Open Soluções Geométrica e Algébrica do Problema de Apolônio

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❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚

❙♦❧✉çõ❡s ●❡♦♠étr✐❝❛ ❡ ❆❧❣é❜r✐❝❛

❞♦ Pr♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦

†

♣♦r

❆❧②ss♦♥ ❊s♣❡❞✐t♦ ❞❡ ▼❡❧♦

❏♦ã♦ P❡ss♦❛ ✲ P❇

†_{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

(2)

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚

❙♦❧✉çõ❡s ●❡♦♠étr✐❝❛ ❡ ❆❧❣é❜r✐❝❛

❞♦ Pr♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦

†

♣♦r

❆❧②ss♦♥ ❊s♣❡❞✐t♦ ❞❡ ▼❡❧♦

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❋❡r♥❛♥❞♦ ❆♥t♦♥✐♦ ❳❛✈✐❡r ❞❡ ❙♦✉s❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

▼❛rç♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇

†_{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

(3)

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❆ ❉❡✉s✱ ♣♦r ❡st❛r s❡♠♣r❡ ♣r❡s❡♥t❡ ❡♠ ♠✐♥❤❛ ✈✐❞❛

❆ ❆♥tô♥✐♦ ❉❛✈✐❞ ❞❡ ▼❡❧♦✱ ♠❡✉ ♣❛✐ q✉❡ ♠❡s♠♦ ♥ã♦ ❡st❛♥❞♦ ♠❛✐s ❡♥tr❡ ♥ós✱ s❡✉ ❡①❡♠♣❧♦ ❞❡ ♣❡rs❡✈❡r❛♥ç❛ ♠❡ ❞❡✉ ♠✉✐t❛ ❢♦rç❛✱ ❡ ❛ ♠✐♥❤❛ ♠ã❡✱ ❏♦s❡❢❛ ▼❛r✐❛ ❞❡ ▼❡❧♦✱ q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❛♦ ❡st✉❞♦✳

❆s ♠✐♥❤❛s q✉❡r✐❞❛s ✜❧❤❛s✱ ❆❧②♥❡ ▼❛r✐❛ ❞❛ ❙✐❧✈❛ ▼❡❧♦ ❡ ❆♥❛ ❇❡❛tr✐③ ❞❛ ❙✐❧✈❛ ▼❡❧♦ q✉❡ sã♦ ♦ ♠♦t✐✈♦ ❞❛ ♠✐♥❤❛ ❡①✐stê♥❝✐❛ ❡ ❞❡❞✐❝❛çã♦ ❛♦ tr❛❜❛❧❤♦✳

❆ ♠✐♥❤❛ ❡s♣♦s❛✱ ▲í✈✐❛ ❞❛ ❙✐❧✈❛ ▼❡❧♦✱ ♣❡❧♦ ❛♣♦✐♦ ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✳

❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡✲ ♠át✐❝❛ q✉❡ ❧❡❝✐♦♥❛r❛♠ ♥♦ P❘❖❋▼❆❚✱ ♣♦✐s ❢♦r❛♠ ❡❧❡s q✉❡♠ ✐♥❝❡♥t✐✈❛r❛♠ á ❜✉s❝❛ ❛♦s ❝❛❜❡❞❛✐s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳

❆♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✱ ♣❡❧❛ ❞✐s♣♦s✐çã♦ ❡♠ ❛✈❛❧✐❛r ❡st❡ ❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ❉r✳ ❋❡r♥❛♥❞♦ ❆♥tô♥✐♦ ❳❛✈✐❡r ❞❡ ❙♦✉③❛ q✉❡ ♦r✐❡♥t♦✉ ❝♦♠ ♠✉✐t❛ ❞❡❞✐❝❛çã♦ ❡ ♣❛❝✐ê♥❝✐❛ t♦❞♦ ♦ ♠❡✉ tr❛❜❛❧❤♦✳

❆♦s ♠❡✉s ❛♠✐❣♦s✱ ❆❧❞❡❝❦ ▼❡♥❡③❡s✱ ❋r❛♥❝✐s❝♦ ▲✐♠❛✱ ▼❛r❝❡❧♦ ❉❛♥t❛s✱ ❉✐ê❣♦ ❆②❧❧♦✱ ❘♦♥❛❧❞♦✱ ❲✐❧❧✐❛♥♦ ❈♦st❛ ❡ ❆❧❡ss❛♥❞r♦ ▼❡❧♦✭✐r♠ã♦✮ ♣❡❧❛s ✈❡r❞❛❞❡✐r❛s ❛♠✐③❛❞❡s ❝✉❧✲ t✐✈❛❞❛s ❣r❛ç❛s ❛♦ ❛❞✈❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛ ❡♠ ♥♦ss❛s ✈✐❞❛s✳

➚ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭❙❇▼✮ ♣❡❧❛ ❝r✐❛çã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐✲ ♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✭P❘❖❋▼❆❚✮ ❞❛♥❞♦ ♦♣♦rt✉♥✐❞❛❞❡ ♣❛r❛ q✉❡ ♣r♦❢❡ss♦r❡s ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ♣♦ss❛♠ ♠❡❧❤♦r❛r ♦s s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❡ ❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ✭❯❋P❇✮✱ ♣♦r ❛❜r❛ç❛r ❡st❛ ✐❞❡✐❛✳

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◆❡st❡ tr❛❜❛❧❤♦✱ ♦ ♥♦ss♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ é ❛♣r❡s❡♥t❛r ✉♠❛ s♦❧✉çã♦ ❣❡♦♠étr✐❝❛ ❡ ❛❧❣é❜r✐❝❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧õ♥✐♦✳ ❖s ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧õ♥✐♦ ❡♥❝♦♥tr❛♠✲s❡ ❝♦♠♦ ❝✐t❛çõ❡s ♥♦s tr❛❜❛❧❤♦s ❞❡ P❛♣♣✉s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❉❛❞♦s três ❡❧❡♠❡♥t♦s✱ ❝❛❞❛ ✉♠ ❞♦s q✉❛✐s ♣♦❞❡ s❡r ♣♦♥t♦s✱ r❡t❛s ♦✉ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❝♦♥str✉✐r ✉♠❛ ❝✐r❝✉♥✲ ❢❡rê♥❝✐❛ q✉❡ ♣❛ss❛ ♣❡❧♦✭s✮ ♣♦♥t♦✭s✮ ❡ s❡❥❛ t❛♥❣❡♥t❡ ❛ ❝❛❞❛ ✉♠❛ ❞❛s ❧✐♥❤❛s ❞❛❞❛s✱ ♠❛s ♥♦ss♦ tr❛❜❛❧❤♦ ✈❛✐ ♠♦str❛r ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❛s s♦❧✉çõ❡s ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ ♦s três ♦❜❥❡t♦s sã♦ três ❝✐r❝✉♥❢❡rê♥❝✐❛s ♥ã♦ s❡❝❛♥t❡s✱ ♥ã♦ t❛♥❣❡♥t❡s ❡ ❝♦♠ r❛✐♦s ❞✐s✲ t✐♥t♦s✳ ❊st❡ ❚r❛❜❛❧❤♦ ❝♦♠❜✐♥❛ ❡❧❡♠❡♥t♦s ❤✐stór✐❝♦s ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✈ár✐♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡st❡✳

P❛❧❛✈r❛s ❝❤❛✈❡✿ ❈♦♥str✉çõ❡s ●❡♦♠étr✐❝❛s✱ Pr♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦✱ ❍♦♠♦t❡t✐❛

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❚❤✐s ✇♦r❦✱ ♦✉r ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ✐s t♦ ♣r❡s❡♥t ❛ ❣❡♦♠❡tr✐❝ ❛♥❞ ❛❧❣❡❜r❛✐❝ s♦❧✉t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❆♣♦❧❧♦♥✐✉s✳ ❚❤❡ ♣r♦❜❧❡♠s ❛r❡ ❛s ❆♣♦❧❧♦♥✐✉s ❝✐t❛t✐♦♥s ✐♥ P❛♣♣✉s ✇♦r❦s ❛s ❢♦❧❧♦✇s✿ ●✐✈❡♥ t❤r❡❡ ❡❧❡♠❡♥ts✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ♠❛② ❜❡ ♣♦✐♥ts✱ ❧✐♥❡s ♦r ❝✐r✲ ❝✉♠❢❡r❡♥❝❡✱ ❝♦♥str✉❝t ❛ ❝✐r❝✉♠❢❡r❡♥❝❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ t❤❡ ♣♦✐♥t ✭s✮ ❛♥❞ ✐s t❛♥❣❡♥t t♦ ❡❛❝❤ ♦❢ t❤❡ ❧✐♥❡s ❣✐✈❡♥✱ ❜✉t ♦✉r ✇♦r❦ ✇✐❧❧ s♣❡❝✐✜❝❛❧❧② s❤♦✇ t❤❡ s♦❧✉t✐♦♥s ❢♦r t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ t❤r❡❡ ♦❜❥❡❝ts ❛r❡ t❤r❡❡ t❛♥❣❡♥t ❝✐r❝✉♠❢❡r❡♥❝❡ ♥♦♥✲❞r②✐♥❣✱ ❛♥❞ ✇✐t❤ ❞✐✛❡r❡♥t r❛❞✐✐✳ ❲❡ ✇✐❧❧ ❛❧s♦ ♣r❡s❡♥t ❤✐st♦r✐❝❛❧ ❡❧❡♠❡♥ts ♦❢ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❆♣♦❧❧♦♥✐✉s✱ ✇❡ ❤❛✈❡ ❞❡✈❡❧♦♣❡❞ s❡✈❡r❛❧ ✐♠♣♦rt❛♥t ♠❛t❤❡♠❛t✐❝❛❧ ❝♦♥❝❡♣ts ❢♦r ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ ❝♦♥str✉❝t✐♦♥s✳

❑❡② ✇♦r❞s✿ ●❡♦♠❡tr✐❝ ❈♦♥str✉❝t✐♦♥s✱ ♣r♦❜❧❡♠ ♦❢ ❆♣♦❧❧♦♥✐✉s ❛♥❞ ❉✐❧❛t❡✳

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✶ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛ ✲ ❆s♣❡❝t♦s ❍✐stór✐❝♦s ✶ ✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✸ ❆♣♦❧ô♥✐♦ ❡ ❛s ❝ô♥✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✹ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✹✳✶ ❚rês ♣♦♥t♦s ❞✐st✐♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✹✳✷ ❚rês r❡t❛s ❞✐st✐♥t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✹✳✸ ❯♠❛ r❡t❛ ❡ ❞♦✐s ♣♦♥t♦s ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛ ❡ss❛ r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹✳✹ ❉✉❛s r❡t❛s ❡ ✉♠ ♣♦♥t♦ ♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛ ♥❡♥❤✉♠❛ ❞❡ss❛s r❡t❛s ✺ ✶✳✹✳✺ ❯♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ❞♦✐s ♣♦♥t♦s ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛ ❡ss❛ ❝✐r✲

❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✹✳✻ ❉✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡ ✉♠ ♣♦♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✹✳✼ ❉✉❛s r❡t❛s ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✹✳✽ ❉✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡ ✉♠❛ r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹✳✾ ❯♠ ♣♦♥t♦✱ ✉♠❛ r❡t❛ ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹✳✶✵ ❚rês ❝✐r❝✉♥❢❡rê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✶✶

✷✳✶ ❊❧❡♠❡♥t♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✸ ❯♠❛ ❝♦♥str✉çã♦ ●❡♦♠étr✐❝❛ ❞♦ Pr♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ✶✽

✹ ❯♠❛ s♦❧✉çã♦ ❛❧❣é❜r✐❝❛ ❞♦ Pr♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ✸✹

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✺

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❆s ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s ❥á ❛♣❛r❡❝❡♠ ♥❛ ●ré❝✐❛ ❛♥t✐❣❛ ❡ ♦s ✜❧ós♦❢♦s ❣r❡❣♦s ❥á t✐♥❤❛♠ ❢❛s❝í♥✐♦ ♣♦r r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ ❣❡♦♠❡tr✐❛ ♣♦r ♠❡✐♦ ❞❡ ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s✳ ❉❡s❞❡ ❛ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛ ❡♥❝♦♥tr❛♠✲s❡ ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❡♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❝♦♥str✉çã♦ ❞♦ ♣❡♥t❛❣r❛♠❛ ♦✉ ♣❡♥tá❣♦♥♦ ❡str❡❧❛❞♦ ❡ ❞♦s ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s✱ ✈❡r ❬✶❪✳ ◆❛ é♣♦❝❛ ❞❡ ❊✉❝❧✐❞❡s✱ ❛s ❣r❛♥❞❡③❛s ♣❛ss❛r❛♠ ❛ s❡r ❛ss♦❝✐❛❞❛s ❛ s❡❣♠❡♥t♦s ❞❡ r❡t❛s✱ ♥❛s❝❡ ❡♥tã♦ à á❧❣❡❜r❛ ❣❡♦♠étr✐❝❛ ♦♥❞❡ ❛ ♣❛❧❛✈r❛ r❡s♦❧✈❡r ❡r❛ s✐♥ô♥✐♠♦ ❞❡ ❝♦♥str✉✐r✱ ✈❡r ❬✷❪✳ P❛r❛ ♦s ❣r❡❣♦s ❛s ❝♦♥str✉çõ❡s sã♦ r❡❛❧✐③❛❞❛s ❝♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ♣❛r❛ ❝♦♥str✉✐r ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝♦♠ ❝❡♥tr♦ ❡ r❛✐♦ ❞❛❞♦s✳

❖ Pr♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ é ✉♠ ❞♦s ❣r❛♥❞❡s ♣r♦❜❧❡♠❛s ❞❛ ❍✐stór✐❛ ❞❛ ●❡♦♠❡tr✐❛✳ ❊st❡✈❡ ♣r❡s❡♥t❡ ❛♦ ❧♦♥❣♦ ❞❛ ❡✈♦❧✉çã♦ ❞❛ ❣❡♦♠❡tr✐❛ ❡ ✈✐✉ ♥❛s❝❡r ♦ q✉❡ s❡ ❝❤❛♠❛ ❞❡ ❣❡♦♠❡tr✐❛ ♠♦❞❡r♥❛ ♣♦r ♦♣♦s✐çã♦ ❛ ❣❡♦♠❡tr✐❛ ❣r❡❣❛✳ ❆♣♦❧ô♥✐♦ ♣r♦♣õ❡ ✉♠ ♣r♦❜❧❡♠❛ q✉❡ ✈✐r✐❛ ❛ s❡r ❝♦♥❤❡❝✐❞♦ ♣♦r s❡✉ ♥♦♠❡✿ ❊♥❝♦♥tr❛r ✉♠ ❝ír❝✉❧♦ t❛♥❣❡♥t❡ ❛ três ♦✉tr♦s ❝ír❝✉❧♦s✱ ♣♦❞❡♥❞♦ ❡st❡s s❡r ❞❡❣❡♥❡r❛❞♦s ❡♠ r❡t❛s ✭❝ír❝✉❧♦ ❞❡ r❛✐♦ ✐♥✜♥✐t♦✮ ♦✉ ♣♦♥t♦s ✭❝ír❝✉❧♦ ❞❡ r❛✐♦ ③❡r♦✮✳ ❉❡s❞❡ ❡♥tã♦✱ ❞✐✈❡rs♦s ♠❛t❡♠át✐❝♦s tê♠ s❡ ❡♠♣❡♥❤❛❞♦ ♥❛ ❜✉s❝❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ♦ ❛tr❛❡♥t❡ ♣r♦❜❧❡♠❛✳

❖ ♥♦ss♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ✈❛✐ s❡ ❝♦♥❝❡♥tr❛r ❡♠ ♠♦str❛r ✉♠❛ s♦❧✉çã♦ ❣❡♦♠étr✐❝❛ ❡ ❛❧❣é❜r✐❝❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ q✉❛♥❞♦ ♦s três ❡❧❡♠❡♥t♦s ❡♠ q✉❡stã♦ sã♦ três ❝✐r❝✉♥❢❡rê♥❝✐❛s ♥ã♦ t❛♥❣❡♥t❡s✱ ♥ã♦ s❡❝❛♥t❡s ❡ ♦s ❝❡♥tr♦s ♥ã♦ ❛❧✐♥❤❛❞♦s✳ ❆ s♦❧✉✲ çã♦ ●❡♦♠étr✐❝❛ s❡rá ♣♦r ♠❡✐♦ ❞❡ ✐♥✈❡rsã♦✳

◆♦ss♦ tr❛❜❛❧❤♦ ❢♦✐ ❞✐✈✐❞♦ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✿

◆♦ ❝❛♣ít✉❧♦ ✶ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞❡ ❆♣♦❧ô♥✐♦ ❡ ❞❡ s✉❛s ❝♦♥✲ tr✐❜✉✐çõ❡s ♣❛r❛ ❛ ❣❡♦♠❡tr✐❛✱ ❢❛❧❛r❡♠♦s ❞❛ r❡❧❛çã♦ ❞❡ ❆♣♦❧ô♥✐♦ ❝♦♠ ❛s ❝ô♥✐❝❛s ❡ t❛♠❜é♠ ❛♣r❡s❡♥t❛r❡♠♦s ♦s ❞❡③ ❝❛s♦s ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ❞❡s❞❡ ♦ ❝❛s♦ ♠❛✐s s✐♠♣❧❡s q✉❡ é ♦ ❝❛s♦ ❞❡ três ♣♦♥t♦ ♥ã♦ ❛❧✐♥❤❛❞♦s ❛té ♦ ❝❛s♦ ❞❛s três ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ❛❧❣✉♠❛s ❞❡ s✉❛s s♦❧✉çõ❡s✳

❈♦♠♦ ❛s ❝♦♥str✉çõ❡s ❡①✐❣❡♠ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ♣ré✈✐♦ ♣❛r❛ ♦ ♣❧❡♥♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ t♦❞♦s ♦s ♣❛ss♦s✱ ♦ ❝❛♣✐t✉❧♦ ✷ é ❞❡st✐♥❛❞♦ ❛s ❛♣r❡s❡♥t❛çõ❡s ❞❡ ❝♦♥✲ ❝❡✐t♦s ❜ás✐❝♦ t❛✐s ❝♦♠♦ ❞❡✜♥✐çõ❡s✱ ♣r♦♣♦s✐çõ❡s ❡ ❧❡♠❛s✳

◆♦ ❈❛♣ít✉❧♦ ✸ ♠♦str❛r❡♠♦s ♣❛ss♦ ❛ ♣❛ss♦ ❛ ❝♦♥str✉çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ❞✉❛s ❞❛s

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◆♦ ❝❛♣ít✉❧♦ ✹ ♠♦str❛r❡♠♦s ❝♦♠♦ ❡♥❝♦♥tr❛r ❛❧❣❡❜r✐❝❛♠❡♥t❡ ♦ ❝❡♥tr♦ ❡ ♦ r❛✐♦ ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s t❛♥❣❡♥t❡s ❛ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❞❛❞❛s ❡ ❡st❡♥❞❡♠♦s ♦ ♠ét♦❞♦ ♣❛r❛ ❡♥❝♦♥tr❛r ♦ ❝❡♥tr♦ ❡ ♦ r❛✐♦ ❞❛s ♦✐t♦ s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦ ♥♦ ❝❛s♦ ❞❡ t❡r♠♦s três ❝✐r❝✉♥❢❡rê♥❝✐❛s✳

❊s♣❡r❛♠♦s q✉❡ ❡st❡ tr❛❜❛❧❤♦ s✐r✈❛ ❞❡ ❛♣♦✐♦ ♣❛r❛ ♦✉tr❛s ❞✐ss❡rt❛çõ❡s✱ ❞❡ ❢❡rr❛✲ ♠❡♥t❛s ❞❡ tr❛❜❛❧❤♦ ♣❛r❛ ❡❧❛❜♦r❛çã♦ ❞❡ ❛✉❧❛s ❞❡ ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s✳

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❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛ ✲ ❆s♣❡❝t♦s

❍✐stór✐❝♦s

✶✳✶ ■♥tr♦❞✉çã♦

❯♠ ❣r❛♥❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❣❡♦♠❡tr✐❛ ❛❝♦♥t❡❝❡✉ ♥❛ ❝❤❛♠❛❞❛ ■❞❛❞❡ ➪✉r❡❛ ❞❛ ♠❛t❡♠át✐❝❛ ❣r❡❣❛✱ ♦✉ ♣❡rí♦❞♦ ❤❡❧❡♥íst✐❝♦✱ ♦✉ ❛✐♥❞❛ ♣❡rí♦❞♦ ❆❧❡①❛♥❞r✐♥♦ ❡ s❡ ❡st❡♥❞❡✉ ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✸✷✹ ❛✳❈ ❛ ✻✵✵ ❞✳❈✳ ❋♦✐ ♥❡st❡ ♣❡rí♦❞♦ q✉❡ ❆♣♦❧ô♥✐♦ ♣r♦♣ôs ✉♠ ♣r♦❜❧❡♠❛✱ q✉❡ ✈❡✐♦ ❛ r❡❝❡❜❡r ♦ s❡✉ ♥♦♠❡✱ ❝♦♥s✐st✐♥❞♦ ❡♠✿ ❊♥❝♦♥tr❛r ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ t❛♥❣❡♥t❡ ❛ três ♦✉tr❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s✱ ♣♦❞❡♥❞♦ ❡st❛s s❡r❡♠ ❞❡❣❡♥❡r❛✲ ❞❛s ❡♠ r❡t❛s ✭❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ ✐♥✜♥✐t♦✮ ♦✉ ♣♦♥t♦s ✭❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ ③❡r♦✮✳ ❊st❡ ♣r♦❜❧❡♠❛ ❞❡s♣❡rt♦✉ ♦ ✐♥t❡r❡ss❡ ❞❡ ✈ár✐♦s ♠❛t❡♠át✐❝♦s ❛♦ ❧♦♥❣♦ ❞♦s sé❝✉❧♦s✱ ❝❛❞❛ q✉❛❧ ❜✉s❝❛♥❞♦ s♦❧✉çõ❡s s❡❣✉♥❞♦ ❛s ♠❛✐s ❞✐✈❡rs❛s ❛❜♦r❞❛❣❡♥s q✉❡ r❡✢❡t✐❛♠ ♦ ✐♥str✉♠❡♥t❛❧ ♠❛t❡♠át✐❝♦ ❞✐s♣♦♥í✈❡❧ ❡♠ ❝❛❞❛ é♣♦❝❛✳ ❆ ♦❜s❡r✈❛çã♦ ❞❡st❡s ❡s❢♦rç♦s ♣❡r♠✐t❡✱ ♣♦✐s✱ ♣❡r❝❡❜❡r ❛ ✐♥t❡r❡ss❛♥t❡ tr❛❥❡tór✐❛ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❣❡♦♠❡tr✐❛✱ ❝♦♥str✉í❞❛ ♣♦r ♠❛t❡♠át✐❝♦s r❡❝♦♥❤❡❝✐❞♦s ❡ r❡❢❡r❡♥❝✐❛❞♦s ♥❛ ❛t✉❛❧✐❞❛❞❡✳

✶✳✷ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛

❚r❛t❛❞♦s s♦❜r❡ ❛s s❡çõ❡s ❝ô♥✐❝❛s sã♦ ❝♦♥❤❡❝✐❞♦s ❛♥t❡s ❞❛ é♣♦❝❛ ❞❡ ❊✉❝❧✐❞❡s ✭ ✸✷✺ ❛ ✷✻✺ ❛✳❈✳✮✳ ❊✱ ❛ss♦❝✐❛❞♦ à ❤✐stór✐❛ ❞❡ss❛s ❝✉r✈❛s✱ t❡♠♦s ❆♣♦❧ô♥✐♦ q✉❡ ♥❛s❝❡✉ ♥❛ ❝✐❞❛❞❡ ❞❡ P❡r❣❛✱ r❡❣✐ã♦ ❞❛ P❛♥❢í❧✐❛ ✭❛t✉❛❧♠❡♥t❡ ❚✉rq✉✐❛✮ ♣♦r ✈♦❧t❛ ❞❡ ✷✻✷ ❛✳❈✳ ❡ ✈✐✈❡✉✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ❛té ✶✾✵ ❛✳❈✳ ❆♣♦❧ô♥✐♦ ❢♦✐ ❝♦♥t❡♠♣♦râ♥❡♦ ❡ r✐✈❛❧ ❞❡ ❆r✲ q✉✐♠❡❞❡s q✉❡ ✈✐✈❡✉✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ❡♥tr❡ ✷✽✼ ❛✳❈✳ ❡ ✷✶✷ ❛✳❈✳ ❡✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❊✉❝❧✐❞❡s✱ ❢♦r♠❛♠ ❛ trí❛❞❡ ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ s❡♥❞♦ ❛ ❞♦s ♠❛✐♦r❡s ♠❛t❡♠át✐❝♦s ❣r❡✲ ❣♦s ❞❛ ❛♥t✐❣✉✐❞❛❞❡✳ ❆♣♦❧ô♥✐♦ ❡st✉❞♦✉ ❝♦♠ ♦s ❞✐s❝í♣✉❧♦s ❞❡ ❊✉❝❧✐❞❡s ❡♠ ❆❧❡①❛♥❞r✐❛ ❡ ❢♦✐ ❛strô♥♦♠♦ ♥♦tá✈❡❧✱ t❛❧✈❡③ ❡❧❡✱ ❡ ♥ã♦ ❊✉❝❧✐❞❡s✱ ♠❡r❡❝❡✉ ❞♦s ❛♥t✐❣♦s ♦ ❛❞❥❡t✐✈♦ ❞❡ ✧♦ ❣r❛♥❞❡ ●❡ô♠❡tr❛ ✧✳ ❆ ♠❛✐♦r ♣❛rt❡ ❞❛s ♦❜r❛s ❞❡ ❆♣♦❧ô♥✐♦ ❞❡s❛♣❛r❡❝❡✉✳ ❖ q✉❡

(13)

s❛❜❡♠♦s ❞❡ss❛s ♦❜r❛s ♣❡r❞✐❞❛s ❞❡✈❡♠♦s ❛ P❛♣♣✉s ❞❡ ❆❧❡①❛♥❞r✐❛ ✭sé❝ ■❱ ❛✳❈✳✮ ✈❡r ❬✹❪✳ ❙✉❛ ♦❜r❛ ♣r✐♠❛ é ❙❡çõ❡s ❈ô♥✐❝❛s ❝♦♠♣♦st❛ ♣♦r ✽ ✈♦❧✉♠❡s ✭❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✹✵✵ ♣r♦♣♦s✐çõ❡s✦✮✳ ❉❛ ♦❜r❛ ♦r✐❣✐♥❛❧ s♦❜r❡✈✐✈❡r❛♠ ✼ ✈♦❧✉♠❡s✱ s❡♥❞♦ ✹ ❡s❝r✐t♦s ❡♠ ❣r❡❣♦ ❡ ✸ tr❛❞✉③✐❞♦s ♣❛r❛ ♦ ár❛❜❡ ♣♦r ❚❤❛❜✐t ■❜♥ ◗✉rr❛ ✭✽✷✻ ❛ ✾✵✶✮ ♥♦ sé❝✳ ■❳✳✳ ❖s três ♣r✐♠❡✐r♦s ✈♦❧✉♠❡s sã♦ ❜❛s❡❛❞♦s ❡♠ tr❛❜❛❧❤♦s ❞❡ ❊✉❝❧✐❞❡s ❡ ♦ ♦✐t❛✈♦ ✈♦❧✉♠❡ ❢♦✐✱ ✐♥❢❡❧✐③♠❡♥t❡✱ ♣❡r❞✐❞♦✳ ❊♠ ✶✳✼✶✵✱ ❊❞♠✉♥❞ ❍❛❧❧❡② tr❛❞✉③✐✉ ♦s s❡t❡ ✈♦❧✉♠❡s s♦✲ ❜r❡✈✐✈❡♥t❡s ❞❡ ❙❡❝çõ❡s ❈ô♥✐❝❛s ♣❛r❛ ♦ ❧❛t✐♠ ❡ t♦❞❛s ❛s ❞❡♠❛✐s tr❛❞✉çõ❡s ♣❛r❛ ❛s ❧í♥❣✉❛s ♠♦❞❡r♥❛s ❢♦r❛♠ ❢❡✐t❛s ❛ ♣❛rt✐r ❞❛ tr❛❞✉çã♦ ❞❡ ❍❛❧❧❡②✳ ❆✜r♠❛✲s❡ q✉❡✱ ❝♦♠ ♦ tr❛❜❛❧❤♦ ✏❆s ❈ô♥✐❝❛s✑✱ ❆♣♦❧ô♥✐♦ t❡♥❤❛ ❝♦♠♣❧❡t❛❞♦ ❡ ❛♠♣❧✐❛❞♦ ❛ ♦❜r❛ ❞❡ ♠❡s♠♦ ♥♦♠❡ ❡s❝r✐t❛ ♣♦r ❊✉❝❧✐❞❡s✱ s❡♥❞♦ ❡st❛ ❝♦♠♣♦st❛ ♣♦r q✉❛tr♦ ❧✐✈r♦s q✉❡✱ ❝♦♠ ♦✉tr♦s tr❛❜❛❧❤♦s✱ é ❝♦♥❤❡❝✐❞❛ ❛♣❡♥❛s ♣♦r r❡❢❡rê♥❝✐❛s ♣♦st❡r✐♦r❡s

✶✳✸

❆♣♦❧ô♥✐♦ ❡ ❛s ❝ô♥✐❝❛s

❖ ❣r❡❣♦ ▼❡♥❛❡❝♠✉s ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❛ ❡st✉❞❛r ❛s ❝ô♥✐❝❛s✱ ✐❞❡♥t✐✜❝❛♥❞♦✲❛s ❝♦♠♦ s❡çõ❡s ❞❡ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ ❝♦♥❡s ❝✐r❝✉❧❛r❡s✳ ❚❛♠❜é♠ s❡ t❡♠ ♥♦tí❝✐❛ ❞❛s ✐♥✈❡st✐❣❛✲ çõ❡s ❞❡ ❈♦♥♦♥ ❞❡ ❙❛♠♦s ✭❝✳ ✷✹✺ ❛✳❈✳✮ ❛❝❡r❝❛ ❞❡st❛s ❝✉r✈❛s✳ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛✱ ♣♦r ✈♦❧t❛ ❞❡ ✷✷✺ ❛✳❈✳ ♣r♦❞✉③✐✉ ✉♠ ❡①t❡♥s♦ ❡st✉❞♦ ❡♠ s❡✉ ❧✐✈r♦ ❆s ❝ô♥✐❝❛s✱ ♠♦str❛♥❞♦ q✉❡ ❛s ❝✉r✈❛s q✉❡ ❞ã♦ ♥♦♠❡ à ♦❜r❛ ♣♦❞✐❛♠ s❡r ♦❜t✐❞❛s ❝♦♠♦ ❞✐❢❡r❡♥t❡s s❡çõ❡s ❞❡ q✉❛❧q✉❡r s✉♣❡r❢í❝✐❡ ❝ô♥✐❝❛ ❝✐r❝✉❧❛r✳ ❋♦✐ ❆♣♦❧ô♥✐♦ t❛❧✈❡③ s❡❣✉✐♥❞♦ s✉❣❡stã♦ ❞❡ ❆r✲ q✉✐♠❡❞❡s✱ q✉❡♠ ♥♦♠❡♦✉ ❛s s❡❝çõ❡s ❝ô♥✐❝❛s ❝♦♠ ♦s t❡r♠♦s ❡❧✐♣s❡✱ ❤✐♣ér❜♦❧❡ ❡ ♣❛rá❜♦❧❛✳ ❆❞♠✐t✐♥❞♦ q✉❡ ❛ s✉♣❡r❢í❝✐❡ ❝ô♥✐❝❛ t❡♠ ❞✉❛s ❢♦❧❤❛s✱ ❆♣♦❧ô♥✐♦ ♠♦str♦✉ q✉❡ ❛ ❤✐♣ér❜♦❧❡ t❡♠ ❞♦✐s r❛♠♦s✳ ❊♠❜♦r❛ à ♣r✐♠❡✐r❛ ✈✐st❛ ❛s três ❝ô♥✐❝❛s ♣♦ss❛♠ ♣❛r❡❝❡r ❡ss❡♥❝✐❛❧♠❡♥t❡ ❞✐st✐♥t❛s✱ q✉❡r ❡♠ s✉❛ ❢♦r♠❛✱ q✉❡r ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ s❛✲ t✐s❢❛③❡♠✱ é ♦ ❢❛t♦ ❞❡ t❡r❡♠ s✐❞♦ ♦❜t✐❞❛s ❝♦♠♦ ❝♦rt❡s ❡♠ ❝♦♥❡s ❞✉♣❧♦s q✉❡ ❛s r❡ú♥❡ ❡♠ ✉♠❛ ♠❡s♠❛ ❢❛♠í❧✐❛✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❬✶❪✱ ❆♣♦❧ô♥✐♦ ❛♥❛❧✐s♦✉ ❛s ♣r♦♣r✐❡❞❛❞❡s ❢✉♥❞❛✲ ♠❡♥t❛✐s ❞❛s ❝ô♥✐❝❛s✱ ♠❛✐s ❝♦♠♣❧❡t❛♠❡♥t❡ ❡ ❝♦♠ ♠❛✐s ❣❡♥❡r❛❧✐❞❛❞❡ q✉❡ ♥♦s ❡s❝r✐t♦s ❞❡ ♦✉tr♦s ❛✉t♦r❡s✳ ❉❡ ❢❛t♦✱ ❥á ♥♦ ♣r✐♠❡✐r♦ ❧✐✈r♦✱ ❆♣♦❧ô♥✐♦ ❞❡s❡♥✈♦❧✈❡ ❛ ❝❤❛♠❛❞❛ t❡♦r✐❛ ❞♦s ❞✐â♠❡tr♦s ❝♦♥❥✉❣❛❞♦s✱ ✉s❛♥❞♦ ✉♠ ♣❛r ❞❡st❡s ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡ ❛♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ♦❜❧íq✉♦s✳ ❆✐♥❞❛ s❡❣✉♥❞♦ ❬✶❪✱ ♦s ♠ét♦❞♦s ❞❡ ❆♣♦❧ô♥✐♦✱ ❡♠ ❆s ❝ô♥✐❝❛s✱ ❡♠ ♠✉✐t♦s ♣♦♥t♦s✱ sã♦ tã♦ s❡♠❡❧❤❛♥t❡s ❛♦s ♠♦❞❡r♥♦s q✉❡ às ✈❡③❡s s❡ ❝♦♥s✐❞❡r❛ s❡✉ tr❛t❛❞♦ ❝♦♠♦ ✉♠❛ ❡s♣é❝✐❡ ❞❡ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✱ ❛♥t❡❝✐♣❛♥❞♦ ♦ tr❛❜❛❧❤♦ ❞❡ ❉❡s❝❛r✲ t❡s ❡♠ ✶✽✵✵ ❛♥♦s✳ ❱❛❧❡ ❞❡st❛❝❛r✱ ♣♦ré♠✱ q✉❡ ♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ✉t✐❧✐③❛❞♦ ♣♦r ❆♣♦❧ô♥✐♦ ❡ s❡✉s ❛♥t❡❝❡ss♦r❡s ❡r❛ s❡♠♣r❡ ❝♦♥s✐❞❡r❛❞♦ ❛ ♣♦st❡r✐♦r✐ s♦❜r❡ ✉♠❛ ❝✉r✈❛ ❞❛❞❛ ❛ ✜♠ ❞❡ s❡ ❡st✉❞❛r s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ◆ã♦ ♣❛r❡❝❡ ❤❛✈❡r ❡①❡♠♣❧♦✱ ♥❛ ❣❡♦✲ ♠❡tr✐❛ ❛♥t✐❣❛✱ ❞❡ ✉♠ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❛ ♣r✐♦r✐ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ✜♥s ❞❡ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❡ ✉♠❛ ❡q✉❛çã♦✳ P♦❞❡♠♦s ❛ss✐♠ ❞✐③❡r q✉❡ ♥❛ ❣❡♦♠❡tr✐❛ ❣r❡❣❛ ❛s ❡q✉❛çõ❡s ❡r❛♠ ❞❡t❡r♠✐♥❛❞❛s ♣❡❧❛s ❝✉r✈❛s✱ ❡ ♥ã♦ ♦ ❝♦♥trár✐♦✳ ❖s tr❛❜❛❧❤♦s ❞❡ ❊✉❝❧✐❞❡s ❡ ❆rq✉✐♠❡❞❡s✱ ❛ss✐♠ ❝♦♠♦ ❛s ❙❡çõ❡s ❝ô♥✐❝❛s ❞❡ ❆♣♦❧ô♥✐♦ ♠❛♥t✐✈❡r❛♠ s✉❛ ✉t✐❧✐❞❛❞❡ ❝♦♠♦ ❛♣♦✐♦ ❵❛ ♣❡sq✉✐s❛ ❛té ♦ sé❝✉❧♦ ❳❱■■✱ q✉❛♥❞♦ ❏♦❤❛♥♥❡s ❑❡♣❧❡r

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✭✶✺✼✶✲✶✻✸✵✮ s❡ ❞❡❞✐❝♦✉ ❛ ✉♠ ❡st✉❞♦ ♣r♦❢✉♥❞♦ ❞♦s ♠♦✈✐♠❡♥t♦s ❞♦s ♣❧❛♥❡t❛s ❡ s✉❛s ór❜✐t❛s

✶✳✹ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦

❉❛❞♦s três ♦❜❥❡t♦s✱ ❝❛❞❛ ✉♠ ♣♦❞❡♥❞♦ s❡r ✉♠ ♣♦♥t♦✱ ✉♠❛ r❡t❛ ♦✉ ✉♠❛ ❝✐r❝✉♥✲ ❢❡rê♥❝✐❛✱ tr❛ç❛r ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ é t❛♥❣❡♥t❡ ❛ t♦❞♦s ❡ss❡s três ♦❜❥❡t♦s✱ ♥❡st❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ é t❛♥❣❡♥t❡ ❛ ✉♠ ♣♦♥t♦ s❡ ❡❧❛ ♣❛ss❛ ♣♦r ❡ss❡ ♣♦♥t♦✳ P♦❞❡♠♦s ❞✐✈✐❞✐r ❡ss❡ ♣r♦❜❧❡♠❛ ❡♠ ❞❡③ ❝❛s♦s✱ ❞❡s❞❡ ♦ ♠❛✐s s✐♠♣❧❡s✱ ♦ ❝❛s♦ ❞❡ três ♣♦♥t♦s✱ ❛té ♦ ♠❛✐s ❝♦♠♣❧✐❝❛❞♦✱ ♦ ❝❛s♦ ❞❡ três ❝✐r❝✉♥❢❡rê♥❝✐❛s✱❝❛s♦ ❡st❡ q✉❡ s❡rá ♦❜❥❡t♦ ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦ ♥♦s ❝❛♣ít✉❧♦s ✸ ❡ ✹✳ ❊st❡ ú❧t✐♠♦ ❢♦✐ ❝♦♥s✐❞❡r❛❞♦ ✉♠ ❞❡s❛✜♦ ♣❛r❛ ♦s ♠❛t❡♠át✐❝♦s ❞♦s sé❝✉❧♦s ❳❱■ ❡ ❳❱■■✱ t❡♥❞♦ s✐❞♦ r❡s♦❧✈✐❞♦ ♣♦r ❆❞r✐❛❛♥ ✈❛♥ ❘♦♦♠❡♥ ♥♦ ♠❡✐♦ ❞♦ sé❝✉❧♦ ❳❱■✱ ✉s❛♥❞♦ ✐♥t❡rs❡❝çã♦ ❞❡ ❝ô♥✐❝❛s✳ P♦✉❝♦ t❡♠♣♦ ❞❡✲ ♣♦✐s✱ ♦ ♠❛t❡♠át✐❝♦ ❋r❛♥ç♦✐s ❱✐èt❡ r❡s♦❧✈❡✉ ❡st❡ ♣r♦❜❧❡♠❛ ✉t✐❧✐③❛♥❞♦ ❛♣❡♥❛s ré❣✉❛ ❡ ❝♦♠♣❛ss♦✳

▼♦str❛r❡♠♦s ❛❜❛✐①♦ ♦s ❉❡③ ❝❛s♦s ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦✱ ❛❧❣✉♥s ❞❡♣❡♥❞❡♥❞♦ ❞❛s ♣♦s✐çõ❡s ❞♦s ♦❜❥❡t♦s✱ s❡♠ ✈✐s✉❛❧✐③❛çã♦ ❞❡ s♦❧✉çõ❡s✳

✶✳✹✳✶ ❚rês ♣♦♥t♦s ❞✐st✐♥t♦s

◆❡st❡ ❝❛s♦ ♣♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ❛♣❡♥❛s ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦✳

✶✳✹✳✷ ❚rês r❡t❛s ❞✐st✐♥t❛s

❚rês r❡t❛s sã♦ ❞✉❛s ❛ ❞✉❛s ❝♦♥❝♦rr❡♥t❡s✱ ♠❛s ♥ã♦ ❝♦♥❝♦rr❡♠ ❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦✱ ♥❡ss❡ ❝❛s♦ ❝♦♥s❡❣✉✐♠♦s ✈✐s✉❛❧✐③❛r q✉❛tr♦ s♦❧✉çõ❡s✿

❉✉❛s ❞❡st❛s r❡t❛s sã♦ ♣❛r❛❧❡❧❛s ❡♥tr❡ s✐✱ ♠❛s ♥ã♦ sã♦ ♣❛r❛❧❡❧❛s à t❡r❝❡✐r❛ r❡t❛✱ ♥❡ss❡ ❝❛s♦ ❝♦♥s❡❣✉✐♠♦s ✈✐s✉❛❧✐③❛r ❞✉❛s s♦❧✉çõ❡s✿

❚❡♠♦s ❚❛♠❜é♠ ❞♦✐s ❝❛s♦s ❡♠ q✉❡ ♥ã♦ ❡♥❝♦♥tr❛♠♦s ♥❡♥❤✉♠❛ s♦❧✉çã♦✿

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✶✳✹✳✸ ❯♠❛ r❡t❛ ❡ ❞♦✐s ♣♦♥t♦s ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛ ❡ss❛ r❡t❛

❖s ❞♦✐s ♣♦♥t♦s ♣❡rt❡♥❝❡♠ ❛♦ ♠❡s♠♦ ❞♦s ❞♦✐s s❡♠✐♣❧❛♥♦ ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛ r❡t❛✱ ♣♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ❞✉❛s s♦❧✉çõ❡s ♥❡ss❡ ❝❛s♦✿

▼❛s s❡ ♦s ❞♦✐s ♣♦♥t♦s ♣❡rt❡♥❝❡r❡♠ ❛ s❡♠✐♣❧❛♥♦s ❞✐❢❡r❡♥t❡s ❡♥tã♦ ♥ã♦ ❝♦♥s❡❣✉✐♠♦s ❡♥❝♦♥tr❛r ♥❡♥❤✉♠❛ s♦❧✉çã♦✿

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✶✳✹✳✹ ❉✉❛s r❡t❛s ❡ ✉♠ ♣♦♥t♦ ♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛ ♥❡♥❤✉♠❛ ❞❡s✲

s❛s r❡t❛s

❆s r❡t❛s sã♦ ♣❛r❛❧❡❧❛s ❡ ♥❡st❡ ❝❛s♦ ♣♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ✉♠❛ ♦✉ ♥❡♥❤✉♠❛ s♦❧✉çã♦✱ ✐ss♦ ❞❡♣❡♥❞❡ ❞❛ ♣♦s✐çã♦ ❞♦ ♣♦♥t♦ ❡♠ r❡❧❛çã♦ ❛s r❡t❛s✿

❙❡ ❛s r❡t❛s ❢♦r❡♠ ❝♦♥❝♦rr❡♥t❡s ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❞✉❛s s♦❧✉çõ❡s✿

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✶✳✹✳✺ ❯♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ❞♦✐s ♣♦♥t♦s ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛

❡ss❛ ❝✐r❝✉♥❢❡rê♥❝✐❛

❙❡ ♦s ❞♦✐s ♣♦♥t♦s sã♦ ❡①t❡r♥♦s à ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♣♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ❞✉❛s s♦❧✉çõ❡s✿

❙❡ ♦s ❞♦✐s ♣♦♥t♦s sã♦ ✐♥t❡r♥♦s à ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♣♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ❞✉❛s s♦❧✉çõ❡s✿

❙❡ ♦s ❞♦✐s ♣♦♥t♦s sã♦ ✉♠ ✐♥t❡r♥♦ ❡ ♦ ♦✉tr♦ ❡①t❡r♥♦ à ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♥ã♦ ✈✐s✉❛❧✐✲ ③❛♠♦s ♥❡♥❤✉♠❛ s♦❧✉çã♦✿

(18)

✶✳✹✳✻ ❉✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡ ✉♠ ♣♦♥t♦

◆❡st❡ ❝❛s♦ ♦ ♣♦♥t♦ t❡♠ ✈❛r✐❛s ♦♣çõ❡s ❞❡ ❝♦❧♦❝❛çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❛q✉✐ ♦ ❝❛s♦ ❡♠ q✉❡ ❛s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ♥ã♦ sã♦ s❡❝❛♥t❡s ❡ ♥❡♠ t❛♥❣❡♥t❡ ❡ ♦ ♣♦♥t♦ é ❡①t❡r♥♦ ❛s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡ ❝♦♥s❡❣✉✐♠♦s ✈✐s✉❛❧✐③❛r q✉❛tr♦ s♦❧✉çõ❡s✿

✶✳✹✳✼ ❉✉❛s r❡t❛s ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛

◆❡st❡ ❝❛s♦ t❡♠♦s ✈ár✐❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ✉♠❛ ❞❡❧❛s é ❛ ✐❧✉str❛❞❛ ❛❜❛✐①♦✱ ♥❛ q✉❛❧ ❝♦♥s❡❣✉✐♠♦s ✈✐s✉❛❧✐③❛r q✉❛tr♦ s♦❧✉çõ❡s✿

(19)

✶✳✹✳✽ ❉✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡ ✉♠❛ r❡t❛

◆❡st❡ ❝❛s♦ t❛♠❜é♠ t❡♠♦s ✈ár✐❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ✉♠❛ ❞❡❧❛s s❡♠ s♦❧✉çã♦ é ❛ ✐❧✉s✲ tr❛❞❛ ❛❜❛✐①♦✿

✶✳✹✳✾ ❯♠ ♣♦♥t♦✱ ✉♠❛ r❡t❛ ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛

(20)

✶✳✹✳✶✵ ❚rês ❝✐r❝✉♥❢❡rê♥❝✐❛s

◆❡st❡ ❝❛s♦✱ t❛♠❜é♠ ♣♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ✈❛r✐❛s s✐t✉❛çõ❡s✱ t♦❞❛s ❞❡♣❡♥❞❡♥❞♦ ❞❛s ♣♦s✐çõ❡s ❞❛s três ❝✐r❝✉♥❢❡rê♥❝✐❛s✳ ❯♠ ❞♦s ❝❛s♦s✱ q✉❡ ❝♦♠♦ ❢❛❧❛♠♦s ❛❝✐♠❛ s❡rá ♦❜❥❡t♦ ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦✱ é q✉❛♥❞♦ ❛s três ❝✐r❝✉♥❢❡rê♥❝✐❛s sã♦ ❞✐st✐♥t❛s ♥ã♦ s❡❝❛♥t❡s✱ ♥ã♦ t❛♥❣❡♥t❡s ❡ q✉❡ t❡♠ s❡✉s ❝❡♥tr♦s ♥ã♦ ❛❧✐♥❤❛❞♦s✱ ✈❡❥❛ ✜❣✉r❛s✿

❯♠❛ s♦❧✉çã♦ é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ t❛♥❣❡♥t❡ ✐♥t❡r✐♦r♠❡♥t❡ ❛s três ❝✐r❝✉♥❢❡rê♥❝✐❛s✿

❖✉ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ t❛♥❣❡♥t❡ ❡①t❡r✐♦r♠❡♥t❡ ❛s três ❝✐r❝✉♥❢❡rê♥❝✐❛s✿

❖✉tr♦ ❝❛s♦ s❡r✐❛ três ❝✐r❝✉♥❢❡rê♥❝✐❛s t❛♥❣❡♥t❡s ✐♥t❡r✐♦r♠❡♥t❡ ❛ ❞✉❛s ❡ ❡①t❡r✐♦r✲ ♠❡♥t❡ ❛s três ❝✐r❝✉♥❢❡rê♥❝✐❛s✿

(21)

▼❛✐s ✉♠ ❝❛s♦ s❡r✐❛ três ❝✐r❝✉♥❢❡rê♥❝✐❛s t❛♥❣❡♥t❡s ✐♥t❡r✐♦r♠❡♥t❡ ❛ ✉♠❛ ❡ ❡①t❡r✐✲ ♦r♠❡♥t❡ ❛ ❞✉❛s ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s✿

(22)

❈♦♥❝❡✐t♦s ❇ás✐❝♦s

◆❡st❡ ❝❛♣✐t✉❧♦ ✐♥tr♦❞✉③✐r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s q✉❡ ♥♦s ❛❥✉❞❛r❛♠ ❛ ❝♦♠♣r❡❡♥✲ ❞❡r ❛ ❝♦♥str✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ♦♥❞❡ ♦s três ♦❜❥❡t♦s sã♦ ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡①t❡r♥❛s ❡♥tr❡ s✐ ❡ s❡♠ ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦✳ ❖ ❧❡✐t♦r ♠❛✐s ✐♥t❡r❡ss❛❞♦ ♣♦❞❡ ✈❡r ❡♠ ❬✸❪✱ ❬✻❪✱ ❬✺❪ ♠❛✐s s♦❜r❡ ♦ t❡♠❛✳

✷✳✶ ❊❧❡♠❡♥t♦s ❜ás✐❝♦s

❉❡✜♥✐çã♦ ✶ ❯♠ ❝♦♥❥✉♥t♦s ❞❡ ♣♦♥t♦s ❝♦♥st✐t✉✐ ✉♠ ❧✉❣❛r ❣❡♦♠étr✐❝♦ q✉❛♥❞♦ s❛t✐s❢❛③ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣r♦♣r✐❡❞❛❞❡ P ❝♦♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

❛✮ ❚♦❞♦ ♣♦♥t♦ q✉❡ ♣❡rt❡♥❝❡ ❛♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ P❀ ❜✮ ❚♦❞♦ ♣♦♥t♦ q✉❡ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ P ♣❡rt❡♥❝❡ ❛♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✳

❉❡✜♥✐çã♦ ✷ ❈♦♥s✐❞❡r❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ C(O, R) ❡ ✉♠ ♣♦♥t♦P q✉❛❧q✉❡r✳ ❆ ♣♦✲

tê♥❝✐❛ ❞❡ p ❡♠ r❡❧❛çã♦ ❛ C é ❞❛❞❛ ♣♦r P otC(P) = P A.P B✳ ♦♥❞❡ ❆ ❡ ❇ sã♦ ♦s

♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡ ✉♠❛ r❡t❛ s❡❝❛♥t❡ ❛ ❈ q✉❡ ♣❛ss❛ ♣♦r P✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P é t❛♥❣❡♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈ ♥♦ ♣♦♥t♦ ❚ t❡♠♦s ❡♥tã♦ q✉❡

P otC(P) = P T

2✳ ❙❡ ♦ ♣♦♥t♦ P é ✐♥t❡r♥♦ à ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈ ❡♥tã♦

P otC(p) < 0✳

❙❡ ♦ ♣♦♥t♦ P é ❡①t❡r♥♦ à ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈ t♦♠❛♠♦s P otC(p) > 0✳ ❙❡ ♦ ♣♦♥t♦ P

♣❡rt❡♥❝❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈✱ ❡♥tã♦ P otC(p) = 0✳

P❛r❛ ✐❧✉str❛r ♠❡❧❤♦r ♦ q✉❡ ❞❡✜♥✐♠♦s ♦❜s❡r✈❡ ❛s ✜❣✉r❛s ❛❜❛✐①♦✿

(23)

Pr♦♣♦s✐çã♦ ✶ ✿ ❆ ♣♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❛❞❛ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♦✉ s❡❥❛✱ P otC(P) ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ ♣♦♥t♦ P ❡ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛

♣♦r P otC(p) = d

2

−r2 ♦♥❞❡ ❞ é ❛ ❞✐stâ♥❝✐❛ ❞❡ P ❛♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ r é ♦ r❛✐♦ ❞♦ ❝✐r❝✉❧♦✳

❉❡♠♦♥str❛çã♦✿ ✶♦ _{❈❛s♦✿ ❖ ♣♦♥t♦ P é ❡①t❡r✐♦r✳}

❖❜s❡r✈❛♥❞♦ ❛ ✜❣✉r❛ t❡♠♦s✿

P otc(P) = P A·P B

= (P I −IA)(P I +IA) = P I2

−IA2

= (P O2

−OI2

)−(OA2

−OI2

) = P O2

−OA2

= d2 −r2

✷♦ _{❈❛s♦✿ ❖ ♣♦♥t♦ P é ✐♥t❡r✐♦r✳}

P otC(P) = P A·P B·cosπ

P otC(P) = −P A·P B

= −(IA−P I)(P I +IA) = (P I −IA)(P I +IA) = P I2

−IA2

= (P O2

−OI2

)−(OA2

−OI2

) = P O2

−OA2

= d2 −r2

(24)

❉❡✜♥✐çã♦ ✸ ✿ ❉❛❞❛ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s C1 ❡ C2 ♥ã♦ ❝♦♥❝ê♥tr✐❝❛s✱ ❝❤❛♠❛♠♦s ❞❡ ❡✐①♦ r❛❞✐❝❛❧ ❛♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s q✉❡ t❡♠ ✐❣✉❛❧ ♣♦tê♥❝✐❛ ❡♠ r❡❧❛çã♦ ❛ C1 ❡ C2✳

Pr♦♣♦s✐çã♦ ✷ ✿ ❖ ❡✐①♦ r❛❞✐❝❛❧ ❞❡ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s C1 ❡ C2 é ✉♠❛ r❡t❛ ♣❡r♣❡♥✲ ❞✐❝✉❧❛r à r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ❝❡♥tr♦s ❞❡ss❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s✳

❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛sC1 ❡C2 ❞❡ ❝❡♥tr♦sO1 ❡O2 ❡ r❛✐♦s r1 ❡ r2 r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♥s✐❞❡r❡♠♦s t❛♠❜é♠ ▼ ❝♦♠♦ s❡♥❞♦ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦O1O2✱ ✉♠ ♣♦♥t♦ P ❞❡ss❛ r❡t❛ ❡ s✉❛ ♣r♦❥❡çã♦P1 s♦❜r❡ ♦ s❡❣♠❡♥t♦O1O2✳ ❖❜s❡r✈❡ ❛ ✜❣✉r❛ ❛❜❛✐①♦✿

P otC1(P) =P otC2(P)

P O2

1 −r

2

1 =P O

2

2 −r

2 2

P O2

1 −P O

2

2 =r

2

1−r

2 2.

❈♦♥s✐❞❡r❛♥❞♦ ♦s tr✐â♥❣✉❧♦s P M O1 ❡ P M O2 ❡ ❛♣❧✐❝❛♥❞♦ ❛ ❧❡✐ ❞♦s ❝♦ss❡♥♦s t❡♠♦s✿

❉♦ tr✐â♥❣✉❧♦ P M C1✿ P C 2

1 =

O₁O₂

2

+m2

+ 2· O1O2

2 ·m·cosα.

❉♦ tr✐â♥❣✉❧♦ P M C2✿ P C 2

2 =

O₁O₂

2

+m2

−2· O1O2

2 ·m·cosα.

❋❛③❡♥❞♦ ❛ s✉❜tr❛çã♦ ❞❛s ❞✉❛s ❡q✉❛çõ❡s✿

P O2

1 −P O

2

2 = 2·O1O2·m·cosα.

(25)

▼❛s✱

cosα= M P1

m ❡ P O

2

1−P O

2

2 =r

2

1 −r

2 2.

❊♥tã♦✱

r2

1 −r

2

2 = 2·O1·O2·M P1.

❖❜s❡r✈❡ q✉❡✱ r2

1 −r

2

2 é ❝♦♥st❛♥t❡✱ ♣♦rt❛♥t♦ 2·O1O2 ·M P11 t❛♠❜é♠ s❡rá✳ ❉❡st❛ ú❧t✐♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ M P1 é ❝♦♥st❛♥t❡✱ ♥ã♦ ❞❡♣❡♥❞❡♥❞♦ ❞❛s ♣♦s✐çõ❡s ❞❡ P✳ ❧♦❣♦ ♦ ❡✐①♦ r❛❞✐❝❛❧ ❞❡ C1 ❡ C2 é ❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ O1O2 ♣❛ss❛♥❞♦ ♣♦r P ❝✉❥❛ ♣♦s✐çã♦ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ♣♦r✿

M P1 =

r2

1−r

2 2

2·O1O2

.

◆♦t❡ q✉❡ q✉❛♥❞♦ ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s sã♦ t❛♥❣❡♥t❡s ♦✉ s❡❝❛♥t❡s ✜❝❛ ❢á❝✐❧ ❞❡t❡r♠✐♥❛r ♦ ❡✐①♦ r❛❞✐❝❛❧✱ ✈❡❥❛✿

◆♦t❡ q✉❡ ♣❛r❛ ❞❡♠♦♥str❛r ❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ♣r❡❝✐s❛♠♦s ❞❡ ✉♠ ♣♦♥t♦ P q✉❡ ♣❡rt❡♥❝❡ ❛ ❡ss❡ ❡✐①♦✳ P❛r❛ ❡♥❝♦♥tr❛r ❡ss❡ ♣♦♥t♦ P ♣❡rt❡♥❝❡♥t❡ ❛♦ ❡✐①♦ r❛❞✐❝❛❧✱ ❝♦♥s✐❞❡r❛✲s❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ C3 q✉❛❧q✉❡r ❡ s❡❝❛♥t❡ ❛ C1 ❡ C2✳ ❙✉♣♦♥❤❛ q✉❡

P1, Q1 ∈C3∩C1 ❡ P2, Q2 ∈C3∩C2✱ ♦❜s❡r✈❡ q✉❡ ♦ ❡✐①♦ r❛❞✐❝❛❧✱ ER1 ❞❡C1 ❡C3 é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P1 ❡ Q1 ❡ q✉❡ ♦ ❡✐①♦ r❛❞✐❝❛❧✱ ER2❞❡ C2 ❡ C3 é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P2 ❡ Q2✳ ▲♦❣♦✱ ♦ ♣♦♥t♦ P ∈ER1∩ER2 ♣❡rt❡♥❝❡ ❛♦ ❡✐①♦ r❛❞✐❝❛❧ ❞❡ C1 ❡C2✳

(26)

Pr♦♣♦s✐çã♦ ✸ ❙❡ ❞❡ ✉♠ ♣♦♥t♦ P q✉❛❧q✉❡r ❞♦ ❡✐①♦ r❛❞✐❝❛❧ ❞❡ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛ ❢♦r tr❛ç❛❞❛s t❛♥❣❡♥t❡s ❛ ❡ss❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s✱ ❡♥tã♦ ♦s s❡❣♠❡♥t♦s t❛♥❣❡♥t❡s ♣♦ss✉❡♠ ❛s ♠❡s♠❛s ♠❡❞✐❞❛s✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s C1 ❡ C2 ❡ ♦ ♣♦♥t♦ P ❞♦ ❡✐①♦ r❛❞✐❝❛❧ ❱❡❥❛ ✜❣✉r❛ ❛❜❛✐①♦✿

❈♦♥s✐❞❡r❡ ♦ s❡❣♠❡♥t♦ P▼ t❛♥❣❡♥t❡ ❛ C1 ❡ ♦ s❡❣♠❡♥t♦ P◆ t❛♥❣❡♥t❡ ❛ C2✳

❧♦❣♦

P otC₁(M) =P otC₂(N)

(P M)2

= (P N)2

P M =P N

(27)

❉❡✜♥✐çã♦ ✹ ✿ ❉❛❞♦s ✉♠ ♥ú♠❡r♦ ❦ ❡ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ❖ ❡ P✱ ❝❤❛♠❛r❡♠♦s ♦ ♣♦♥t♦ P′_{✱ t❛❧ q✉❡} _OP′ ₌_k.OP_{✱ ❤♦♠♦tét✐❝♦ ❞♦ ♣♦♥t♦ P✳ ❋✐①❛❞♦s ✉♠ ❡s❝❛❧❛r ❦ ❡ ✉♠}

♣♦♥t♦ ❖ ❛ tr❛♥s❢♦r♠❛çã♦ ❣❡♦♠étr✐❝❛ q✉❡ ❧❡✈❛ ✉♠ ♣♦♥t♦ P ♥♦ ♣♦♥t♦ P′ _{é ❝❤❛♠❛❞❛}

❤♦♠♦t❡t✐❛ ❞✐r❡t❛ s❡ ❦ ❃ ✵ ❡ ❞❡ ❤♦♠♦t❡t✐❛ ✐♥✈❡rs❛ s❡ ❦ ❁ ✵✱ ❡ ♦ ♣♦♥t♦ ❖ ✜①❛❞♦ é ❝❤❛♠❛❞♦ ❝❡♥tr♦ ❞❡ ❤♦♠♦t❡t✐❛✳

❉❡✜♥✐çã♦ ✺ ✿ ❯♠❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r ❞♦✐s ❝❡♥tr♦s ❞❡ ❤♦♠♦t❡t✐❛ é ❝❤❛♠❛❞❛ ❞❡ ❡✐①♦ ❞❡ ❤♦♠♦t❡t✐❛s✱ s❡ ❡ss❡s ❞♦✐s ❝❡♥tr♦s sã♦ ❞❡ ❤♦♠♦t❡t✐❛ ❞✐r❡t❛ ❝❤❛♠❛♠♦s ❞❡ ❡✐①♦ ❞❡ ❤♦♠♦t❡t✐❛s ❞✐r❡t❛s✱ s❡ sã♦ ❞♦✐s ❝❡♥tr♦s ❞❡ ❤♦♠♦t❡t✐❛ ✐♥✈❡rs❛ ❝❤❛♠❛♠♦s ❞❡ ❡✐①♦ ❞❡ ❤♦♠♦t❡t✐❛s ✐♥✈❡rs❛s✳

❉❡✜♥✐çã♦ ✻ ❉✐③❡♠♦s q✉❡ ✉♠ ♣♦♥t♦ P é ♦ ❝❡♥tr♦ ❞❡ ❤♦♠♦t❡t✐❛ ❞❡ ❞✉❛s ❝✐r❝✉♥❢❡✲ rê♥❝✐❛s s❡ ✉♠❛ ❞❡❧❛s é ❧❡✈❛❞❛ ♥❛ ♦✉tr❛ ♣♦r ✉♠❛ ❤♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ P

Pr♦♣♦s✐çã♦ ✹ ✿ ❯♠❛ ❤♦♠♦t❡t✐❛ tr❛♥s❢♦r♠❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡♠ ✉♠❛ ❝✐r❝✉♥❢❡✲ rê♥❝✐❛✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ P ✉♠ ♣♦♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈ ❝♦♠ ❝❡♥tr♦ ❡♠ ✉♠ ♣♦♥t♦ ❆✳ ❙❡❥❛♠ P′ _❡ _A′ _{♦s ❤♦♠♦tét✐❝♦s ❞♦s ♣♦♥t♦s P ❡ ❆ ♣♦r ✉♠❛ ❤♦♠♦t❡t✐❛ ❞❡}

❝❡♥tr♦ ❖ ❡ r❛③ã♦ ❦✳ ❈♦♠♦ ❛ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ P❆ é ❝♦♥st❛♥t❡✱ s❡ ♦ ♣♦♥t♦ P ❣✐r❛r ❡♠ t♦r♥♦ ❞❡ ❆ ❞❡s❝r❡✈❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈✱ ♦ ♣♦♥t♦P′ _{❞❡s❝r❡✈❡rá ✉♠ ❝✐r❝✉♥❢❡rê♥❝✐❛}

C′_{❞❡ ❝❡♥tr♦} _A′ _{❡ r❛✐♦} _A′_P′_✳

Pr♦♣♦s✐çã♦ ✺ ✿ ❉✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s α ❡ β ❝♦♠ ❝❡♥tr♦s ❡ r❛✐♦s ❞✐st✐♥t♦s ❝♦♠♦

♠♦str❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦ ♣♦ss✉❡♠ ❞♦✐s ❝❡♥tr♦s ❞❡ ❤♦♠♦t❡t✐❛✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ ♦s ♣♦♥t♦s A1 ❡ A2 ♦s ❝❡♥tr♦s ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s α ❡

β✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛ ♦ s❡❣♠❡♥t♦ A1B1 ✉♠ r❛✐♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ α ♣❛r❛❧❡❧♦ ❛ ✉♠ r❛✐♦ A2B2 ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ β✳ ❈♦♠♦ A1B1 ❡ A2B2 sã♦ s❡❣♠❡♥t♦ ❞✐s♣♦st♦s

(28)

♣❛r❛❧❡❧❛♠❡♥t❡✱ ❡❧❡s sã♦ ❤♦♠♦tét✐❝♦s✳ ❙❡❥❛ k = A2B2

A1B1

s✉❛ r❛③ã♦ ❡ O1 s❡✉ ❝❡♥tr♦✳ ❙❡ ♦s s❡❣♠❡♥t♦s A1B3 ❡ A2B2 sã♦ ❞✐s♣♦st♦s ♣❛r❛❧❡❧❛♠❡♥t❡✱ ♠❛s ♣♦ss✉❡♠ s❡♥t✐❞♦s ❝♦♥trár✐♦s ❡❧❡s sã♦ ❤♦♠♦tét✐❝❛s ♣♦r ✉♠❛ r❛③ã♦ −k = A2B2

A1B3

❝✉❥♦ ❝❡♥tr♦ é ♦ ♣♦♥t♦

O2✳

❉❡✜♥✐çã♦ ✼ ✿ ❉✐③❡♠♦s q✉❡ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s sã♦ ♦rt♦❣♦♥❛✐s s❡ s❡✉s r❛✐♦s sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ♥♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦✳

❉❡✜♥✐çã♦ ✽ ✿ ❙❡❥❛♠ ❈ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ❖ ❡ P ✉♠ ♣♦♥t♦ ❡①t❡r♥♦ ❛ ❈✳ ❙❡❥❛ ❆P ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♣❛ss❛♥❞♦ ♣♦r P ❡ t❛♥❣❡♥t❡ ❛ ❈ ♥♦ ♣♦♥t♦ ❆✳ ❖ ♣♦♥t♦ ♦❜t✐❞♦ ♣❡❧❛ ✐♥t❡rs❡❝çã♦ ❞❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ s❡❣♠❡♥t♦ ❖P q✉❡ ♣❛ss❛ ♣♦r ❆ é ❝❤❛♠❛❞♦ ✐♥✈❡rs♦ ❞♦ ♣♦♥t♦ P✱ ❞❡♥♦t❛r❡♠♦s ❡ss❡ ♣♦♥t♦ ♣♦r P1 ✈❡❥❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✳ ❙❡ ♦ ♣♦♥t♦ P ❢♦r ✐♥t❡r♥♦ ❛ ❈✱ ❝♦♠ P 6= O✱ ❡♥tã♦ ♣❡❣❛♠♦s ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛

♣❡r♣❡♥❞✐❝✉❧❛r ❛ ❖P ♣❛ss❛♥❞♦ ♣♦r P q✉❡ ✐♥t❡r❝❡♣t❛ ❈ ♥✉♠ ♣♦♥t♦ ❇✳ ◆❡st❡ ❝❛s♦ P1 é ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛ ❈ ♥♦ ♣♦♥t♦ ❇ ❝♦♠ ♦ s❡❣♠❡♥t♦ ❖P✱ ♦❜s❡r✈❡ ❛ ✜❣✉r❛ ❛❜❛✐①♦✳ ❋✐①❛❞❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❛ tr❛♥s❢♦r♠❛çã♦ ❣❡♦♠étr✐❝❛ q✉❡ ❧❡✈❛ ✉♠ ♣♦♥t♦ P ♥♦ ♣♦♥t♦ P1 é ❝❤❛♠❛❞❛ ✐♥✈❡rsã♦✱ ❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✜①❛❞❛ é ❞✐t❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦✳ ❙❡ P ∈C t♦♠❛♠♦s P =P1✳

(29)

❯♠❛ ❝♦♥str✉çã♦ ●❡♦♠étr✐❝❛ ❞♦

Pr♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦

❈♦♠ ♦s ❡❧❡♠❡♥t♦s ❛♣r❡s❡♥t❛❞♦s ♥♦ ❝❛♣✐t✉❧♦ ✷✱ t❡r❡♠♦s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥str✉✐r ♥❡st❡ ❝❛♣✐t✉❧♦ ❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ❡♠ q✉❡ ♦s três ❡❧❡♠❡♥t♦s sã♦ ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡①t❡r♥❛s ❡♠ s✐✱ s❡♠ ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❡ ❝♦♠ ♦s ❝❡♥tr♦s ♥ã♦ ❛❧✐♥❤❛❞♦s✱ ♥♦s s❡r✈✐✉ ❝♦♠♦ ❜❛s❡ ♣❛r❛ ❡ss❛ ❝♦♥str✉çã♦ ♦s t❡①t♦s ✈❡r ❡♠ ❬✺❪✱ ❬✻❪ ❡ ❬✼❪ ✈❡r ✜❣✉r❛ ❛❜❛✐①♦✿

❖❜s❡r✈❡ q✉❡ ❛ ♣♦tê♥❝✐❛ ❞♦ ❝❡♥tr♦ r❛❞✐❛❧ P✱ ❞❛s três ❝✐r❝✉♥❢❡rê♥❝✐❛sC1 ✱C2 ❡ C3✱ é ♣♦s✐t✐✈❛✱ ❧♦❣♦ ❡❧❡ é ❝❡♥tr♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈ ❞❡ ❝❡♥tr♦ ❖ ❡ r❛✐♦ r✱ ♦rt♦❣♦♥❛❧ às ❝✐r❝✉♥❢❡rê♥❝✐❛s C1 ✱C2 ❡ C3✳ ❙❡ t♦♠❛r♠♦s ❈ ❝♦♠♦ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦✱ ❞❡✜♥✐çã♦ ✼ ❞♦ ❝❛♣✳ ✷✱ ❡♥tã♦ C′ ₌_{C, C}′

1 =C1✱C2′ =C2 ❡ C3′ =C3✳

(30)

❙✉♣♦♥❤❛ q✉❡ ❙ s❡❥❛ ✉♠❛ ❞❛s s♦❧✉çõ❡s ❞♦ Pr♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ❡♥✈♦❧✈❡♥❞♦ ❛s ❝✐r✲ ❝✉♥❢❡rê♥❝✐❛s C1, C2 ❡ C3 ❡ ❝♦♥s✐❞❡r❡ ❈ ❝♦♠♦ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦✱ ❝♦♠♦ ❢♦✐ ♦❜s❡r✈❛❞♦ ❛❝✐♠❛✱ ❛ ✐♥✈❡rsã♦ ♠❛♥té♠ C1 ✱C2 ❡ C3✱ ❧♦❣♦ q✉❛tr♦ s♦❧✉çõ❡s ❞❡ss❡ ♣r♦✲ ❜❧❡♠❛ sã♦ ❧❡✈❛❞❛s ♣❡❧❛ ✐♥✈❡rsã♦ ♥❛s ♦✉tr❛s q✉❛tr♦✱ ♦✉ s❡❥❛✱ ❛s ♦✐t♦ s♦❧✉çõ❡s sã♦ ❞✉❛s ❛ ❞✉❛s ✐♥✈❡rs❛s✳ ❈♦♠♦ t❡♠♦s três ❝✐r❝✉♥❢❡rê♥❝✐❛s ❞❡ ❝❡♥tr♦s ❡ r❛✐♦s ❞✐st✐♥t♦s✱ ❡♥tã♦ ❡①✐st❡♠ s❡✐s ❝❡♥tr♦s ❞❡ ❤♦♠♦t❡t✐❛✱ ♦s q✉❛✐s ❞❡t❡r♠✐♥❛♠ q✉❛tr♦ ❡✐①♦s ❞❡ ❤♦♠♦t❡t✐❛✳ ❯♠ ❞❡❧❡s é ✉♠ ❡✐①♦ ❞❡ ❤♦♠♦t❡t✐❛ ❞✐r❡t❛ ❡ ♦s ♦✉tr♦s três sã♦ ❡✐①♦s ❞❡ ❤♦♠♦t❡t✐❛s ✐♥✈❡rs❛s✳

❉❡✜♥✐çã♦ ✾ ❋✐①❛❞❛s ✉♠❛ r❡t❛ r ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈✱ ❝❤❛♠❛♠♦s ❞❡ ❢❡✐①❡ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s rC ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s q✉❡ ❛❞♠✐t❡♠✱ ❝♦♠ ❈✱ ❛ r❡t❛ r

❝♦♠♦ ❡✐①♦ r❛❞✐❝❛❧✳

▲❡♠❛ ✾✳✶ ❉✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ♥ã♦ ❝♦♥❝ê♥tr✐❝❛s C1 ❡ C2 ❞❡t❡r♠✐♥❛♠ ❥✉♥t❛♠❡♥t❡ ❝♦♠ s❡✉ ❡✐①♦ r❛❞✐❝❛❧ ✉♠ ❢❡✐①❡ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s✱ ♦ q✉❛❧ ❞❡♥♦t❛r❡♠♦s ♣♦r C1C2✳

❉❡♠♦♥str❛çã♦✿ ❉✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ♥ã♦ ❝♦♥❝ê♥tr✐❝❛s C1 ❡ C2 ❞❡t❡r♠✐♥❛♠ ✉♠ ❡✐①♦ r❛❞✐❝❛❧ ER✱ ♣♦rt❛♥t♦✱ ❞❡t❡r♠✐♥❛♠ ✉♠ ❢❡✐①❡ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛sC1C2✳

▲❡♠❛ ✾✳✷ ❙❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈ ✐♥t❡r❝❡♣t❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ α ❡ s✉❛ ✐♥✈❡rs❛ α′ _{s♦❜ ✉♠ ♠❡s♠♦ â♥❣✉❧♦✱ ❡♥tã♦ ❈ é ♦rt♦❣♦♥❛❧ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦✳}

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ σ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦✳ ❙❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈ ♥ã♦

✐♥t❡r❝❡♣t❛ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ α ❡ α′ _{s♦❜ ✉♠ ♠❡s♠♦ â♥❣✉❧♦✱ ❡♥tã♦} _C ₆₌ _C′_{✳ ▲♦❣♦✱ ❈}

♥ã♦ é ♦rt♦❣♦♥❛❧ ❛ σ ✱ ♣♦✐s s❡ ❢♦ss❡ C =C′_✳

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Pr♦♣♦s✐çã♦ ✻ ❈♦♥s✐❞❡r❡ três ❝✐r❝✉♥❢❡rê♥❝✐❛s C1✱C2 ❡ C✳ ❙❡ C ✐♥✈❡rt❡ C1 ❡♠ C2✱ ❡♥tã♦ C ♣❡rt❡♥❝❡ ❛♦ ❢❡✐①❡ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s C1C2✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ❈ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ✐♥✈❡rt❡ C1 ❡♠ C2✳ ❙❡❥❛ ER1 ♦ ❡✐①♦ r❛❞✐❝❛❧ ❡♥tr❡ C1 ❡ C2✳ ❙❡❥❛ O1 ❡ O2✱ ♦s ❝❡♥tr♦s ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s C1 ❡ C2 r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛ α ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♦rt♦❣♦♥❛❧ ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s C1 ❡ C2✳ ❖ ❡✐①♦ r❛❞✐❝❛❧ ❊❘✶ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ α ❡ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ r❡t❛ O1O2✳ ❈♦♠♦ ✐♥t❡r❝❡♣t❛C1 ❡ C2 s♦❜ ♦ ♠❡s♠♦ â♥❣✉❧♦✱ ♣❡❧♦ ❧❡♠❛ ✸✳✷✱αé ♦rt♦❣♦♥❛❧ à ❈✳ ❙❡❥❛ ER2 ♦ ❡✐①♦ r❛❞✐❝❛❧ ❡♥tr❡ ❈ ❡ C1✳ ❈♦♠♦ ♦ ❡✐①♦ r❛❞✐❝❛❧ ER2 ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ α ❡ é ♣❡r♣❡♥❞✐❝✉❧❛r à r❡t❛ O1O2✱ ER1 =ER2✳

▲❡♠❛ ✾✳✸ ❙❡❥❛♠ C1✱ C2 ❡ C3 três ❝✐r❝✉♥❢❡rê♥❝✐❛s ♥ã♦ ❝♦♥❝ê♥tr✐❝❛s✳ ❊♥tã♦ ♦s ❡✐①♦s r❛❞✐❝❛✐s ❞❡ss❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s t♦♠❛❞♦s ❞♦✐s ❛ ❞♦✐s sã♦ ♣❛r❛❧❡❧♦s ♦✉ ❝♦♥❝♦rr❡♥t❡s✳ ◗✉❛♥❞♦ sã♦ ❝♦♥❝♦rr❡♥t❡s✱ ❛ ✐♥t❡rs❡çã♦ ❞❡ss❡s ❡✐①♦s é ✉♠ ú♥✐❝♦ ♣♦♥t♦ P ❞♦ ♣❧❛♥♦ t❛❧

q✉❡ P otC1(P) =P otC2(P) =P otC3(P)✳ ❈❤❛♠❛♠♦s ❡st❡ ♣♦♥t♦P ❞❡ ❝❡♥tr♦ r❛❞✐❝❛❧ ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s C1✱C2 ❡ C3✳

❉❡♠♦♥str❛çã♦✿ ❙❡ t♦♠❛r♠♦s ♦s três ❝❡♥tr♦s ❞❡ C1, C2 ❡ C3 ❛❧✐♥❤❛❞♦s✱ ❡♥tã♦ ♦s ❡✐①♦s r❛❞✐❝❛✐s sã♦ ♣❛r❛❧❡❧♦s✱ ✈❡❥❛ ✜❣✉r❛✿

❙❡ t♦♠❛r♠♦s ♦s três ❝❡♥tr♦s ❞❡C1, C2 ❡ C3 ♥ã♦ ❛❧✐♥❤❛❞♦s✱ t❡r❡♠♦s✿

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❊✐①♦ r❛❞✐❝❛❧ ER1 ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s C1 ❡C2✿

❊✐①♦ r❛❞✐❝❛❧ ER2 ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s C1 ❡C3✿

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❋❛③❡♥❞♦ ❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ ♦s ❡✐①♦s r❛❞✐❝❛✐sER1 ❡ER2✱ ❡♥❝♦♥tr❛♠♦s ♦ ♣♦♥t♦ P✱ ✈❡❥❛ ✜❣✉r❛✿

◆♦t❡ q✉❡ s❡ ♦s ❝❡♥tr♦s ❞❛s três ❝✐r❝✉♥❢❡rê♥❝✐❛s ♥ã♦ ❡stã♦ ❛❧✐♥❤❛❞♦s ❡♥tã♦ ❡ss❡ ♣♦♥t♦ P é ✉♠ ♣♦♥t♦ ❜❡♠ ❞❡✜♥✐❞♦✳

❈♦♠ ♠❛✐s ❡ss❡s ❧❡♠❛s✱ ♣r♦♣♦s✐çõ❡s ❡ ❞❡✜♥✐çõ❡s✱ ✈❛♠♦s ❝♦♠❡ç❛r ❛s ❝♦♥str✉çõ❡s ❝♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ❞❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ❡♠ q✉❡ ♦s três ♦❜❥❡✲ t♦s sã♦ três ❝✐r❝✉♥❢❡rê♥❝✐❛s ♥ã♦ t❛♥❣❡♥t❡s✱ ♥ã♦ s❡❝❛♥t❡s ❡ ❝♦♠ ❝❡♥tr♦s ♥ã♦ ❛❧✐♥❤❛❞♦s✳

✶♦ _{P❛ss♦✿ ❊♥❝♦♥tr❛♠♦s ♦ ❝❡♥tr♦ r❛❞✐❝❛❧ ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s}_C

1, C2 ❡C3 q✉❡ ❝❤❛♠❛✲ r❡♠♦s ❞❡ P✳ ❈♦♠♦ ♠♦str❛❞♦ ❡♠ ❧❡♠❛ ✾✳✸✱ ❡ss❡ ♣♦♥t♦ ♣♦❞❡ s❡r ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ ♦s ❡✐①♦s r❛❞✐❝❛✐s ER1 ❞❡ C1 ❡ C2 ❡ ER2 ❞❡ C1 ❡ C3 ❡ ❞❡♣♦✐s ❝♦♥str✉✐r ❛ ❝✐r✲ ❝✉♥❢❡rê♥❝✐❛ ❝❡♥tr❛❞❛ ❡♠ P ❡ ♦rt♦❣♦♥❛❧ ❛ C1, C2 ❡C3✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✿

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✷♦ _{P❛ss♦✿ ❈♦♥str✉✐r❡♠♦s ❛❣♦r❛ ♦s ❝❡♥tr♦s ❞❡ ❤♦♠♦t❡t✐❛ ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s} _C₁_✱

C2 ❡ C3✱ q✉❡ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣r♦♣✳✺ ❞♦ ❝❛♣ ✷✱ ❡♥❝♦♥tr❛r❡♠♦s s❡✐s ❝❡♥tr♦s ❞❡ ❤♦♠♦✲ t❡t✐❛✳

Pr✐♠❡✐r❛♠❡♥t❡✱ ❝♦♠❡ç❛♠♦s ❝♦♥str✉✐♥❞♦ ♦s ❝❡♥tr♦s ❞❡ ❤♦♠♦t❡t✐❛ ❞❡ C1 ❡ C2✳

P❛r❛ ✐ss♦ ❝♦♥str✉í♠♦s ❛ r❡t❛ t✱ q✉❡ ♣❛ss❛ ♣♦r O1 ❡ O2✱ ✉♠ r❛✐♦ r1 ❞❡ C1✱ ✉♠ r❛✐♦ r2 ❞❡ C2 ❝♦♠ ❡ss❡s r❛✐♦s ♣❛r❛❧❡❧♦s ❡♥tr❡ s✐✳

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❈♦♥str✉í♠♦s ❛❣♦r❛ ✉♠❛ r❡t❛ s ♣❛ss❛♥❞♦ ♣♦rO2 ❡ ♣❡❧♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡

r1 ❡ C2 q✉❡ ❝❤❛♠❛♠♦s ❞❡ H1✱ ❡ss❛ r❡t❛ ✈❛✐ ✐♥t❡rs❡❝t❛r t❛♠❜é♠ C2 ❡♠ ✉♠ ♦✉tr♦ ♣♦♥t♦ q✉❡ ❞❡♥♦♠✐♥❛♠♦s ❞❡ H2✳

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❋✐♥❛❧♠❡♥t❡ ❝♦♥str✉í♠♦s ❛s r❡t❛s u1✱ ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ H1 ❡ ♣❡❧♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ C1 ❡ r1 ❡ u2 ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ H2 ❡ ♣❡❧♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ C1 ❡ r1✳

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❉❛í✱ ♦ ❝❡♥tr♦ ❞❡ ❤♦♠♦t❡t✐❛ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛C1s❡rá ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ ❛ r❡t❛ t ❡ ❛ r❡t❛u2 q✉❡ ❞❡♥♦♠✐♥❛r❡♠♦s ♣♦r ❊ ❡ ♦ ❝❡♥tr♦ ❞❡ ❤♦♠♦t❡t✐❛ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛

C2s❡rá ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ ❛s r❡t❛s t ❡u1 ❞❡ ❞❡♥♦♠✐♥❛r❡♠♦s ♣♦r ❋✱ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✿

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❆♥❛❧♦❣❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❝♦♥str✉✐r ♦s ❝❡♥tr♦s ❞❡ ❤♦♠♦t❡t✐❛ ❞❡ C1 ❡ C3✱ q✉❡ ❞❡✲ ♥♦♠✐♥❛r❡♠♦s ❞❡ ♣♦♥t♦s ● ❡ ❍✳

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❉❛ ♠❡s♠❛ ❢♦r♠❛ ♦s ❝❡♥tr♦s ❞❡ ❤♦♠♦t❡t✐❛ ❞❡ C2 ❡ C3 q✉❡ ❞❡♥♦♠✐♥❛r❡♠♦s ❞❡ ♣♦♥t♦s ■ ❡ ❏✳

(40)

❊♥❝♦♥tr❛♠♦s ❡♥tã♦✱ t♦❞♦s ♦s ❝❡♥tr♦s ❞❡ ❤♦♠♦t❡t✐❛ ♣r♦❝✉r❛❞♦s✳

(41)

❖ ♣ró①✐♠♦ ♣❛ss♦ ❛❣♦r❛ s❡rá ❝♦♥str✉✐r ❞✉❛s s♦❧✉çõ❡s ❡ ♣❛r❛ ✐ss♦ ✈❛♠♦s ❡s❝♦❧❤❡r ♦ ❡✐①♦ r❛❞✐❝❛❧ ❋● q✉❡ ❞❡♥♦♠✐♥❛r❡♠♦s ❞❡ r❡t❛ ❜✱ ♣❛r❛ ❝♦♥str✉✐r ❛s ♦✉tr❛s s♦❧✉çõ❡s ❜❛st❛ ❢❛③❡r ♦s ♠❡s♠♦s ♣❛ss♦s ✉t✐❧✐③❛♥❞♦ ♦✉tr♦ ❡✐①♦✳ ❚❡♠♦s q✉❡ t♦❞❛s ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❞♦ ❢❡✐①❡ bC ♣❛ss❛♠ ♣❡❧♦s ♣♦♥t♦s ❞❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ ❛ r❡t❛ ❜ ❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❈✱

✈❛♠♦s ❝❤❛♠❛r ❡ss❡s ♣♦♥t♦s ❞❡ ❆ ❡ ❇✳

(42)

❖❜s❡r✈❡ ❛❣♦r❛ q✉❡ ❛ s♦❧✉çã♦ ❞♦ ♥♦ss♦ ♣r♦❜❧❡♠❛ s❡ r❡s✉♠❡ ❛ ❡♥❝♦♥tr❛r ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ♣❛ss❛ ♣♦r ❆ ❡ ❇ ❡ é t❛♥❣❡♥t❡ ❛ ✉♠❛ ❞❛s três ❝✐r❝✉♥❢❡rê♥❝✐❛s✳

P❛r❛ ❝♦♥str✉✐r ❡ss❛s ❞✉❛s s♦❧✉çõ❡s ♣r❡❝✐s❛♠♦s ❡♥❝♦♥tr❛r ♣♦♥t♦s q✉❡ ❡st❡❥❛♠ s♦❜r❡ ♦ ❡✐①♦ r❛❞✐❝❛❧ F G ❡ q✉❡ t❡♥❤❛♠ ❛ ♠❡s♠❛ ♣♦tê♥❝✐❛ ♣❛r❛ C1, C2, C3 ❡ ♣❛r❛ ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❞♦ ❢❡✐①❡✳ ❊s❝♦❧❤❛ ❡♥tã♦ ❛ ❝♦r❞❛ ❢♦r♠❛❞❛ ♣❡❧♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡C1 ❝♦♠ C ❡ ❞❡♣♦✐s✱ ♣r♦❧♦♥❣❛✲❧❛ ♣❛r❛ ♦❜t❡r ✉♠ ♣♦♥t♦ ◆ ❞♦ ❡✐①♦ r❛❞✐❝❛❧F G❝♦♠♦ ✐❧✉str❛ ❛ ✜❣✉r❛✿

(43)

❯s❛♠♦s ❛❣♦r❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❛✉①✐❧✐❛r C4 ❝♦♠ r❛✐♦ ✐❣✉❛❧ ❛ ♠❡t❛❞❡ ❞♦ s❡❣✲ ♠❡♥t♦ O1N ♦♥❞❡ ❡♥❝♦♥tr❛♠♦s ♦s ♣♦♥t♦s T1 ❡ T2✳ ❊ss❡s ♣♦♥t♦s T1 ❡ T2 sã♦ ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛✱ ♣♦✐s ♦s tr✐â♥❣✉❧♦s O1T1N ❡ O1T2N sã♦ ✐♥s❝r✐t♦s ❡♠ ✉♠❛ s❡♠✐ ❝✐r❝✉♥✲ ❢❡rê♥❝✐❛✳

(44)

❊♥❝♦♥tr❛♠♦s ❡♥tã♦ ♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ C1✱ q✉❡ ❝♦♥té♠ ❛ ❝♦r❞❛ ❡s❝♦❧❤✐❞❛✱ ❡ ❛ss✐♠ ♣♦❞❡♠♦s ❝♦♥str✉✐r ❛s ❞✉❛s s♦❧✉çõ❡sS1q✉❡ é ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ❆✱ ❇ ❡ T1 ❡S2 ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ❆✱ ❇ ❡ T2✳

(45)

❯♠❛ s♦❧✉çã♦ ❛❧❣é❜r✐❝❛ ❞♦ Pr♦❜❧❡♠❛

❞❡ ❆♣♦❧ô♥✐♦

❆ ❝♦♥str✉çã♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ♠♦str❛r ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r t❡♠ ♠✉✐t❛ ✉t✐❧✐❞❛❞❡✱ ♠❛s s❡ t✐✈❡r♠♦s ✐♥t❡r❡ss❛❞♦s ♥♦ ❝❡♥tr♦ ❡ ♥♦ r❛✐♦ ❞❛s s♦❧✉çõ❡s ❡❧❛ s❡ ♠♦str❛ ♠✉✐t♦ ✈❛❣❛✳ P❛r❛ ❡♥❝♦♥tr❛r ❡ss❡ ❝❡♥tr♦ ❡ ❡ss❡ r❛✐♦ t❡♠♦s q✉❡ ♦❧❤❛r ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❢♦r♠❛ ♠❛✐s ❛❧❣é❜r✐❝❛✳ ❖❜s❡r✈❡ ❛♥❛❧✐t✐❝❛♠❡♥t❡ q✉❡ ♣♦❞❡♠♦s ❞❡s❝r❡✈❡r ❝❛❞❛ ❝✐r❝✉♥✲ ❢❡rê♥❝✐❛ ♣❡❧❛ ❛ss♦❝✐❛çã♦✿

Ω :C3 _→_C ₌_C _⊂C2_/❈ é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛

c1 = (x1, y1, R1)→C1 : (X−x1) 2

+ (Y −y1) 2

=R12

❚❡♠♦s q✉❡✱ Ω ♣♦❞❡ s❡r ❞❡✜♥✐❞❛ t❛♠❜é♠ q✉❛♥❞♦ R1 = 0✱ ♥❡st❡ ❝❛s♦ ❡st❛r❡♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡❣❡♥❡r❛❞❛✱ ♦✉ s❡❥❛ ✉♠ ♣♦♥t♦✳

❖❜s❡r✈❡ ❛s ❝♦♥❞✐çõ❡s ♣❛r❛ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s s❡r❡♠ t❛♥❣❡♥t❡s

❙❡❥❛♠ ❞❛❞❛s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s C1 ❡ C2 ❝♦♠ ❝❡♥tr♦s O1 = (x1, y1) ❡ O2 =

(x2, y2)❡ r❛✐♦s R1 ❡ R2 r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ss❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s sã♦✿

■ ✲ ❚❛♥❣❡♥t❡s ❡①t❡r♥❛♠❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❝❡♥tr♦s ❢♦r ✐❣✉❛❧ ❛ s♦♠❛ ❞❡ s❡✉s r❛✐♦s✱ ♦✉ s❡❥❛✿

d(O1, O2) = R1+R2

■■ ✲ ❚❛♥❣❡♥t❡s ✐♥t❡r♥❛♠❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❝❡♥tr♦s ❢♦r ✐❣✉❛❧ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ s❡✉s r❛✐♦s✱ ♦✉ s❡❥❛✿

d(O1, O2) =R1−R2 ♦✉ d(O1, O2) =R2−R1

(46)

❖❜s❡r✈❡ t❛♠❜é♠ q✉❡ ❞❛❞❛s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❞❡ ❡q✉❛çõ❡s✿

C1 : (X−x1) 2

+ (Y −y1) 2

=R2

1 ❡ C2 : (X−x2)

2

+ (Y −y2) 2

=R2 2.

❖s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ C1 ❡ C2 sã♦ ❞❛❞♦s ♣❡❧❛ r❡s♦❧✉çã♦ ❞♦ s✐st❡♠❛✿

(X−x1) 2

+ (Y −y1) 2

= R2

1 (I)

(X−x2) 2

+ (Y −y2) 2

= R2

2 (II)

❉❡s❡♥✈♦❧✈❡♥❞♦ ❛s ♣♦tê♥❝✐❛s ❞❛s ❞✉❛s ❡q✉❛çõ❡s✱ t❡♠♦s✿

(I) X2

+Y2

−2x1X−2y1Y + (x 2

1+y

2

1−R

2 1) = 0

(II) X2

+Y2

−2x2X−2y2Y + (x 2

2+y

2

2 −R

2 2) = 0 ❙✉❜tr❛✐♥❞♦ ❛ ❡q✉❛çã♦ (I) ❞❛ ❡q✉❛çã♦ (II)✱ ❖❜t❡♠♦s ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ r✿

r: 2(x1−x2)X+ 2(y1−y2)Y + (x 2

2−x

2

1+y

2

2−y

2

1 +R

2

1−R

2 2) = 0.

❊♥tã♦ ♣♦❞❡♠♦s t❛♠❜é♠ ❡♥❝♦♥tr❛r ♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ C1 ❡ C2 ♣❡❧❛ r❡s♦✲ ❧✉çã♦ ❞♦ s✐st❡♠❛✿

(X−x1) 2

+ (Y −y1) 2

= R2 1

r : 2(x1−x2)X + 2(y1−y2)Y + (x 2

2−x

2

1+y

2

2 −y

2

1 +R

2

1−R

2

2) = 0

❖❜s❡r✈❡ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✿