# INVERSÃO GEOMÉTRICA APLICADA À RESOLUÇÃO DOS PROBLEMAS DE APOLÔNIO

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P♦r

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## ❆♣♦❧ô♥✐♦

♣♦r

### ❈r✐st✐❛♥♦ ❇❡♥❡✈✐❞❡s ❞❡ ❙♦✉s❛

s♦❜ ♦r✐❡♥t❛çã♦ ❞❛

### ✳ ❊❧✐s❛♥❞r❛ ❞❡ ❋át✐♠❛ ●❧♦ss ❞❡ ▼♦r❛❡s

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

❙❡t❡♠❜r♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇

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❆ ❉❡✉s✱ ♠❡✉ ú♥✐❝♦ ❡ s✉✜❝✐❡♥t❡ ❝♦♠♣❛♥❤❡✐r♦ ♥❛s ❧♦♥❣❛s ♠❛❞r✉❣❛❞❛s ❡♠ q✉❡ ♣❛ss❡✐ ❞❡❞✐❝❛❞♦ ❛♦s ❡st✉❞♦s✳ ❆ ❊❧❡✱ t♦❞❛ ❛ ❤♦♥r❛ ❡ t♦❞❛ ❛ ❣❧ór✐❛✳

❆ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ❊❧✐s❛♥❞r❛ ❞❡ ❋át✐♠❛ ●❧♦ss ❞❡ ▼♦r❛✐s ✱ ♣♦r t♦❞❛ ❛ s✉❛ ❝♦♥tr✐❜✉✐çã♦ ✐♥t❡❧❡❝t✉❛❧ ❡ ♣❡❧❛ s✉❛ ❝♦♠♣r❡❡♥sã♦ ❡ ♣❛❝✐ê♥❝✐❛ ❞❡♠♦♥str❛❞❛s q✉❛♥❞♦ ❡①t❡r♥❛✈❛ ♠✐♥❤❛s ❧✐♠✐t❛çõ❡s✳

❆ ♠✐♥❤❛ ♠ã❡✱ ❈❧❡♠✐❧❞❛ ▲✐♠❛ ❇❡♥❡✈✐❞❡s✱ q✉❡ s❡♠♣r❡ ❛❝r❡❞✐t♦✉ ❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❛ ✐r ♠❛✐s ❧♦♥❣❡✳

❆♦s ♠❡✉s ✐r♠ã♦s✱ ●✐s❡✉❞❛✱ ●✐s❡❧❡ ❡ ❈❧❡❞s♦♥ q✉❡ ✈❡♥❝❡r❛♠ ♥❛ ✈✐❞❛ ❡ q✉❡ t❛♠❜é♠ ♠❡ ❛❥✉❞❛r❛♠ ❛ ✈❡♥❝❡r ❡ ❛ s✉♣❡r❛r ❛ ❞♦r ❞❛ ❛✉sê♥❝✐❛ ❞♦ ♥♦ss♦ ♣❛✐✳

❆ ♠✐♥❤❛ ❡s♣♦s❛✱ ❋r❛♥s✉❡❧❛ ▲✉❝❡♥❛ ❇❡♥❡✈✐❞❡s✱ ♣❡❧❛s ♣❛❧❛✈r❛s ❞❡ ✐♥❝❡♥t✐✈♦✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ♣♦r s❡r ❝♦♠♣r❡❡♥s✐✈❛ ♥❛s ♠✐♥❤❛s ♠✉✐t❛s ❛✉sê♥❝✐❛s ❞♦ ❧❛r✳

❆ ♠✐♥❤❛ ✜❧❤❛✱ ❑é③✐❛ ▲✉❝❡♥❛ ❇❡♥❡✈✐❞❡s✱ ♣❡❧❛s ❛❧❡❣r✐❛s ❝♦♥t❛❣✐❛♥t❡s q✉❡ ❞❡♠♦♥str❛✈❛ ♥❛s ♠✐♥❤❛s ❛✢✐çõ❡s✳

❆♦s ♠❡✉s ✐♥s❡♣❛rá✈❡✐s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ❡st✉❞♦✱ ❆❧❛♥ ❡ ●❡♥❛❧❞♦✱ q✉❡ ♠✉✐t♦ ♠❡ ❛❥✉❞❛r❛♠ ❝♦♠ s✉❛s s✉❣❡stõ❡s ❡ ✐♥❝❡♥t✐✈♦s✳

❆♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚✱ ♣❡❧❛ ❞✐s♣♦s✐çã♦ ❡♠ ♥♦s ❡♥s✐♥❛r ❡ ♥♦s ❢❛③❡r ❛❝r❡❞✐t❛r q✉❡ ér❛♠♦s ❝❛♣❛③❡s✳

❆ t✉r♠❛ ✷✵✶✷ ❞❛ ❯❋P❇✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ s♦❧✐❞❛r✐❡❞❛❞❡ ❞❡♠♦♥str❛❞❛ ♥❛s ♠✐♥❤❛s ♥❡❝❡ss✐❞❛❞❡s✳

❆ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

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➚ ♠✐♥❤❛ ♠ã❡✱ ❈❧❡♠✐❧❞❛✱ q✉❡ t❛♠❜é♠ ❛ss✉♠✐✉ ❝♦♠ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ♦ ♣❛♣❡❧ ❞❡ ♣❛✐ ❡♠ ♥♦ss❛s ✈✐❞❛s✳

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❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❛♣r❡s❡♥t❛r ✉♠❛ ♥♦✈❛ ❛❜♦r❞❛❣❡♠ ❞❡♥tr♦ ❞❛ ●❡♦♠❡tr✐❛❀ ❛ ■♥✈❡rsã♦✳ ❆ ●❡♦♠❡tr✐❛ ■♥✈❡rs✐✈❛ é ✉♠❛ ●❡♦♠❡tr✐❛ ♥ã♦ ❊✉❝❧✐❞✐❛♥❛ q✉❡ ♣♦ss✉✐ ✐♥ú♠❡r❛s ❛♣❧✐❝❛çõ❡s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛ ❛ ♣r♦❜❧❡♠❛s ❞❡ t❛♥❣ê♥❝✐❛✳ ❊ss❛ ✏♥♦✈❛✑ ●❡♦♠❡tr✐❛ é ❛♣r❡s❡♥t❛❞❛ ❛♦ ❧♦♥❣♦ ❞❡ss❡ tr❛❜❛❧❤♦ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ s♦❧✉❝✐♦♥❛r ♦s ❞❡③ ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦✳ ❚♦❞❛s ❛s ❝♦♥str✉çõ❡s sã♦ r❡❛❧✐③❛❞❛s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ✉♠ s♦❢t✇❛r❡ ❞❡ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛❀ ♦ ●❡♦❣❡❜r❛✳ ❈♦♠♦ ♦ tr❛❜❛❧❤♦ é ❞✐r❡❝✐♦♥❛❞♦ ♣❛r❛ ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦✱ ❡♥tã♦ ❤á ✉♠❛ ♣r♦♣♦st❛ ❞❡ r♦t❡✐r♦ ♣❛r❛ q✉❡ ♦ ❧❡✐t♦r ♣♦ss❛ ♣❛rt✐❝✐♣❛r ❞♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥str✉çã♦ ❞❛s s♦❧✉çõ❡s ❞♦s r❡❢❡r✐❞♦s ♣r♦❜❧❡♠❛s✱ ♦ q✉❡ ♣♦ss✐❜✐❧✐t❛rá ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❝r✐❛t✐✈✐❞❛❞❡✱ ❞♦ ♣❡♥s❛♠❡♥t♦ ❧ó❣✐❝♦✱ ❞❛ ❛r❣✉♠❡♥t❛çã♦ ❡ ❞❛ ♣rát✐❝❛ ❡♠ ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s✳

P❛❧✈r❛s✲❝❤❛✈❡✿ ●❡♦♠❡tr✐❛ ■♥✈❡rs✐✈❛✳ Pr♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦✳ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛✳ ❈♦♥str✉çõ❡s ●❡♦♠étr✐❝❛s✳

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❚❤✐s ✇♦r❦ ✇❛s ❞❡✈❡❧♦♣❡❞ ✇✐t❤ t❤❡ ❛✐♠ ♦❢ ♣r❡s❡♥t✐♥❣ ❛ ♥❡✇ ❛♣♣r♦❛❝❤ ✇✐t❤✐♥ t❤❡ ●❡♦♠❡tr②✱ t❤❡ ■♥✈❡rs✐♦♥✳ ❚❤❡ ■♥✈❡rs✐✈❡ ●❡♦♠❡tr② ✐s ❛ ♥♦♥✲❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② t❤❛t ❤❛s s❡✈❡r❛❧ ❛♣♣❧✐❝❛t✐♦♥s✱ ♠❛✐♥❧② r❡❧❛t❡❞ t♦ ♣r♦❜❧❡♠s ♦❢ t❛♥❣❡♥❝②✳ ❚❤✐s ✏♥❡✇✑ ●❡♦♠❡tr② ✐s ♣r❡s❡♥t❡❞ t❤r♦✉❣❤♦✉t t❤✐s ✇♦r❦ ✐♥ ♦r❞❡r t♦ s♦❧✈❡ t❤❡ t❡♥ ♣r♦❜❧❡♠s ♦❢ ❆♣♦❧❧♦♥✐✉s✳ ❆❧❧ ❝♦♥str✉❝t✐♦♥s ❛r❡ ❝❛rr✐❡❞ ♦✉t ✇✐t❤ t❤❡ ❛✐❞ ♦❢ ❛ ❉②♥❛♠✐❝ ●❡♦♠❡tr② s♦❢t✇❛r❡✱ ●❡♦❣❡❜r❛✳ ❙✐♥❝❡ t❤❡ ✇♦r❦ ✐s ❞✐r❡❝t❡❞ t♦ t❡❛❝❤❡rs ❛♥❞ st✉❞❡♥ts ♦❢ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥✱ t❤❡♥ t❤❡r❡ ✐s ❛ ♣r♦♣♦s❡❞ r♦❛❞♠❛♣ ❢♦r t❤❡ r❡❛❞❡r t♦ ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡s❡ ♣r♦❜❧❡♠s ♣r♦❝❡ss✱ ✇❤✐❝❤ ✇✐❧❧ ❡♥❛❜❧❡ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❝r❡❛t✐✈✐t②✱ ❧♦❣✐❝❛❧ t❤✐♥❦✐♥❣✱ r❡❛s♦♥✐♥❣ ❛♥❞ ♣r❛❝t✐❝❡ ♦❢ ❣❡♦♠❡tr✐❝ ❝♦♥str✉❝t✐♦♥s✳

❑❡②✇♦r❞s✿ ■♥✈❡rs✐✈❡ ●❡♦♠❡tr②❀ Pr♦❜❧❡♠s ♦❢ ❆♣♦❧❧♦♥✐✉s❀ ❉②♥❛♠✐❝ ●❡♦♠❡tr②❀ ●❡♦♠❡tr✐❝ ❝♦♥str✉❝t✐♦♥s✳

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■♥tr♦❞✉çã♦ ✶

✶ ❆❧❣✉♥s ❈♦♥❝❡✐t♦s ●❡♦♠étr✐❝♦s ✹

✶✳✶ ▲✉❣❛r ●❡♦♠étr✐❝♦ ❡ P♦♥t♦s ◆♦tá✈❡✐s ❞❡ ✉♠ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❚❛♥❣ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷ ●❡♦♠❡tr✐❛ ■♥✈❡rs✐✈❛ ✶✹

✷✳✶ ❯♠ ♣♦✉❝♦ ❞❛ ❍✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ❉❡✜♥✐çã♦ ❞❡ ■♥✈❡rsã♦ ●❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸ Pr♦♣r✐❡❞❛❞❡s ❞❛ ■♥✈❡rsã♦ ●❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✸ ❖s Pr♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦ ✸✶

✸✳✶ ❈♦♥t❡①t♦ ❍✐stór✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷ ❘❡s♦❧✉çã♦ ❞♦s Pr♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦ ✉s❛♥❞♦ ■♥✈❡rsã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷✳✶ Pr♦❜❧❡♠❛ ✶ ✭PPP✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷✳✷ Pr♦❜❧❡♠❛ ✷ ✭❘❘❘✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷✳✸ Pr♦❜❧❡♠❛ ✸ ✭PP❘✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷✳✹ Pr♦❜❧❡♠❛ ✹ ✭PP❈✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✷✳✺ Pr♦❜❧❡♠❛ ✺ ✭P❘❘✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷✳✻ Pr♦❜❧❡♠❛ ✻ ✭P❈❈✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷✳✼ Pr♦❜❧❡♠❛ ✼ ✭❈❘❘✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

(9)

✸✳✷✳✶✵ Pr♦❜❧❡♠❛ ✶✵ ✭❈❈❈✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✻

(10)

❆ ♣❛❧❛✈r❛ ●❡♦♠❡tr✐❛✱ ❞❡r✐✈❛❞❛ ❞♦ ❣r❡❣♦ ❣❡♦♠❡tr❡✐♥✱ s✐❣♥✐✜❝❛ ♠❡❞✐çã♦ ❞❛ t❡rr❛ ✭❣❡♦ = t❡rr❛ ❡ ♠❡tr❡✐♥ = ♠❡❞✐❞❛✮✳ ❆ ●❡♦♠❡tr✐❛ é ♦ r❛♠♦ ❞❛ ▼❛t❡♠át✐❝❛ q✉❡ ❡st✉❞❛ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❡s♣❛ç♦✱ ♥♦r♠❛❧♠❡♥t❡ ❡♠ t❡r♠♦s ❞❡ ✜❣✉r❛s ❞♦ ♣❧❛♥♦ ✭❜✐❞✐♠❡♥s✐♦♥❛❧✮ ❡ só❧✐❞❛s ✭tr✐❞✐♠❡♥s✐♦♥❛❧✮✳ ❊❧❛ ❞✐✈✐❞❡✲s❡ ❡♠ ●❡♦♠❡tr✐❛ ❙✐♥tét✐❝❛✱ ❛ q✉❛❧ é ❛①✐♦♠át✐❝❛ ❡ ❡st✉❞❛ ♦ ♣❧❛♥♦ ❡ ♦s só❧✐❞♦s✱ t❡♠❛ tr❛❜❛❧❤❛❞♦ ♥❛ ♠❛✐s ❝♦♥❤❡❝✐❞❛ ♦❜r❛ ❞❡ ❊✉❝❧✐❞❡s✿ ❖s ❊❧❡♠❡♥t♦s❀ ❡ ❡♠ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ✭●❡♦♠❡tr✐❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✮✱ ♥❛ q✉❛❧ ♦s ♣r♦❜❧❡♠❛s sã♦ r❡s♦❧✈✐❞♦s ❝♦♠ ♠ét♦❞♦s ❛❧❣é❜r✐❝♦s✳

❯♠ ❣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 ❞❡③ ♣r♦❜❧❡♠❛s✱ q✉❡ ✈✐❡r❛♠ ❛ r❡❝❡❜❡r s❡✉ ♥♦♠❡✳ ❊st❡s ♣r♦❜❧❡♠❛s✱ ❡♠ ❡s♣❡❝✐❛❧ ♦ ú❧t✐♠♦✱ ❞❡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❛ ♠✉✐t❛s s♦❧✉çõ❡s t❡♥❤❛♠ s✐❞♦ ♣r♦❞✉③✐❞❛s ♥❛ ❡r❛ ✏♠♦❞❡r♥❛✑✱ ❛✐♥❞❛ ♥ã♦ ❛t❡♥❞✐❛♠ ❛♦ ❛❧♠❡❥❛❞♦ ♥♦✈♦ ♣❛r❛❞✐❣♠❛✿ ❡♥❝♦♥tr❛r ✉♠ ♠ét♦❞♦ ❞✐r❡t♦✱ ❛♣❧✐❝á✈❡❧ ❛ t♦❞♦s ♦s ❝❛s♦s✱ ❡ s❡ ♣♦ssí✈❡❧✱ q✉❡ ♣❡r♠✐t✐ss❡ ❞❡t❡r♠✐♥❛r ♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ❛❞♠✐ssí✈❡✐s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✳

❍✐st♦r✐❝❛♠❡♥t❡✱ ♦ ❡st✉❞♦ ❞❛ ●❡♦♠❡tr✐❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♥tr❡ ♦s ❣r❡❣♦s✱ ❢♦✐ ❝♦♥s✐❞❡r❛❞♦ ✐♥❞✐s♣❡♥sá✈❡❧ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ ❡ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ✐♥t❡❧❡❝t✉❛❧ ❞♦ ✐♥❞✐✈í❞✉♦✳ ❊♥tã♦✱ ♣♦r q✉❡ ❡ss❛ ❝✐ê♥❝✐❛ t❡♠ s✐❞♦ tr❛t❛❞❛✱ ♥♦

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❡s❝♦❧❛s❄ ❆❝r❡❞✐t♦ q✉❡ ♥ã♦✳ ▼❛s ❛♣❡s❛r ❞❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡♥tr♦ ❞❛ ▼❛t❡♠át✐❝❛✱ ❛ ♠❛♥❡✐r❛ ❞❡ s❡ tr❛❜❛❧❤❛r ❛ ●❡♦♠❡tr✐❛ t❡♠ ❛♣r❡s❡♥t❛❞♦ ♣r♦❜❧❡♠❛s r❡❧❛❝✐♦♥❛❞♦s t❛♥t♦ ❛♦ s❡✉ ❡♥s✐♥♦ q✉❛♥t♦ à s✉❛ ❛♣r❡♥❞✐③❛❣❡♠✳ ❈♦♥t✉❞♦✱ ❛ ♣r❡♦❝✉♣❛çã♦ ❡♠ s❡ r❡s❣❛t❛r ❛ ●❡♦♠❡tr✐❛ ❝♦♠♦ ✉♠❛ ❞❛s ár❡❛s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ▼❛t❡♠át✐❝❛ t❡♠ ❧❡✈❛❞♦ ♠✉✐t♦s ♣r♦❢❡ss♦r❡s ❡ ♣❡sq✉✐s❛❞♦r❡s ❛♣♦✐❛❞♦s ❡♠ t❡♦r✐❛s ❝♦❣♥✐t✐✈❛s ❛ s❡ ❞❡❞✐❝❛r❡♠ à r❡✢❡①ã♦ ❡ à ❡❧❛❜♦r❛çã♦✱ ✐♠♣❧❡♠❡♥t❛çã♦ ❡ ❛✈❛❧✐❛çã♦ ❞❡ ❛❧t❡r♥❛t✐✈❛s✱ q✉❡ ❜✉sq✉❡♠ s✉♣❡r❛r ❛s ❞✐✜❝✉❧❞❛❞❡s q✉❡ sã♦ ❢r❡q✉❡♥t❡s ♥❛ ❛❜♦r❞❛❣❡♠ ❞❡ss❡ t❡♠❛✳

❊♥t❡♥❞❡♥❞♦ q✉❡ ❛ ❛rt❡ ❞❛ ▼❛t❡♠át✐❝❛ ❡stá ❡♠ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s✱ é q✉❡ ♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❜✉s❝❛ ❞❡s❡♥✈♦❧✈❡r ✉♠ r♦t❡✐r♦ q✉❡ ♣♦ss✐❜✐❧✐t❡ s♦❧✉❝✐♦♥❛r ♣r♦❜❧❡♠❛s ❝❧áss✐❝♦s ❞❡ t❛♥❣ê♥❝✐❛ ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✿ ❖s ❉❡③ Pr♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦✱ ❛tr❛✈és ❞❛ ✉t✐❧✐③❛çã♦ ❞❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ●❡♦♠❡tr✐❛ ■♥✈❡rs✐✈❛✳ P❡❧♦ ❢❛t♦ ❞❡ s❡r ✉♠❛ ❣❡♦♠❡tr✐❛ ♥ã♦✲❊✉❝❧✐❞✐❛♥❛✱ ♦ ♠ét♦❞♦ ✐♥✈❡rs✐✈♦ ♣♦ss✐❜✐❧✐t❛rá ♥♦✈❛s ❡str❛té❣✐❛s ❞❡ r❡s♦❧✉çã♦✱ ✐♥st✐❣❛♥❞♦ ❞❡st❛ ♠❛♥❡✐r❛ ♦ r❛❝✐♦❝í♥✐♦ ♥❛ ❜✉s❝❛ ❞❡ ♥♦✈❛s s♦❧✉çõ❡s ♣❛r❛ t❛✐s ♣r♦❜❧❡♠❛s✳

❘❡s♦❧✈❡r ♦s ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦ ♣♦r ♠❡✐♦ ❞❛ ✐♥✈❡rsã♦ ❡①✐❣❡ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ♣ré✈✐♦ q✉❡ ♥♦s ❞❛rá s✉♣♦rt❡ ♣❛r❛ ❝❤❡❣❛r♠♦s ❛ t❛✐s s♦❧✉çõ❡s✳ ❙❡♥❞♦ ❛ss✐♠✱ ♦s ❞♦✐s ❝❛♣ít✉❧♦s q✉❡ ❛♥t❡❝❡❞❡♠ ❛ r❡s♦❧✉çã♦ ❞❡ t❛✐s ♣r♦❜❧❡♠❛s t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ tr❛③❡r ♦ ❡♠❜❛s❛♠❡♥t♦ ♥❡❝❡ssár✐♦ ♣❛r❛ q✉❡ ❤❛❥❛ ❝♦♠♣r❡❡♥sã♦ às ❝♦♥str✉çõ❡s ❢❡✐t❛s ♥♦ ❈❛♣ít✉❧♦ ✸✳

❖ ❈❛♣ít✉❧♦ ✶ é ❝♦♠♣♦st♦ ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❛ ●❡♦♠❡tr✐❛ P❧❛♥❛ ❞❛❞♦s ❡♠ ❢♦r♠❛ ❞❡ ❞❡✜♥✐çõ❡s✱ ♣r♦♣♦s✐çõ❡s ❡ s✉❛s r❡s♣❡❝t✐✈❛s ❞❡♠♦♥str❛çõ❡s✳ ❉❡♥tr❡ ❡st❡s ❞❡st❛❝❛♠✲ s❡ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❡ ❛ t❛♥❣ê♥❝✐❛ ❡♥tr❡ ❞♦✐s ❡❧❡♠❡♥t♦s✳ ❈♦♥❝❡✐t♦s ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞♦s ♥❛ ❝♦♥str✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ sã♦ ❛♣r❡s❡♥t❛❞❛s ❞❡✜♥✐çõ❡s ❡ ♣r♦♣♦s✐çõ❡s ❞❛ ●❡♦♠❡tr✐❛ ■♥✈❡rs✐✈❛ ♥✉♠❛ s❡q✉ê♥❝✐❛ ❧ó❣✐❝♦✲❞✐❞át✐❝❛✱ s❡♠♣r❡ ✉t✐❧✐③❛♥❞♦ ❝♦♠♦ s✉♣♦rt❡ ✉♠ ❛♠❜✐❡♥t❡

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❖ t❡r❝❡✐r♦ ❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦ é ❞❡❞✐❝❛❞♦ ❡①❝❧✉s✐✈❛♠❡♥t❡ às ❝♦♥str✉çõ❡s ❞❛s s♦❧✉çõ❡s ❞♦s ❞❡③ ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦✳ ◆♦ ❡♥t❛♥t♦✱ ❛♥t❡r✐♦r♠❡♥t❡ ❛ ✐st♦✱ é ❢❡✐t❛ ✉♠❛ ❜r❡✈❡ ❛♣r❡s❡♥t❛çã♦ ❞❛ ✈✐❞❛ ❡ ♦❜r❛ ❞♦ r❡❢❡r✐❞♦ ❛✉t♦r✱ ❜❡♠ ❝♦♠♦ ❞♦s ❞❡s❛✜♦s ❛♦ ❧♦♥❣♦ ❞♦s t❡♠♣♦s ♣❛r❛ s❡ s♦❧✉❝✐♦♥❛r ♦ ❞é❝✐♠♦ ❡ ♠❛✐s ✐♥tr✐❣❛♥t❡ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦ ♣♦r ❡❧❡✳

❆ ♣r♦♣♦st❛ ❞❡st❡ tr❛❜❛❧❤♦ é ♦❢❡r❡❝❡r ✉♠❛ ♦✉tr❛ ❛❧t❡r♥❛t✐✈❛✱ ❛❧é♠ ❞❛ ré❣✉❛ ❡ ❞♦ ❝♦♠♣❛ss♦✱ ♣❛r❛ ❝♦♥str✉✐r ❛s s♦❧✉çõ❡s ❞♦s ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦✳ ❈♦♥t✉❞♦✱ ❡s♣❡r❛✲s❡ ❞♦ ❧❡✐t♦r q✉❡ ♣♦ss❛ ❤❛✈❡r ❢❛♠✐❧✐❛r✐❞❛❞❡ ❝♦♠ ✉♠ s♦❢t✇❛r❡ ❞❡ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛✳

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## ❆❧❣✉♥s ❈♦♥❝❡✐t♦s ●❡♦♠étr✐❝♦s

❖❜❥❡t✐✈❛♥❞♦ r❡s♦❧✈❡r ♦s ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦✱ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ❛❜♦r❞❛❣❡♠ ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❣❡♦♠étr✐❝♦s q✉❡ s❡rã♦ ✐♥❞✐s♣❡♥sá✈❡✐s ♥❛ ❝♦♥str✉çã♦ ❞❛s s♦❧✉çõ❡s ❞♦s r❡❢❡r✐❞♦s ♣r♦❜❧❡♠❛s✳

### ❚r✐â♥❣✉❧♦

❈♦♠❡❝❡♠♦s ❝♦♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s✳ ❱❡❥❛ ❬✽❪ ♣❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✿ M ediatriz ❞❡ ✉♠ s❡❣♠❡♥t♦AB é ❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛AB q✉❡ ❝♦♥té♠ ♦ s❡✉ ♣♦♥t♦ ♠é❞✐♦✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✿ Bissetriz ❞❡ ✉♠ â♥❣✉❧♦∠AOB é ❛ s❡♠✐rr❡t❛ OC t❛❧ q✉❡∠AOC ❂ ∠COB✳

❉❡✜♥✐çã♦ ✶✳✶✳✸✿ Lugar Geometrico ❝♦♥s✐st❡ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ q✉❡ ❣♦③❛♠ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣r♦♣r✐❡❞❛❞❡✳

❆s ❞✉❛s ♣r♦♣♦s✐çõ❡s ❛ s❡❣✉✐r ❝❛r❛❝t❡r✐③❛♠ ❛ ♠❡❞✐❛tr✐③ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❡ ❛ ❜✐ss❡tr✐③ ❞❡ ✉♠ â♥❣✉❧♦ ❝♦♠♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✱ s❡❣✉♥❞♦ ❝♦♥st❛ ❡♠ ❬✻❪ ❡ ❬✾❪✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✿ ❉❛❞♦s ♣♦♥t♦s A ❡ B ♥♦ ♣❧❛♥♦✱ ❛ ♠❡❞✐❛tr✐③ ❞❡ AB é ♦ ❧✉❣❛r

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❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ ❡q✉✐❞✐st❛♠ ❞❡ A ❡ ❞❡ B✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ M ♦ ♣♦♥t♦ ♠é❞✐♦ ❡ m ❛ ♠❡❞✐❛tr✐③ ❞❡ AB ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✶✮✳ ❙❡ P ∈ m✱ ❡♥tã♦✱ ♥♦ tr✐â♥❣✉❧♦ P AB✱ P M é ♠❡❞✐❛♥❛ ❡ ❛❧t✉r❛✱ ♦ q✉❡ ❣❛r❛♥t❡ q✉❡ P AB é tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❞❡ ❜❛s❡ AB✳ ▲♦❣♦ P A=P B✳

❋✐❣✉r❛ ✶✳✶✿ P ∈ ✭♠❡❞✐❛tr✐③ ❞❡ ❆❇✮ ⇔P A=P B

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ P ✉♠ ♣♦♥t♦ ♥♦ ♣❧❛♥♦ t❛❧ q✉❡ P A =P B✳ ❊♥tã♦ ♦ tr✐â♥❣✉❧♦ P AB é ✐sós❝❡❧❡s ❞❡ ❜❛s❡ AB✱ ❞♦♥❞❡ s❡❣✉❡ q✉❡ ❛ ♠❡❞✐❛♥❛ ❡ ❛ ❛❧t✉r❛ r❡❧❛t✐✈❛s ❛ AB ❝♦✐♥❝✐❞❡♠✳ ❈♦♠♦ ❛ ♠❡❞✐❛♥❛ ❞❡ P AB r❡❧❛t✐✈❛ ❛ AB é ♦ s❡❣♠❡♥t♦ P M✱ s❡❣✉❡ q✉❡ P M⊥AB✱ ♦ q✉❡ é ♦ ♠❡s♠♦ q✉❡ ❞✐③❡r q✉❡ ❛ r❡t❛ ←P M−→ é ♠❡❞✐❛tr✐③ ❞❡ AB✳ ❈♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✷✿ ❙❡❥❛ ∠AOB ✉♠ â♥❣✉❧♦ ❞❛❞♦✳ ❙❡ P é ✉♠ ♣♦♥t♦ ❞♦ ♠❡s♠♦✱ ❡♥tã♦ d(P,−→AO) = d(P,−BO)−→ ⇔P ∈ ✭❜✐ss❡tr✐③ ❞❡∠AOB✮✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛✱ ♣r✐♠❡✐r♦✱ q✉❡ P ♣❡rt❡♥❝❡ à ❜✐ss❡tr✐③ ❞❡ ∠AOB ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✷✮ ❡ s❡❥❛♠M ❡N✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ♣és ❞❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❜❛✐①❛❞❛s ❞❡ P às r❡t❛s ←AO→ ❡ ←→BO✳ ❈♦♠♦MOPb =NOPb ✱ OM Pc =ON Pb = 90◦

❡ OP é ❝♦♠✉♠✱ s❡❣✉❡ q✉❡ ♦s tr✐â♥❣✉❧♦s OM P ❡ON P sã♦ ❝♦♥❣r✉❡♥t❡s ♣♦rLAAo✳ ❉❛í✱P M =P N✱ ♦✉ s❡❥❛✱ d(P,←AO) =→ d(P,←→BO✮✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛P ✉♠ ♣♦♥t♦ ♥♦ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦∠AOB✱ t❛❧ q✉❡P M =P N✱ ♦♥❞❡ M ❡ N sã♦ ♦s ♣és ❞❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❜❛✐①❛❞❛s ❞❡ P r❡s♣❡❝t✐✈❛♠❡♥t❡ às r❡t❛s ←AO→ ❡ ←→BO✳ ❊♥tã♦✱ ♦s tr✐â♥❣✉❧♦s M OP ❡ N OP sã♦ ♥♦✈❛♠❡♥t❡ ❝♦♥❣r✉❡♥t❡s

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❋✐❣✉r❛ ✶✳✷✿ P ∈ ❛ ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ AOB

✭P M = P N ❡ OP ❝♦♠✉♠ ✲ ❝❛s♦ Cateto, Hipotenusa✮✳ ▼❛s ❛í MOPb = NOPb ✱ ❞♦♥❞❡ P ❡stá s♦❜r❡ ❛ ❜✐ss❡tr✐③ ❞❡∠AOB✳ ❈♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✸✿ ❊♠ t♦❞♦ tr✐â♥❣✉❧♦ ❛s ♠❡❞✐❛tr✐③❡s ❞♦s ❧❛❞♦s ♣❛ss❛♠ t♦❞❛s ♣♦r ✉♠ ♠❡s♠♦ ♣♦♥t♦ ❝❤❛♠❛❞♦ ❞❡ ❝✐r❝✉♥❝❡♥tr♦✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ ABC ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✱ r, s ❡t✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛s ♠❡❞✐❛tr✐③❡s ❞♦s ❧❛❞♦s BC, CA❡ AB✱ ❡ O ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s r❡t❛sr ❡s ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✸✮✳

❋✐❣✉r❛ ✶✳✸✿ ❈✐r❝✉♥❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦

P❡❧❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ♠❡❞✐❛tr✐③ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❝♦♠♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✱ t❡♠♦s OB = OC ✭♣♦✐s O ∈ r✮ ❡ OC = OA ✭♣♦✐s O ∈ s✮✳ P♦rt❛♥t♦✱ OB = OA ❡ s❡❣✉❡ ♥♦✈❛♠❡♥t❡ ❞❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ♠❡❞✐❛tr✐③ ❝♦♠♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ q✉❡O ∈t✳ ❈♦♠♦

q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

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Pr♦♣♦s✐çã♦ ✶✳✶✳✹✿ ❆s ❜✐ss❡tr✐③❡s ✐♥t❡r♥❛s ❞❡ t♦❞♦ tr✐â♥❣✉❧♦ ❝♦♥❝♦rr❡♠ ❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦✱ ♦ ✐♥❝❡♥tr♦ ❞♦ tr✐â♥❣✉❧♦✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ r, s ❡ t✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛s ❜✐ss❡tr✐③❡s ✐♥t❡r♥❛s ❞♦s â♥❣✉❧♦s ∠BAC,∠ABC ❡ ∠BCA ❞♦ tr✐â♥❣✉❧♦ ABC ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✹✮ ❡ I ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s r❡t❛s r ❡ s✳ ❈♦♠♦ I ∈ r✱ s❡❣✉❡ ❞❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s ❜✐ss❡tr✐③❡s ❝♦♠♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✱ q✉❡I ❡q✉✐❞✐st❛ ❞♦s ❧❛❞♦sAB ❡AC❞❡ABC✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ I ∈ s ❣❛r❛♥t❡ q✉❡ I ❡qü✐❞✐st❛ ❞♦s ❧❛❞♦s AB ❡ BC✳ P♦rt❛♥t♦✱ I ❡qü✐❞✐st❛ ❞❡ AC ❡

BC ❡✱ ✉s❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❛ r❡❢❡r✐❞❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s ❜✐ss❡tr✐③❡s✱ ❝♦♥❝❧✉í♠♦s q✉❡I ♣❡rt❡♥❝❡ à ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ ∠BCA✱ ♦✉ s❡❥❛✱ à r❡t❛ t✳ ❆ss✐♠✱ r, s ❡ t ❝♦♥❝♦rr❡♠ ❡♠ I✳

❋✐❣✉r❛ ✶✳✹✿ ■♥❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦

Pr♦♣♦s✐çã♦ ✶✳✶✳✺✿ ❚♦❞♦ tr✐â♥❣✉❧♦ ❛❞♠✐t❡ ✉♠ ú♥✐❝♦ ❝ír❝✉❧♦ ♣❛ss❛♥❞♦ ♣♦r s❡✉s ✈ért✐❝❡s✳ ❚❛❧ ❝ír❝✉❧♦ é ❞✐t♦ ❝✐r❝✉♥s❝r✐t♦ ❛♦ tr✐â♥❣✉❧♦ ❡ s❡✉ ❝❡♥tr♦ é ♦ ❝✐r❝✉♥❝❡♥tr♦ ❞♦ ♠❡s♠♦✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ❆❇❈ ✉♠ tr✐â♥❣✉❧♦ ❞❡ ❝✐r❝✉♥❝❡♥tr♦ O✱ ✭❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✶✳✺✮✳ ❈♦♠♦O é ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s ♠❡❞✐❛tr✐③❡s ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦✱ t❡♠♦s OA = OB = OC✳ ❉❡♥♦t❛♥❞♦ ♣♦r R t❛❧ ❞✐stâ♥❝✐❛ ❝♦♠✉♠✱ s❡❣✉❡ q✉❡ ♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ R ♣❛ss❛ ♣♦rA, B ❡ C✳ ❊①✐st❡✱ ♣♦rt❛♥t♦✱ ✉♠ ❝ír❝✉❧♦ ♣❛ss❛♥❞♦ ♣❡❧♦s ✈ért✐❝❡s ❞❡ ABC✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ♦ ❝❡♥tr♦ ❞❡ ✉♠ ❝ír❝✉❧♦ q✉❡ ♣❛ss❛ ♣❡❧♦s ✈ért✐❝❡s

(17)

❋✐❣✉r❛ ✶✳✺✿ ❈ír❝✉❧♦ ❝✐r❝✉♥s❝r✐t♦ ❛ ✉♠ tr✐â♥❣✉❧♦✳

❞❡ ABC ❞❡✈❡ ❡q✉✐❞✐st❛r ❞♦s ♠❡s♠♦s✳ P♦rt❛♥t♦✱ ♦ ❝❡♥tr♦ ♣❡rt❡♥❝❡ às ♠❡❞✐❛tr✐③❡s ❞♦s ❧❛❞♦s ❞❡ ABC ❞♦♥❞❡ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s ♠❡s♠❛s✱ q✉❡ é ♦ ❝✐r❝✉♥❝❡♥tr♦ ❖✳ P♦r ✜♠✱ ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦✱ s❡♥❞♦ ❛ ❞✐stâ♥❝✐❛ ❞❡ O ❛♦s ✈ért✐❝❡s✱ é ✐❣✉❛❧ ❛ R✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✻✿ ❚♦❞♦ tr✐â♥❣✉❧♦ ❛❞♠✐t❡ ✉♠ ú♥✐❝♦ ❝ír❝✉❧♦ ❝♦♥t✐❞♦ ♥♦ ♠❡s♠♦ ❡ t❛♥❣❡♥t❡ ❛ s❡✉s ❧❛❞♦s✳ ❚❛❧ ❝ír❝✉❧♦ é ❞✐t♦ ✐♥s❝r✐t♦ ♥♦ tr✐â♥❣✉❧♦ ❡ s❡✉ ❝❡♥tr♦ é ♦ ✐♥❝❡♥tr♦ ❞♦ ♠❡s♠♦✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ I ♦ ✐♥❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ABC ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✻✮✳ ❈♦♠♦

❋✐❣✉r❛ ✶✳✻✿ ❈ír❝✉❧♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ tr✐â♥❣✉❧♦

(18)

I é ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s ❜✐ss❡tr✐③❡s ✐♥t❡r♥❛s ❞❡ABC✱ t❡♠♦s q✉❡I ❡q✉✐❞✐st❛ ❞♦s ❧❛❞♦s ❞❡ABC✳ ❙❡♥❞♦rt❛❧ ❞✐stâ♥❝✐❛ ❝♦♠✉♠ ❛♦s ❧❛❞♦s✱ s❡❣✉❡ q✉❡ ♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ I ❡ r❛✐♦ r ❡stá ❝♦♥t✐❞♦ ❡♠ ABC ❡ t❛♥❣❡♥❝✐❛ s❡✉s ❧❛❞♦s✳ ❆ ✉♥✐❝✐❞❛❞❡ ❞♦ ❝ír❝✉❧♦ ✐♥s❝r✐t♦ ♣♦❞❡ s❡r ❡st❛❜❡❧❡❝✐❞❛ ♠❡❞✐❛♥t❡ ✉♠ ❛r❣✉♠❡♥t♦ ❛♥á❧♦❣♦ ❛♦ ❞❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❝ír❝✉❧♦ ❝✐r❝✉♥s❝r✐t♦✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✼✿ ❊♠ t♦❞♦ tr✐â♥❣✉❧♦ ABC✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❝ír❝✉❧♦ t❛♥❣❡♥t❡ ❛♦ ❧❛❞♦BC ❡ ❛♦s ♣r♦❧♦♥❣❛♠❡♥t♦s ❞♦s ❧❛❞♦sAB❡AC✳ ❚❛❧ ❝ír❝✉❧♦ é ♦ ❝ír❝✉❧♦ ❡①✲✐♥s❝r✐t♦ ❛♦ ❧❛❞♦ BC ❡ s❡✉ ❝❡♥tr♦ é ♦ ❡①✲✐♥❝❡♥tr♦ ❞❡ABC r❡❧❛t✐✈♦ ❛ BC ✭♦✉ ❛♦ ✈ért✐❝❡ A✮✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠r❡s❛s ❜✐ss❡tr✐③❡s ❡①t❡r♥❛s ❞♦s ✈ért✐❝❡sB ❡C❞♦ tr✐â♥❣✉❧♦ ABC ❡ Ie s❡✉ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦✳ ❈♦♠♦ Ie ∈ r ❡ r é ❜✐ss❡tr✐③✱ s❡❣✉❡ q✉❡

d(Ie,−BC) =−→ d(Ie,−→AB)✳ ❉♦ ♠❡s♠♦ ♠♦❞♦✱ ✉♠❛ ✈❡③ q✉❡ Ie ∈ s✱ ❝♦♥❝❧✉í♠♦s q✉❡

d(Ie,

−−→

BC) =d(Ie,

−→

AC)✳ ❉❡♥♦t❛♥❞♦ ♣♦rre ❛ ❞✐stâ♥❝✐❛ ❝♦♠✉♠ ❞❡Ie às r❡t❛s s✉♣♦rt❡s ❞♦s ❧❛❞♦s✱ s❡❣✉❡ q✉❡ ♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦Ie❡ r❛✐♦ret❛♥❣❡♥❝✐❛BC❡ ♦s ♣r♦❧♦♥❣❛♠❡♥t♦s ❞❡ AB ❡ AC ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✼✮✳

❋✐❣✉r❛ ✶✳✼✿ ❈ír❝✉❧♦ ❡①✲✐♥s❝r✐t♦ ❛♦ ❧❛❞♦BC ❞♦ tr✐â♥❣✉❧♦ ABC✳

### ✶✳✷ ❚❛♥❣ê♥❝✐❛

❆❧é♠ ❞❡ss❛s ♣r♦♣♦s✐çõ❡s ❞❡♠♦♥str❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✱ ♣❛r❛ q✉❡ ♣♦ss❛♠♦s ❞❛r s❡❣✉✐♠❡♥t♦ ❛♦ ♥♦ss♦ ❡st✉❞♦ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❡♥✉♥❝✐❛r♠♦s t❛♠❜é♠ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s

(19)

✐♠♣♦rt❛♥t❡s s♦❜r❡ ❛ r❡❧❛çã♦ ❞❡ t❛♥❣ê♥❝✐❛ ❡♥tr❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ r❡t❛✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✿ ❙❡❥❛♠ C ❡ D ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s q✉❡ s❡ ✐♥t❡r❝❡♣t❛♠ ❡♠ ✉♠ ♣♦♥t♦ P✳ ❙❡ C ❡D♣♦ss✉❡♠ t❛♥❣❡♥t❡s s ❡ t✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♠P✱ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ ♦ â♥❣✉❧♦ ❡♥tr❡ C ❡ D ❡♠ P é ♦ â♥❣✉❧♦ ❡♥tr❡ s ❡ t✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❞✐③❡♠♦s q✉❡ C ❡ D sã♦ ♦rt♦❣♦♥❛✐s ❡♠P s❡s ❡ t sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳

❋✐❣✉r❛ ✶✳✽✿ â♥❣✉❧♦ ❡♥tr❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s

▲❡♠❜r❡♠♦s q✉❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛C ❡ ✉♠❛ r❡t❛rsã♦ ❞✐t♦s t❛♥❣❡♥t❡s✱ s❡ ❡①✐st✐r ✉♠ ú♥✐❝♦ ♣♦♥t♦ P ❝♦♠✉♠ ❛C ❡ ❛ r✳

❉❛s q✉❛tr♦ ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❉❡s❡♥❤♦ ●❡♦♠étr✐❝♦ q✉❡ s❡ s❡❣✉❡♠✱ ❞❡♠♦♥str❛r❡♠♦s ❛♣❡♥❛s ❛ t❡r❝❡✐r❛✳ ❖ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦s ♦✉tr♦s ✐t❡♥s ♣♦❞❡ ❡♥❝♦♥trá✲❧❛ ❡♠ ❬✶❪✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✶✿ ❙❡❥❛♠ C ❡ D ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s s❡❝❛♥t❡s ♥♦s ♣♦♥t♦s P ❡ Q✳ ❊♥tã♦✱ ♦ â♥❣✉❧♦ ❡♥tr❡C ❡ D♥♦ ♣♦♥t♦P é ✐❣✉❛❧ ❛♦ â♥❣✉❧♦ ❡♥tr❡C ❡D♥♦ ♣♦♥t♦Q✳ ✭❱❡❥❛ ❋✐❣✉r❛ ✶✳✾✮✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✷✿ ❙❡❥❛♠ C ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ r ✉♠❛ r❡t❛ s❡❝❛♥t❡ ❛ C ♥♦s ♣♦♥t♦s P ❡Q✳ ❊♥tã♦✱ ♦ â♥❣✉❧♦ ❡♥tr❡ C ❡ r ♥♦ ♣♦♥t♦ P é ✐❣✉❛❧ ❛♦ â♥❣✉❧♦ ❡♥tr❡ C ❡ r ♥♦ ♣♦♥t♦ Q✳ ✭❱❡❥❛ ❋✐❣✉r❛ ✶✳✶✵✮✳

(20)

❋✐❣✉r❛ ✶✳✾✿ ➶♥❣✉❧♦ ❡♥tr❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s s❡❝❛♥t❡s(α=β)

❋✐❣✉r❛ ✶✳✶✵✿ ➶♥❣✉❧♦ ❡♥tr❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ r❡t❛ s❡❝❛♥t❡

(21)

Pr♦♣♦s✐çã♦ ✶✳✷✳✸✿ ❙❡❥❛ C ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O✱ ❡ P ✉♠ ♣♦♥t♦ ❞❡ C✳ ❙❡ t é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P ❡ é ♣❡r♣❡♥❞✐❝✉❧❛r à −→OP✱ ❡♥tã♦ t é t❛♥❣❡♥t❡ ❛ C✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ R ♦ r❛✐♦ ❞❡ C✳ ❙❡ Q 6= P é ♦✉tr♦ ♣♦♥t♦ ❞❡ t ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✶✶✮✱ ❡♥tã♦ t❡♠♦s QO > P O = R✱ ✉♠❛ ✈❡③ q✉❡ QP Ob = 90◦ é ♦ ♠❛✐♦r â♥❣✉❧♦ ❞♦

tr✐â♥❣✉❧♦ OP Q✳ P♦rt❛♥t♦ Q /∈ C ❡✱ ❛ss✐♠✱ P é ♦ ú♥✐❝♦ ♣♦♥t♦ ❡♠ ❝♦♠✉♠ ❛ t ❡ C✳ ▲♦❣♦✱ t é ❛ r❡t❛ t❛♥❣❡♥t❡ ❛C ❡♠ P✳

❋✐❣✉r❛ ✶✳✶✶✿ ❘❡t❛ t❛♥❣❡♥t❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛

Pr♦♣♦s✐çã♦ ✶✳✷✳✹✿ ❈♦♠ ❝❡♥tr♦ ❡♠ ✉♠ ♣♦♥t♦P ❢♦r❛ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛C♣❛ss❛ ✉♠❛ ú♥✐❝❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ D ♦rt♦❣♦♥❛❧ àC✳

❯♠ ♦✉tr♦ ❝♦♥❝❡✐t♦ ✐♠♣♦rt❛♥t❡ ❞❛ ●❡♦♠❡tr✐❛ P❧❛♥❛ q✉❡ s❡rá ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛ ♣❛r❛ ❡st❡ tr❛❜❛❧❤♦ é ♦ ❞❡ ❈♦♥❥✉❣❛❞♦ ❍❛r♠ô♥✐❝♦✳

❉❡✜♥✐çã♦ ✶✳✷✳✷✿ ❉❛❞♦s três ♣♦♥t♦s ❝♦❧✐♥❡❛r❡s A, B ❡ P✱ ❝♦♠A6=B✱ ❞✐③❡♠♦s q✉❡ P′ é ♦ ❝♦♥❥✉❣❛❞♦ ❤❛r♠ô♥✐❝♦ ❞❡ P ❡♠ r❡❧❛çã♦ ❛♦ s❡❣♠❡♥t♦AB s❡

AP /BP =AP′/BP.

P❛r❛ ♠❡❧❤♦r ♥♦s ❛♣r♦♣r✐❛r♠♦s ❞❡st❛ ❞❡✜♥✐çã♦✱ ❡♥❝♦♥tr❛r❡♠♦s ♦ ❝♦♥❥✉❣❛❞♦ ❤❛r♠ô♥✐❝♦ ❞❡ ✉♠ ♣♦♥t♦ ❛tr❛✈és ❞❛ s❡❣✉✐♥t❡ ❝♦♥str✉çã♦✿ ❉❛❞♦s ♦s ♣♦♥t♦s A, B ❡ P s♦❜r❡ ✉♠❛ r❡t❛ r✱ tr❛ç❛♠♦s ❞✉❛s r❡t❛s q✉❛✐sq✉❡ra1 ❡ a2 ♣❛ss❛♥❞♦ ♣♦r A ❡ ✉♠❛

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❋✐❣✉r❛ ✶✳✶✷✿ ❈✐r❝✉♥❢❡rê♥❝✐❛ ♦rt♦❣♦♥❛❧ ❛C ❝♦♠ ❝❡♥tr♦ P

r❡t❛ c ♣❛ss❛♥❞♦ ♣♦r P✳ ❯♥✐♥❞♦ ❛ B ♦s ♣♦♥t♦s ❞❡ ✐♥❝✐❞ê♥❝✐❛ ❞❡ c ❝♦♠ a1 ❡ a2 ✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦❜t❡♠♦s ❛s r❡t❛s b1 ❡ b2✳ ❋✐❝❛ ❡♥tã♦ ❢♦r♠❛❞♦ ✉♠ q✉❛❞r✐❧át❡r♦

✭EF HG✱ ♥❛ ❋✐❣✉r❛ ✶✳✶✸✮ t❛❧ q✉❡ ♦s ❧❛❞♦s ♦♣♦st♦s ❝♦♥❝♦rr❡♠ ❡♠ A ❡ B✱ ❡ t❛❧ q✉❡ ✉♠❛ ❞❡ s✉❛s ❞✐❛❣♦♥❛✐s ♣❛ss❛ ♣♦r P✳ ❙❡❥❛ P′ ♦ ♣♦♥t♦ ❞❡ ❡♥❝♦♥tr♦ ❞❡ r ❝♦♠ ❛ ♦✉tr❛

❞✐❛❣♦♥❛❧ ❞♦ q✉❛❞r✐❧át❡r♦✳ ❊♥tã♦P′ é ♦ ❝♦♥❥✉❣❛❞♦ ❤❛r♠ô♥✐❝♦ ❞❡P ❡♠ r❡❧❛çã♦ ❛AB✳

❋✐❣✉r❛ ✶✳✶✸✿ ❈♦♥❥✉❣❛❞♦ ❤❛r♠ô♥✐❝♦ ❞❡P ❡♠ r❡❧❛çã♦ ❛AB

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## ●❡♦♠❡tr✐❛ ■♥✈❡rs✐✈❛

### ✷✳✶ ❯♠ ♣♦✉❝♦ ❞❛ ❍✐stór✐❛

❆ ●❡♦♠❡tr✐❛ ■♥✈❡rs✐✈❛ t❡✈❡ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♥✉♠ ♣❡rí♦❞♦ ❜❛st❛♥t❡ r❡❝❡♥t❡ ❡♠ r❡❧❛çã♦ ❛ t♦❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ▼❛t❡♠át✐❝❛✱ q✉❡ é ♠✐❧❡♥❛r✳ ❉❛t❛✲s❡ ❞♦ sé❝✉❧♦ ❳■❳✱ ❡ t❡♠ ❝♦♠♦ r❡s♣♦♥sá✈❡❧ ❏❛❝♦❜ ❙t❡✐♥❡r ✭✶✼✾✻✲✶✽✻✸✮✱ ♠❛t❡♠át✐❝♦ s✉íç♦ q✉❡ só ❢♦✐ ♣❛r❛ ❛ ❡s❝♦❧❛ ❛♣ós ♦s ✶✽ ❛♥♦s✱ ♠❛s q✉❡ ❛♣❡s❛r ❞✐ss♦ ❢♦✐ ❝♦♥s✐❞❡r❛❞♦ ♦ ♠❛✐♦r ❣❡ô♠❡tr❛ s✐♥tét✐❝♦ ❞♦s t❡♠♣♦s ♠♦❞❡r♥♦s✳ ❖ s❡✉ tr❛❜❛❧❤♦ s❡r✈✐✉ ♣❛r❛ s♦❧✉❝✐♦♥❛r ♣r♦❜❧❡♠❛s q✉❡ ❛té ❡♥tã♦ ♥ã♦ ❡r❛♠ ♣♦ssí✈❡✐s ❞❡ s❡r❡♠ r❡s♦❧✈✐❞♦s ♣❡❧❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ ❞❡♥tr❡ ❡st❡s ❡stá ♦ ❢❛♠♦s♦ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ q✉❡ ❝♦♥s✐st❡ ❡♠ tr❛ç❛r ❝✐r❝✉♥❢❡rê♥❝✐❛s t❛♥❣❡♥t❡s ❛ ♦✉tr❛s três ❝✐r❝✉♥❢❡rê♥❝✐❛s ❞❛❞❛s✱ ✈❡❥❛ ❬✹❪✳

### ✷✳✷ ❉❡✜♥✐çã♦ ❞❡ ■♥✈❡rsã♦ ●❡♦♠étr✐❝❛

P♦r s❡ tr❛t❛r ❞❡ ✉♠❛ ●❡♦♠❡tr✐❛ ♥ã♦✲❊✉❝❧✐❞✐❛♥❛ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♥ã♦ s❡r ❢❛♠✐❧✐❛r ❛♦s ♥♦ss♦s ❡st✉❞♦s ♠❛t❡♠át✐❝♦s ✭❥á q✉❡ ♦s ♥♦ss♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ♥ã♦ ❛ ❛❜♦r❞❛♠✮✱ ❝♦♠❡ç❛r❡♠♦s ❛ ❡①♣❧♦rá✲❧❛ ♣❡❧❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ♥❡❝❡ssár✐❛ ❛♦ s❡✉ ❡♥t❡♥❞✐♠❡♥t♦✳

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❉❡✜♥✐çã♦ ✷✳✷✳✶✿ ❉❛❞❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ❡♠ O ❡ r❛✐♦ r✱ ❞✐③❡♠♦s q✉❡ ❛ ✐♥✈❡rsã♦ é ❛ tr❛♥s❢♦r♠❛çã♦ q✉❡ ❢❛③ ❝♦rr❡s♣♦♥❞❡r✱ ❛ ❝❛❞❛ ♣♦♥t♦P ❞♦ ♣❧❛♥♦✱ ❞✐❢❡r❡♥t❡ ❞❡ O✱ ♦ ♣♦♥t♦ P′ q✉❡ s❡ ❡♥❝♦♥tr❛ s♦❜r❡ ❛ s❡♠✐rr❡t❛−→OP ❡ q✉❡ s❛t✐s❢❛③

OP .OP′ =r2

P♦❞❡♠♦s ❡st❡♥❞❡r ❡st❛ ❞❡✜♥✐çã♦ ❢❛③❡♥❞♦ ✉s♦ ❞❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ q✉❡ r❡❧❛❝✐♦♥❛ ✐♥✈❡rsã♦ ❞❡ ✉♠ ♣♦♥t♦ ❝♦♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♥❥✉❣❛❞♦s ❤❛r♠ô♥✐❝♦s✳

Pr♦♣♦s✐çã♦ ✷✳✷✳✶✿ ❖s ♣♦♥t♦s P ❡ P′ sã♦ ✐♥✈❡rs♦s ❡♠ r❡❧❛çã♦ à ❝✐r❝✉♥❢❡rê♥❝✐❛ C

❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ sã♦ ❝♦♥❥✉❣❛❞♦s ❤❛r♠ô♥✐❝♦s ❡♠ r❡❧❛çã♦ ❛♦ ❞✐â♠❡tr♦ AB ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ ✐♥t❡rs❡çã♦ ❞❛ r❡t❛ ←→OP ❝♦♠ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ C✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❛♥❞♦ ♦ ❛❧✐♥❤❛♠❡♥t♦ ♥❛ ♦r❞❡♠ A✱ P′✱ BP✱ ❝♦♠♦ ♥❛

❋✐❣✉r❛ ✷✳✶✱ t❡♠♦s✿

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❋✐❣✉r❛ ✷✳✶✿ ■♥✈❡rsã♦ ❞❡ ♣♦♥t♦ ❡ ❝♦♥❥✉❣❛❞♦ ❤❛r♠ô♥✐❝♦

AP BP =

AP′

BP′ ⇔

(AO+OP) (OP −OB) =

(AO+OP′)

(OB−OP′)

⇔ (r+OP)

(OP −r) =

(r+OP′)

(r−OP′)

⇔ (r+OP).(r−OP′) = (OP r).(r+OP)

⇔ r2

−r.OP′+r.OPOP .OP=r.OP +OP .OPr2

−r.OP′

⇔ 2.r2

= 2.OP .OP′ OP .OP=r2

.

■st♦ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❛ ❞❡♠♦♥str❛çã♦✳

P❛r❛ ❢❛❝✐❧✐t❛r ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ■♥✈❡rsã♦ ●❡♦♠étr✐❝❛✱ ✉s❛r❡♠♦s ♦ r❡❝✉rs♦ ❞❛ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♦ ●❡♦❣❡❜r❛✱ ♣❛r❛ ♦❜t❡r♠♦s ❛ ❧♦❝❛❧✐③❛çã♦ ❞❡ ✉♠ ♣♦♥t♦ ✐♥✈❡rs♦ ❞❡ ✉♠ ❞❛❞♦ ♣♦♥t♦✱ ❞✐❢❡r❡♥t❡ ❞❡ O ✭❋✐❣✉r❛ ✷✳✷✮✳ ❱❛♠♦s ❛ ❝♦♥str✉çã♦✿

✶✳ ❉❡s❡♥❤❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦ λ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r❀

✷✳ ▼❛rq✉❡ ✉♠ ♣♦♥t♦ P ❡ ❞❡s❡♥❤❡ ❛ s❡♠✐rr❡t❛ −→OP❀

✸✳ P❛ss❛♥❞♦ ♣♦r O✱ tr❛ç❛r ❛ ♣❡r♣❡♥❞✐❝✉❧❛r à s❡♠✐rr❡t❛−→OP ❡ ❛ss✐♥❛❧❛r ♦s ♣♦♥t♦s A ❡ B ❞❡ ✐♥t❡rs❡❝çã♦ ❝♦♠ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛❀

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✹✳ ❚r❛ç❛r ❛ s❡♠✐rr❡t❛ −→AP ❡ ❛ss✐♥❛❧❛r ♦ ♣♦♥t♦ C✱ ♦✉tr♦ ♣♦♥t♦ q✉❡ ✐♥t❡rs❡❝t❛ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛❀

✺✳ ❚r❛ç❛r ❛ s❡♠✐rr❡t❛ −BC❀−→

✻✳ ❆ss✐♥❛❧❛r ♦ ♣♦♥t♦P′✱ ✐♥✈❡rs♦ ❞❡P ❡♠ r❡❧❛çã♦ ❛λ ✱ q✉❡ r❡s✉❧t❛ ❞❛ ✐♥t❡rs❡❝çã♦

❞❛ s❡♠✐rr❡t❛ −→OP ❡ −BC✳−→

❱❡❥❛ q✉❡ P′ ❛ss✐♠ ❝♦♥str✉í❞♦✱ ❞❡ ❢❛t♦✱ é ♦ ✐♥✈❡rs♦ ❞♦ ♣♦♥t♦ P✱ ♣♦✐s ♦s tr✐â♥❣✉❧♦s

r❡tâ♥❣✉❧♦s AOP ❡ BOP′ sã♦ s❡♠❡❧❤❛♥t❡s ♣❡❧♦ ❝❛s♦ AA✱ ❧♦❣♦✱ OA OP′ =

OP

OB ✱ ♦✉ s❡❥❛✱ OA.OB =OP .OP′✳ ❈♦♠♦ OA=OB =r2✱ ❡♥tã♦✱

OP .OP′ =r2✳

❋✐❣✉r❛ ✷✳✷✿ ❈♦♥str✉çã♦ ❞♦ ✐♥✈❡rs♦ ❞❡ ✉♠ ♣♦♥t♦

❖❜s❡r✈❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❛❞❛ ❛❝✐♠❛✱ q✉❡ P′ é ✐♥✈❡rs♦ ❞❡ P ❡♠ r❡❧❛çã♦ ❛ λ✱ s❡ ❡

s♦♠❡♥t❡ s❡ P é ✐♥✈❡rs♦ ❞❡ P′ r❡❧❛t✐✈❛♠❡♥t❡ ❛ λ✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡ s❡ A 6=O✱

(A′)=A ❡✱ ♣♦rt❛♥t♦✱ ❛ ✐♥✈❡rsã♦ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❥❡t♦r❛ ❞♦ ♣❧❛♥♦✱ ❝♦♠ ❡①❝❡çã♦ ❞❡

O✳ ❉✐❛♥t❡ ❞❛ ❝♦♥str✉çã♦ ❛♣r❡s❡♥t❛❞❛✱ s✉r❣❡ ♦ s❡❣✉✐♥t❡ q✉❡st✐♦♥❛♠❡♥t♦✿ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ♦ ✐♥✈❡rs♦ ❞❡ q✉❛❧q✉❡r ♣♦♥t♦ ❞♦ ♣❧❛♥♦❄ ❆ r❡s♣♦st❛ é ♥ã♦✦ P♦✐s ✈❡r✐✜❝❛✲s❡ q✉❡ ♦ ❛❢❛st❛♠❡♥t♦ ❞❡ q✉❛❧q✉❡r ♣♦♥t♦ ❞♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦ ❧❡✈❛

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♦ s❡✉ ✐♥✈❡rs♦ ❛ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞♦ ♠❡s♠♦ ❝❡♥tr♦ ❡ ✈✐❝❡✲✈❡rs❛✳ ◆♦ ❡♥t❛♥t♦✱ ♣❛r❛ q✉❡ ❛ ✐♥✈❡rsã♦ s❡❥❛ ✈❡r❞❛❞❡✐r❛♠❡♥t❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❣❡♦♠étr✐❝❛ s❡ ❢❛③ ♥❡❝❡ssár✐♦ ❡st❛❜❡❧❡❝❡r ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ❝❛❞❛ ♣♦♥t♦ ❞♦ ♣❧❛♥♦ ❡ s❡✉ r❡s♣❡❝t✐✈♦ ✐♥✈❡rs♦✳ ▼❛s ❝♦♠♦ ♦ ✐♥✈❡rs♦ ❞❡ O ❡stá ✐♥✜♥✐t❛♠❡♥t❡ ❧♦♥❣❡✱ ❝♦♥✈❡♥❝✐♦♥❛♠♦s ✉♠ ♣♦♥t♦ ♥♦ ✐♥✜♥✐t♦ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r Ω✱ ❞❡ ♠♦❞♦ q✉❡ O é ♦ ✐♥✈❡rs♦ ❞❡ Ω ❡ Ωé ♦ ✐♥✈❡rs♦ ❞❡ O✳ ❱❡❥❛ ❬✼❪✳

❆♥t❡s ❞❛ ♣ró①✐♠❛ ❞❡✜♥✐çã♦✱ é ♥❡❝❡ssár✐♦ ♣♦st✉❧❛r q✉❡Ω ❡stá ❡♠ t♦❞❛s ❛s r❡t❛s✱ ♣♦✐sOP ♣♦❞❡ ❝r❡s❝❡r ✐♥✜♥✐t❛♠❡♥t❡ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ r❡t❛ q✉❛❧q✉❡r✳ ▲♦❣♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ t♦❞❛s ❛s r❡t❛s ❞♦ ♣❧❛♥♦ ♣♦ss✉❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠✳ ❊ ❛ ❡ss❡ ♥♦✈♦ ♣❧❛♥♦ ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ P❧❛♥♦ ■♥✈❡rs✐✈♦✳

❉❡✜♥✐çã♦ ✷✳✷✳✷✿ ❙❡❥❛ λ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r✳ ❆ tr❛♥s❢♦r♠❛çã♦ ❣❡♦♠étr✐❝❛ ♥♦ P❧❛♥♦ ■♥✈❡rs✐✈♦ q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ ♣♦♥t♦ P 6= O ♦ s❡✉ ✐♥✈❡rs♦ P′ ❡♠

r❡❧❛çã♦ àλ❡ é t❛❧ q✉❡ ❖ é ♦ ✐♥✈❡rs♦ ❞❡Ω❡Ωé ♦ ✐♥✈❡rs♦ ❞❡O é ❞❡♥♦♠✐♥❛❞❛ ✐♥✈❡rsã♦✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ✐♥✈❡rsã♦ t♦r♥❛✲s❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡✜♥✐❞❛ ❡♠ t♦❞♦ ♦ ♣❧❛♥♦ ❡ ❝♦♠ ✈❛❧♦r❡s ❡♠ t♦❞♦ ♦ ♣❧❛♥♦✳

❆❣♦r❛ ❝♦♠ ✉♠❛ ❞❡✜♥✐çã♦ ♣r❡❝✐s❛ ❞❡ ✐♥✈❡rsã♦✱ ♣♦❞❡♠♦s ❡①♣❧♦r❛r ❡ss❛ ✏♥♦✈❛✑ ❣❡♦♠❡tr✐❛ ♥✉♠❛ ♣❡rs♣❡❝t✐✈❛ ❞❡ ❝♦♥str✉çã♦ ❞❡ ♥♦✈♦s ❝♦♥❝❡✐t♦s q✉❡ ♥♦s ♣♦ss✐❜✐❧✐t❛rã♦ ❞❡s✈❡♥❞❛r s♦❧✉çõ❡s ♣❛r❛ ♣r♦❜❧❡♠❛s q✉❡✱ ❛té ❡♥tã♦✱ t♦r♥❛✈❛♠✲s❡ tr❛❜❛❧❤♦s♦s s♦❜r❡ ❛ ót✐❝❛ ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ♦✉ ❛té ♠❡s♠♦ ✐rr❡s♦❧ú✈❡✐s✳ P❛r❛ ❡st❡ ✜♠✱ s❡rã♦ ❝♦♥str✉í❞❛s✱ ❛tr❛✈és ❞❡ ✉♠ r❡❝✉rs♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✿ ♦ ●❡♦❣❡❜r❛✱ t♦❞❛s ❛s ✜❣✉r❛s q✉❡ t❡rã♦ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ❞❛r ♦ s✉♣♦rt❡ ✈✐s✉❛❧ às ❞❡✜♥✐çõ❡s✱ t❡♦r❡♠❛s✱ ♣r♦♣♦s✐çõ❡s✱ ❧❡♠❛s✱ ❝♦r♦❧ár✐♦s ❡ ❞❡♠♦♥str❛çõ❡s ♣r❡s❡♥t❡s ♥❡st❡ tr❛❜❛❧❤♦✳ ❊st❡ r❡❝✉rs♦ t♦r♥❛✲s❡ út✐❧ ♣♦r ♣♦ss✉✐r ✉♠❛ ❢❡rr❛♠❡♥t❛ ❝❤❛♠❛❞❛ ✐♥✈❡rsã♦✱ q✉❡ ❢❛❝✐❧✐t❛rá ❛ ❝♦♥str✉çã♦ ❞❛s r❡❢❡r✐❞❛s ✜❣✉r❛s✳

❙❡r✐❛ ❛ ✐♥✈❡rsã♦ ✉♠❛ ✐s♦♠❡tr✐❛❄ ◗✉❛❧ é✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ✐♥✈❡rs♦ ❞❡ ✉♠❛ ❝♦r❞❛ AB ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦❄ ❱❡❥❛ ❛ ❝♦♥str✉çã♦ ❛❜❛✐①♦✿

❖❜s❡r✈❡ q✉❡ ♦ ✐♥✈❡rs♦ ❞❡ AB é ♦ ❛r❝♦ ACd ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❝♦♥❝❧✉✐✲s❡ q✉❡

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❋✐❣✉r❛ ✷✳✸✿ ✐♥✈❡rs♦ ❞❡ AB ❡♠ r❡❧❛çã♦ ❛ λ

❛ ✐♥✈❡rsã♦ ♥ã♦ é ✉♠❛ ✐s♦♠❡tr✐❛✱ ♣♦✐s ♥ã♦ ♣r❡s❡r✈❛ ❞✐stâ♥❝✐❛s✳

❈♦♠ ❛s ❝♦♥❝❧✉sõ❡s ❞❛❞❛s ❛❝✐♠❛✱ s♦♠♦s ✐♥st✐❣❛❞♦s ❛ ♣❡r❣✉♥t❛r✿ q✉❛❧ s❡r✐❛ ❛ ✜❣✉r❛ ❢♦r♠❛❞❛ ♣❡❧❛ ✐♥✈❡rsã♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛❄ ❊ ❞❡ ✉♠ q✉❛❞r❛❞♦❄ ❊ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ q✉❛❧q✉❡r❄ ▼❛✐s ✉♠❛ ✈❡③ ♦ ✉s♦ ❞❛ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛ s❡rá ❡ss❡♥❝✐❛❧ ♣❛r❛ ♦❜t❡r♠♦s ❡ss❛s r❡s♣♦st❛s✳ ❆ ♣r✐♥❝í♣✐♦ ❞❡s❝r❡✈❡♠♦s ❛ s❡❣✉✐r ♦ r♦t❡✐r♦ q✉❡ ♥♦s ♣♦ss✐❜✐❧✐t❛rá t✐r❛r ✉♠❛ ❝♦♥❝❧✉sã♦ ❡♠ r❡❧❛çã♦ ❛♦ ✐♥✈❡rs♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦✿

✶✳ ❈♦♥str✉❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦C ❞❡ ❝❡♥tr♦ O❀

✷✳ ❈♦♥str✉❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ C1 ❞❡ ❝❡♥tr♦O′ ♣❛ss❛♥❞♦ ♣♦r O❀

✸✳ ❈♦♥str✉❛ ✉♠ ♣♦♥t♦P s♦❜r❡ C1❀

✹✳ ❈♦♥str✉❛P′✱ ♦ ✐♥✈❡rs♦ ❞❡ P ❡♠ r❡❧❛çã♦ àC

✺✳ ▲✐❣✉❡ ♦ r❛str♦ ❞♦ ♣♦♥t♦P′ ❡ ♠♦✈✐♠❡♥t❡ ♦ ♣♦♥t♦P s♦❜r❡ t♦❞❛ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛

C1✭❋✐❣✉r❛ ✷✳✹✮✳

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❋✐❣✉r❛ ✷✳✹✿ ❘❛str♦ ❞❡ P′

❖❜s❡r✈❡ q✉❡ ♦ r❛str♦ ❞❡ P′ ❢♦r♠❛ ✉♠❛ r❡t❛✳ ▼❛✐s ❞♦ q✉❡ ✐st♦✱ ❛ r❡t❛ ❢♦r♠❛❞❛ é

♣❡r♣❡♥❞✐❝✉❧❛r ❛ r❡t❛ OO′✳ ▲♦❣♦✱ ♣♦❞❡♠♦s ❡♥✉♥❝✐❛r ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿

Pr♦♣♦s✐çã♦ ✷✳✷✳✷✿ ❙❡❥❛ C ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦r✳ ❆ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ àC ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛C1✱ ❞❡ ❝❡♥tr♦O′✱ q✉❡ ♣❛ss❛ ♣♦rO é ✉♠❛ r❡t❛ q✉❡

♥ã♦ ♣❛ss❛ ♣♦r O ❛ q✉❛❧ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ OO′

❉❡♠♦♥str❛çã♦✿ ▼♦str❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ♣♦♥t♦ P s♦❜r❡ C1✱

s❡✉ ✐♥✈❡rs♦ P′ ❡st❛rá s♦❜r❡ ❛ r❡t❛ r✱ ♣❡r♣❡♥❞✐❝✉❧❛r à OO✱✭✈❡❥❛ ❋✐❣✉r❛ ✷✳✺✮✳ P♦r

❝♦♥str✉çã♦✱ C é ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r ❡ C1 é ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O′ ♣❛ss❛♥❞♦

♣♦r O✳ ❈♦♥s✐❞❡r❡ P ❞✐❛♠❡tr❛❧♠❡♥t❡ ♦♣♦st♦ à ❖✳ ❙❡❥❛ P✬ ✐♥✈❡rs♦ ❞❡ P r❡❧❛t✐✈♦ à ❈✳ ❈♦♥s✐❞❡r❡ ◗ s♦❜r❡ C1 ❡ s❡❥❛ Q′ s❡✉ ✐♥✈❡rs♦ r❡❧❛t✐✈♦ à C✳ ❖s â♥❣✉❧♦s ∠P OQ ❡

∠P′OQsã♦ ❝♦♥❣r✉❡♥t❡s✳ ❆❧é♠ ❞✐ss♦ t❡♠♦sOQ.OQ=OP .OP′✱ ❛♠❜♦s ✐♥✈❡rs♦s ♣♦r

❝♦♥str✉çã♦✳ ▲♦❣♦✱OQ/OP′ =OP /OQ′✳ ❖❜s❡r✈❡ t❛♠❜é♠ q✉❡ ♦s tr✐â♥❣✉❧♦sP OQ

Q′OPsã♦ s❡♠❡❧❤❛♥t❡s ✭❝❛s♦LAL✮✳ ❊ ❝♦♠♦ ♦ â♥❣✉❧♦OQP é r❡t♦ ✭tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦

❡♠ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡♠ q✉❡ ✉♠ ❞♦s ❧❛❞♦s (OP) é ❞✐â♠❡tr♦✮✱ ❡♥tã♦ ♦ â♥❣✉❧♦ OP′Q

t❛♠❜é♠ é r❡t♦ ♣♦✐s é ❝♦♥❣r✉❡♥t❡ ❛ OQP✳ ▲♦❣♦✱ Q′Pé r❡t❛ ✭❧❛❞♦ ❞♦ tr✐â♥❣✉❧♦✮

♣❡r♣❡♥❞✐❝✉❧❛r ❛ OO′✳

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❋✐❣✉r❛ ✷✳✺✿ ■♥✈❡rsã♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ❞❡✈❡♠♦s ♠♦str❛r t❛♠❜é♠ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ♣♦♥t♦ U′ s♦❜r❡ r

❡①✐st❡ s✉❛ ✐♠❛❣❡♠ U s♦❜r❡ C1✳ P♦r ❝♦♥str✉çã♦✱ C é ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦r ❡

C1 é ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦O′ ♣❛ss❛♥❞♦ ♣♦r O✳ ❈♦♥s✐❞❡r❡P ∈C1 ❞✐❛♠❡tr❛❧♠❡♥t❡ ♦♣♦st♦

à O✳ ❈♦♥str✉❛♠♦s P′ ✐♥✈❡rs♦ ❞❡ P r❡❧❛t✐✈♦ à C ❡ s❡❥❛ r ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P

❡ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ←→OP✳ ❈♦♥s✐❞❡r❡ U′ ♣♦♥t♦ s♦❜r❡ r✳ ❆ s❡♠✐rr❡t❛ OU❝♦rt❛ C 1

❡♠ ✉♠ ♣♦♥t♦ U✳ ❖s tr✐â♥❣✉❧♦s P OU ❡ U′OPsã♦ s❡♠❡❧❤❛♥t❡s ✭❝❛s♦ ❆❆❆✮✳ ▲♦❣♦✱

OP/OU′ = OU/OP✳ P♦rt❛♥t♦ (OP).(OP) = (OU).(OU)✳ ▼❛s✱ ♣♦r ❝♦♥str✉çã♦

(OU).(OU′) = r2 ♣♦✐s

(OP).(OP′) = r2✳ ▲♦❣♦

U ❡ U′ sã♦ ✐♥✈❡rs♦s r❡❧❛t✐✈♦s ❛ C✱

❝♦♠♦ q✉❡rí❛♠♦s ♠♦str❛r✳

❆♣ós ✈❡r♠♦s ❡st❡ r❡s✉❧t❛❞♦ ✜❝❛ ❛ s❡❣✉✐♥t❡ ♣❡r❣✉♥t❛✿ ♦ ✐♥✈❡rs♦ ❞❡ q✉❛❧q✉❡r ❝✐r❝✉♥❢❡rê♥❝✐❛ r❡s✉❧t❛ ❡♠ ✉♠❛ r❡t❛❄ ❱❡❥❛♠♦s ♦ q✉❡ ❛❝♦♥t❡❝❡ s❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♥ã♦ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦✳ ❙❡❣✉✐♥❞♦ ♦s ♠❡s♠♦s ♣❛ss♦s ❞❛ ❝♦♥str✉çã♦ ❛♥t❡r✐♦r✱ t❡♠♦s ❛ ❋✐❣✉r❛ ✷✳✻✿

P❛rt✐♥❞♦ ❞❛s ❝♦♥❝❧✉sõ❡s ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞♦ r❛str♦ ❞❡ P✬✱ ♥❛ ✜❣✉r❛ ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❡♥✉♥❝✐❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

Pr♦♣♦s✐çã♦ ✷✳✷✳✸✿ ❙❡❥❛ C ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦r✳ ❆ ✐♥✈❡rsã♦ ❡♠

(31)

❋✐❣✉r❛ ✷✳✻✿ ❘❛str♦ ❞❡ P✬

r❡❧❛çã♦ àC ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛C1 q✉❡ ♥ã♦ ♣❛ss❛ ♣♦rO é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡

♥ã♦ ♣❛ss❛ ♣♦r O✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ C ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r ❡ C1 ✉♠❛

❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ♥ã♦ ♣❛ss❛ ♣♦r O✳ ❙❡❥❛ s ✉♠❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦rO ❡ é s❡❝❛♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ C1✱ ✈❡❥❛ ❋✐❣✉r❛ ✷✳✼✳ ❙❡♥❞♦ A 6=B ♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❞❡ s ❝♦♠

C1✱ ❡♥tã♦✱ ♦s ✐♥✈❡rs♦sA′ ❡B′ ❞❡A❡B✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♠ r❡❧❛çã♦ àC✱ ♣❡rt❡♥❝❡♠

à r❡t❛ s✳ ❆❣♦r❛✱ s❡❥❛P /∈s t❛❧ q✉❡ P ∈C1✳ ❙❡ P′ é ♦ ✐♥✈❡rs♦ ❞❡ P ❡♠ r❡❧❛çã♦ à C✱

❡♥tã♦✱ t❡♠♦s q✉❡

r2

=OA.OA′ =OP .OP.

▲♦❣♦✱ OA/OP′ = OP /OA′✳ P♦rt❛♥t♦✱ ❝♦♠♦ ♦s â♥❣✉❧♦s POAb ❡ POAb ′ ❝♦✐♥❝✐❞❡♠✱

t❡♠♦s q✉❡ ♦s tr✐â♥❣✉❧♦s OAP ❡ OP′Asã♦ s❡♠❡❧❤❛♥t❡s✳ ❆ss✐♠✱

OAPb = OcP′A′ ✭✷✳✶✮

❆♥❛❧♦❣❛♠❡♥t❡✱ t❡♠♦s q✉❡

(32)

❋✐❣✉r❛ ✷✳✼✿ ■♥✈❡rsã♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ♥ã♦ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✳

OBPb = OcP′B✭✷✳✷✮

❆❣♦r❛✱ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ s♦♠❛ ❞♦s â♥❣✉❧♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ t❡♠♦s q✉❡✿

OAPb =ABPb +AP Bb =OP Bb +AP B.b

▲♦❣♦✱

AP Bb =OAPb −OBPb ✭✷✳✸✮

P♦r ♦✉tr♦ ❧❛❞♦✱ ❖ cPA=OcPB+A′cPB✱ ❧♦❣♦✱

A′cP′B

=OcP′A

OcP′B✭✷✳✹✮

(33)

❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛s ❡q✉❛çõ❡s (2.1),(2.2),(2.3)❡ (2.4)✱ t❡♠♦s q✉❡

AP Bb =A′cPB.

P♦rt❛♥t♦✱ ❝♦♠♦ P ❞❡s❝r❡✈❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ C1✱ ❡♥tã♦✱ P′ ❞❡s❝r❡✈❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛

C′

1✱ ❝♦♠♦ q✉❡rí❛♠♦s ♠♦str❛r✳

❆ ♣❛rt✐r ❞❡ss❡ ♠♦♠❡♥t♦✱ ❞❡❞✐❝❛r❡♠♦s ♥♦ss♦s ❡s❢♦rç♦s ♣❛r❛ t❡♥t❛r♠♦s ❞❡s✈❡♥❞❛r q✉❡ t✐♣♦ ❞❡ ✜❣✉r❛ s❡rá ❢♦r♠❛❞❛ ❛♦ ❝❛❧❝✉❧❛r♠♦s ♦ ✐♥✈❡rs♦ ❞❡ ✉♠❛ r❡t❛ ♣❛ss❛♥❞♦ ♦✉ ♥ã♦ ♣❡❧♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦✳ ❋❡✐t♦ ✐st♦✱ ❡st❛r❡♠♦s ❡♠ ❝♦♥❞✐çõ❡s ❞❡ r❡s♦❧✈❡r ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ♣♦r ♠❡✐♦ ❞❛ ✐♥✈❡rsã♦✳

Pr♦♣♦s✐çã♦ ✷✳✷✳✹✿ ❙❡❥❛ C ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦r✳ ❆ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ C ❞❡ ✉♠❛ r❡t❛ s q✉❡ ♣❛ss❛ ♣♦rO é ❛ ♣ró♣r✐❛ r❡t❛ s✳

❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ♠♦str❛r q✉❡ ♦s ✐♥✈❡rs♦s ❞♦s ♣♦♥t♦s ❞❡ st❛♠❜é♠ ❡stã♦ ❡♠ s✳ ❱❡❥❛ ❋✐❣✉r❛ ✷✳✽✳

❋✐❣✉r❛ ✷✳✽✿ ■♥✈❡rsã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✳

❙❡P 6=O é ✉♠ ♣♦♥t♦ ❞❛ r❡t❛ s✱ ❡♥tã♦✱ ♦ ✐♥✈❡rs♦ P′ ❞❡P ♣❡rt❡♥❝❡ à s❡♠✐rr❡t❛ −→OP

❧♦❣♦✱ ❝♦♠♦ P ❡ O ♣❡rt❡♥❝❡♠ à s✱ ❡♥tã♦✱ P′ s✳ ❙❡ P =O✱ ❡♥tã♦✱ s❡✉ ✐♥✈❡rs♦ q✉❡ é

Ω♣❡rt❡♥❝❡ às✱ ♣♦✐s ♣❡rt❡♥❝❡ ❛ t♦❞❛s ❛s r❡t❛s✳ ❆❣♦r❛✱ r❡st❛ ♠♦str❛r q✉❡ ❝❛❞❛ ♣♦♥t♦

(34)

Q❞❡ s é ✐♥✈❡rs♦ ❞❡ ❛❧❣✉♠ ♣♦♥t♦ ❞❡s✳ ❈♦♠♦Q∈s✱ ❡♥tã♦✱ ❝❤❛♠❛♥❞♦ ❞❡P ♦ ✐♥✈❡rs♦ ❞❡ Q t❡♠♦s q✉❡ P = Q′ s✳ ▲♦❣♦✱ P s ❡✱ ❝♦♠♦ P= Q′′ = Q✱ ❡♥tã♦✱ Q é ♦

✐♥✈❡rs♦ ❞❡ ✉♠ ♣♦♥t♦ P ❞❡s✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡ s′ =s✱ ❝♦♠♦ q✉❡rí❛♠♦s ♠♦str❛r✳

Pr♦♣♦s✐çã♦ ✷✳✷✳✺✿ ❙❡❥❛ C ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦r✳ ❆ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ à C ❞❡ ✉♠❛ r❡t❛ s q✉❡ ♥ã♦ ♣❛ss❛ ♣♦rO é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ♣❛ss❛ ♣♦r O✳

❉❡♠♦♥str❛çã♦✿ ❉❛❞❛ ✉♠❛ r❡t❛ s q✉❡ ♥ã♦ ♣❛ss❛ ❡♠ O✱ tr❛❝❡ ❛ s❡♠✐rr❡t❛ −→OA ♣❡r♣❡♥❞✐❝✉❧❛r ❛ s♥♦ ♣♦♥t♦A ❡ ♦ ♣♦♥t♦A′✱ ✐♥✈❡rs♦ ❞❡A✱ ✈❡❥❛ ❋✐❣✉r❛ ✷✳✾✳ P❛r❛ ❝❛❞❛

P ∈s✱ P 6=A✱ tr❛❝❡ s❡✉ ✐♥✈❡rs♦ P′✳ ❉❡ss❛ ❢♦r♠❛✱ t❡♠✲s❡

OA.OA′ =OP .OP=r2

.

❊ ❛ss✐♠ ❝♦♠♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✷✳✷✳✷ ✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦s tr✐â♥❣✉❧♦s ∆OA′P′ ❡ ∆OP A sã♦ s❡♠❡❧❤❛♥t❡s ✭♣❡❧♦ ❝❛s♦ LAL ❞❡ s❡♠❡❧❤❛♥ç❛✮ ❡✱ ♣♦rt❛♥t♦✱ ♦

tr✐â♥❣✉❧♦ ∆OA′Pé r❡tâ♥❣✉❧♦ ❡♠ P

❋✐❣✉r❛ ✷✳✾✿ ■♥✈❡rsã♦ ❞❛ r❡t❛ q✉❡ ♥ã♦ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✳

(35)

▼❛s ❝♦♠♦ t♦❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ é ✐♥s❝r✐tí✈❡❧ ♥✉♠❛ s❡♠✐❝✐r❝✉♥❢❡rê♥❝✐❛✱ s❡❣✉❡ q✉❡ P′ ❡stá ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❞✐â♠❡tr♦ OA✳ ❈♦♠♦ q✉❡rí❛♠♦s ♠♦str❛r✳

### ✷✳✸ Pr♦♣r✐❡❞❛❞❡s ❞❛ ■♥✈❡rsã♦ ●❡♦♠étr✐❝❛

❆♦ ❢♦❝❛r♠♦s ♥❛ r❡s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦✱ t♦r♥❛✲s❡ ✐♥❞✐s♣❡♥sá✈❡❧ ❛❜♦r❞❛r♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ t❛♥❣ê♥❝✐❛ ❡♥✈♦❧✈❡♥❞♦ ✐♥✈❡rsã♦✳

Pr♦♣♦s✐çã♦ ✷✳✸✳✶✿ ❙❡❥❛ C✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦O ❡ r❛✐♦r✳ ❙❡té ✉♠❛ r❡t❛ q✉❡ ♥ã♦ ♣❛ss❛ ♣♦r O✱ ❡♥tã♦ ❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ t′ ✭✐♥✈❡rs❛ ❞❡ t✮ ❡♠ O

é ♣❛r❛❧❡❧❛ à r❡t❛ t✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛♠♦s✱ ♣♦r ❝♦♥tr❛❞✐çã♦✱ q✉❡ ❛ r❡t❛ t❛♥❣❡♥t❡ t0 à

❝✐r❝✉♥❢❡rê♥❝✐❛ t′ ♥♦ ♣♦♥t♦ O ✐♥t❡r❝❡♣t❡ ❛ r❡t❛ t ♥✉♠ ♣♦♥t♦ P q✉❡ ♥ã♦ é ✳ ❙❡

P = t0∩t✱ ❡♥tã♦✱ P 6=O✱ ♣♦✐s O /∈ t, ❡ P′ ∈ t′0∩t

✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✳✹✱ t❡♠♦s

t′

0 =t0 ❡ ❡♥tã♦ ♦s ♣♦♥t♦sP′ ❡O ♣❡rt❡♥❝❡♠ àt0 ❡ t❛♠❜é♠ à ❝✐r❝✉♥❢❡rê♥❝✐❛t′✳ ▲♦❣♦✱

t0 ♥ã♦ é t❛♥❣❡♥t❡ ❛ t′✳ ❱❡❥❛ ❛ ❋✐❣✉r❛ ✷✳✶✵✳

❋✐❣✉r❛ ✷✳✶✵✿ ❆ r❡t❛t0 t❛♥❣❡♥t❡ ❛ t′ ❡ ♣❛r❛❧❡❧❛ ❛ t✳

(36)

Pr♦♣♦s✐çã♦ ✷✳✸✳✷✿ ❙❡❥❛ C ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ✐♥✈❡rsã♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r✳ ❙❡❥❛♠ m ❡ n ❞✉❛s r❡t❛s ❝♦♥❝♦rr❡♥t❡s ❡♠ P ❞✐❢❡r❡♥t❡ ❞❡ Ω ✳ ❊♥tã♦✱ ♦ â♥❣✉❧♦ ❡♥tr❡ ❛s r❡t❛s m ❡ n é ♦ ♠❡s♠♦ â♥❣✉❧♦ ❡♥tr❡ m′ n

❉❡♠♦♥str❛çã♦✿ ❉✐✈✐❞✐r❡♠♦s ❡♠ três ❝❛s♦s ♣♦ssí✈❡✐s✿

I)❙✉♣♦♥❤❛♠♦s q✉❡ ❛s r❡t❛sm❡n♣❛ss❛♠ ♣♦rO✳ ❊♥tã♦✱ ❝♦♠♦m=m′ n =n

♦ â♥❣✉❧♦ ❡♥tr❡ m ❡ n é ♦ ♠❡s♠♦ â♥❣✉❧♦ ❡♥tr❡ m′ n

II) ❙✉♣♦♥❤❛♠♦s q✉❡ m ❡ n ♥ã♦ ♣❛ss❡♠ ♣♦r O ✭✈❡❥❛ ❋✐❣✉r❛ ✷✳✶✶✮✳ ❊♥tã♦✱ m′

n′ sã♦ ❝✐r❝✉♥❢❡rê♥❝✐❛s q✉❡ ♣❛ss❛♠ ♣♦r O✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✸✳✶✱ ❛ r❡t❛ t❛♥❣❡♥t❡ à

m′ ❡♠ O é ♣❛r❛❧❡❧❛ à m ❡ ❛ r❡t❛ t❛♥❣❡♥t❡ à n❡♠ O é ♣❛r❛❧❡❧❛ à n✳ ▲♦❣♦✱ ♦ â♥❣✉❧♦

❡♥tr❡ m′ né ✐❣✉❛❧ ❛♦ â♥❣✉❧♦ ❡♥tr❡ m n✳

❋✐❣✉r❛ ✷✳✶✶✿ ➶♥❣✉❧♦s ❡♥tr❡ r❡t❛s ❝♦♥❝♦rr❡♥t❡s ❡ s✉❛s ✐♥✈❡rs❛s

III) ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ r❡t❛ m ♣❛ss❡ ♣♦r O ❡ n ♥ã♦ ♣❛ss❡ ♣♦r O ✭✈❡❥❛ ❋✐❣✉r❛ ✷✳✶✷✮✳ ❊♥tã♦✱ m′ = m né ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ♣❛ss❛ ♣♦r O✱ ❧♦❣♦✱ m n

✐♥t❡r❝❡♣t❛♠✲s❡ ❡♠ O✳ ❆ss✐♠✱ ♦ â♥❣✉❧♦ ❡♥tr❡ m′ n♥♦ ♣♦♥t♦ Pé ✐❣✉❛❧ ❛♦ â♥❣✉❧♦

♥♦ ♣♦♥t♦O.❆❧é♠ ❞✐ss♦✱ ❛ r❡t❛ t❛♥❣❡♥t❡ à m′ =m❡♠ O é ❛ ♣ró♣r✐❛ r❡t❛ m ❡ ❛ r❡t❛

t❛♥❣❡♥t❡ à n′ ❡♠ O é ♣❛r❛❧❡❧❛ à n✳ P♦rt❛♥t♦✱ ♦ â♥❣✉❧♦ ❡♥tr❡ ❛s r❡t❛s t❛♥❣❡♥t❡s à m