invese points if U. U is the invese of nd is the invese of U. ' ' = Invese points Definition. Let l e line in the plne of ile Ÿ, ente dius. Let µ l with on l. The invese of with espet to is the pole (point) of the pol (line l).
4 l (pol) ' (pole) Hee is Gegonne s onstution fo the iles of Apollonius:. Constut the powe ente of iles,,.. Constut simility is D of,,.. Constut the poles,, of D with espet to,, espetively. 4. Constut points of intesetion of. y with ile, sy p nd in the ode p if ) nd the ode p if ) ",. y with ile,sy q nd in the ode q if * nd the ode q if * ", nd. y with ile, sy nd in the ode if + nd the ode if + ". 5. The iles though,,, nd though p, q, e tngent to iles,,. 6. epet steps though 5 with nothe simility is (until ll fou hve een used). p q ,, e poles of +++ wt s,,. 4
5 + p  q ,, e poles of + wt s,,. + p q  +,, e poles of + wt s,,. q p  ,, e poles of + wt s,,. + Note 4. The onstution of ouse equies poof. Detils n e found in Döie. Note 5. ne of moe of the iles,, my e point o line. See fo emple Geomety: Eulid nd Beyond y. Htshone, Spinge,
r (1+cos(θ)) sin(θ) C θ 2 r cos θ 2
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