PARTICULARIDADES DO TEOREMA DE PONCELET

P❛rt✐❝✉❧❛r✐❞❛❞❡s ❞♦ ❚❡♦r❡♠❛ ❞❡

P♦♥❝❡❧❡t

♣♦r

▼❛r❝♦s ❆♥t♦♥✐♦ ❋❡❧✐① ❞❡ ❆❧♠❡✐❞❛

P❛rt✐❝✉❧❛r✐❞❛❞❡s ❞♦ ❚❡♦r❡♠❛ ❞❡

P♦♥❝❡❧❡t

♣♦r

▼❛r❝♦s ❆♥t♦♥✐♦ ❋❡❧✐① ❞❡ ❆❧♠❡✐❞❛

s♦❜ ♦r✐❡♥t❛çã♦ ❞❛

Pr♦❢

_{✳ ❉r}

_{✳ ❊❧✐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✐❝❛✳

❆❣♦st♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇

❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❡✉s✱ ♣♦r t❡r ♠❡ ❞❛❞♦ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❡st❛r r❡❛❧✐③❛♥❞♦ ❡st❡ tr❛❜❛❧❤♦✱ ✐❧✉♠✐♥❛♥❞♦ ♠✐♥❤❛ ♠❡♥t❡ ❡ ♠❡✉ ❝❛♠✐♥❤♦ ❞✉r❛♥t❡ t♦❞❛ ❛ ❝❛♠✐♥❤❛❞❛❀

❆♦s ♠❡✉s ♣❛✐s ▼ár✐♦ ❋❡❧✐① ❞❡ ❆❧♠❡✐❞❛ ❡ ▼❛r✐❛ ❏♦sé ❞❡ ❆❧♠❡✐❞❛✱ q✉❡ ❡♠❜♦r❛ ♥ã♦ ❡st❛♥❞♦ ♠❛✐s ❝♦♠✐❣♦✱ ✐❧✉♠✐♥❛r❛♠ ❞❡ ❢♦r♠❛ ❜r✐❧❤❛♥t❡ ❡ ❡s♣❡❝✐❛❧ ♦s ♠❡✉s ♣❛ss♦s ❡ ♣❡♥s❛♠❡♥t♦s ♠❡ ❧❡✈❛♥❞♦ ❛ ✈❡♥❝❡r ❡st❛ ❝❛♠✐♥❤❛❞❛ ❡ ♣❛r❛ ♦s q✉❛✐s ❡✉ r♦❣♦ t♦❞❛s ❛s ♥♦✐t❡s ❛ ♠✐♥❤❛ ❡①✐stê♥❝✐❛❀

❆ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ❝♦❧❛❜♦r❛çã♦ ♥❛s ❤♦r❛s ❞✐❢í❝❡✐s ❡ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ❖s♠❛r✐♥❛ ❊✈❛r✐st♦ ❞❡ ❆❧♠❡✐❞❛✱ q✉❡ ❞❡ t❛❧ ❢♦r♠❛ ♠❡ ❞❡✉ ❢♦rç❛s✱ ❝♦♥✜❛♥ç❛ ❡ ❝♦r❛❣❡♠ ♣❛r❛ ❝♦♥t✐♥✉❛r❀

❆ t♦❞♦s ♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ q✉❡ ❞❡ ♠❛♥❡✐r❛ ❝❛r✐♥❤♦s❛✱ ♠❡ ❡♥❝♦r❛❥❛r❛♠ ❝♦♠ ♣❛❧❛✈r❛s ❞❡ ❝♦♥❢♦rt♦ ❡ ❛♠✐❣❛s ❡ q✉❡ ♥❛s ❤♦r❛s ❞✐❢í❝❡✐s s♦✉❜❡r❛♠ ♠❡ ❛✉①✐❧✐❛r ♥♦s tr❛❜❛❧❤♦s ❡ ❞✐✜❝✉❧❞❛❞❡s ❡ ♣♦r ❡st❛r❡♠ ❝♦♠✐❣♦ ♥❡st❛ ❝❛♠✐♥❤❛❞❛ t♦r♥❛♥❞♦✲❛ ♠❛✐s ❢á❝✐❧✱ ❛❧❡❣r❡ ❡ ❛❣r❛❞á✈❡❧✱ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛ ❘❡♥❛rt ❡ ▼❛②❛♥❛❀

❆ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ♣r♦❢❛ _{❊❧✐s❛♥❞r❛ ●❧♦ss ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❝♦♠ q✉❡ ♠❡ ♦r✐❡♥t♦✉}

❞✉r❛♥t❡ ❛ ❡❧❛❜♦r❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧❛ ❝♦♥st❛♥t❡ ♠♦t✐✈❛çã♦ ❡ ♣❛❝✐ê♥❝✐❛❀

❆ ❏❡❛♥ ❱✐❝t♦r P♦♥❝❡❧❡t❀

➚ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

❉❡❞✐❝❛tór✐❛

❉❡❞✐❝♦ ❡st❡ ❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛❀

❉❡✉s ♣❡❧❛ ♠✐♥❤❛ ❡①✐stê♥❝✐❛❀

❆ ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣❡❧❛ ❢é ❡ ❝♦♥✜❛♥ç❛ ❞❡♠♦♥str❛❞❛❀

❆♦s ♠❡✉s ❛♠✐❣♦s ♣❡❧♦ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧❀ ❆♦s ♠❡✉s ♣❛✐s ♣♦r ✐❧✉♠✐♥❛r❡♠ ♠❡✉ ❝❛♠✐♥❤♦ ❡ ♠✐♥❤❛ ♠❡♥t❡❀

❆♦s ♣r♦❢❡ss♦r❡s ♣❡❧♦ s✐♠♣❧❡s ❢❛t♦ ❞❡ ❡st❛r❡♠ ❞✐s♣♦st♦s ❛ ❡♥s✐♥❛r❀

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞♦ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t à ❣❡♦♠❡tr✐❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❯♠❛ ❞❛s ♥♦ss❛s ♣r✐♥❝✐♣❛✐s ♠♦t✐✈❛çõ❡s é q✉❡ ♥♦s ❝✉rs♦s ❞❡ ▼❛t❡♠át✐❝❛ ❛ ♥í✈❡❧ ❞❡ ❊♥s✐♥♦ ▼é❞✐♦ ❛ ●❡♦♠❡tr✐❛ é ♣♦✉❝♦ ✉t✐❧✐③❛❞❛ ❡ ❡♠ ❛❧❣✉♠❛s ❝✐r❝✉♥stâ♥❝✐❛s ♦s t❡♦r❡♠❛s ♥ã♦ sã♦ ❞❡♠♦♥str❛❞♦s ♣❛r❛ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛s t❡♦r✐❛s ❛❜♦r❞❛❞❛s✳ ❋✐♥❛❧♠❡♥t❡✱ ✉♠❛ ❧✐st❛ ❞❡ ❡①❡r❝í❝✐♦s é ♣r♦♣♦st❛✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❚r✐â♥❣✉❧♦s✱ ❈✐r❝✉♥❢❡rê♥❝✐❛s✱ ❊❧✐♣s❡s✱ ❚❛♥❣❡♥t❡s✳

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ P♦♥❝❡❧❡t✬s ❚❤❡♦r❡♠ ❢♦r t❡❛❝❤✐♥❣ ❣❡♦♠❡tr②✳ ❖♥❡ ♦❢ ♦✉r ♠❛✐♥ ♠♦t✐✈❛t✐♦♥s ❢♦r t❤✐s ✇♦r❦ ✐s t❤❛t ✐♥ s♦♠❡ t❤❡ ❤✐❣❤ s❝❤♦♦❧ ❧❡✈❡❞ ♠❛t❤ ❝♦✉rs❡s t❤❡ st✉❞② ♦❢ ❣❡♦♠❡tr② ✐s s❧✐❣❤t❧② ✉s❡❞ ❛♥❞ ✐♥ ❝❡rt❛✐♥ ❝✐r❝✉♥st❛♥❝❡s t❤❡ t❤❡♦r❡♠s ❛r❡ ♥♦t ❞❡♠♦♥str❛t❡❞ t♦ t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ t❤❡♦r✐❡s ❞✐s❝✉ss❡❞✳ ❋✐♥❛❧❧②✱ ❛ ❧✐st ♦❢ ❡①❡r❝✐s❡s ✐s ♣r♦♣♦s❡❞✳

❑❡②✇♦r❞s✿ ❚r✐❛♥❣❧❡s✱ ❈✐r❝❧❡s✱ ❊❧❧✐♣s❡s✱ ❚❛♥❣❡♥ts✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✸

✶✳✶ ♥♦ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦s ◗✉❛❞r✐❧át❡r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✷ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❊❧✐♣s❡ ✷✷

✷✳✶ ❘❡❞✉çã♦ ❛♦ ❝❛s♦ ❝ír❝✉❧♦ ❡ ❡❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷ Pr♦✈❛ ❞♦ P♦r✐s♠❛ ❞❡ P♦♥❝❡❧❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✸ ❙✉❣❡stã♦ ❞❡ ❊①❡r❝í❝✐♦s ✹✷

❆ ❆❧❣✉♠❛s ❘❡❧❛çõ❡s ❚r✐❣♦♥♦♠étr✐❝❛s ✹✾

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✸

◆♦t❛çõ❡s

◆❡st❛ ❞✐ss❡rt❛çã♦ ✉s❛r❡♠♦s ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿

OP ✲ ❙❡❣♠❡♥t♦ ❞❡ ❡①tr❡♠✐❞❛❞❡s ♥♦s ♣♦♥t♦sO ❡ P❀ OP ✲ ❈♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦OP❀

−→

OP ✲ ❙❡♠✐rr❡t❛ ❞❡ ♦r✐❣❡♠ ❡♠ O ❡ ❝♦♥t❡♥❞♦ P❀

←→

OP ✲ ❘❡t❛ ❝♦♥t❡♥❞♦ ♦s ♣♦♥t♦s O ❡ P❀

ø

BC ✲ ❆r❝♦ ❞❡ ❡①tr❡♠✐❞❛❞❡s B ❡C❀

BACÒ ✲ ➶♥❣✉❧♦ ❝♦♠ ✈ért✐❝❡ ❡♠ A ❡ ❧❛❞♦s ♥❛s s❡♠✐rr❡t❛s −→AB❀ −→AC❀ r _⊥s ✲ ❆s r❡t❛sr ❡s sã♦ ♦rt♦❣♦♥❛✐s❀

d(P,−→OA) ✲ ❉✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦P à s❡♠✐rr❡t❛ ❞❡ ♦r✐❣❡♠ ❡♠ O ❡ ❝♦♥t❡♥❞♦ A❀

Γ(O, r) ✲ ❈✐r❝✉♥❢❡rê♥❝✐❛ Γ ❞❡ ❝❡♥tr♦ ♥♦ ♣♦♥t♦ O ❡ r❛✐♦ r✳

■♥tr♦❞✉çã♦

❆ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ ❡❞✉❝❛çã♦ ❞❡ q✉❛❧✐❞❛❞❡ s❡ ❞á q✉❛♥❞♦ r❡✢❡t✐♠♦s s♦❜r❡ ❛s ♥❡❝❡ss✐❞❛❞❡s ❞♦s ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ❡ ❞❡ ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ ❡❞✉❝❛❝✐♦♥❛❧ ❛❜r❛♥❣❡♥t❡✳ ❆♣r❡♥❞❡r é ✈✐✈❡♥❝✐❛r ❡ ❛❞q✉✐r✐r ❡①♣❡r✐ê♥❝✐❛s✱ é ❡♥❢r❡♥t❛r ❞❡s❛✜♦s✱ ❞❡s❝♦❜r✐r ❝♦✐s❛s ♥♦✈❛s✱ ❜✉s❝❛r ❝♦♥❤❡❝✐♠❡♥t♦s✳

❊ss❡ tr❛❜❛❧❤♦ ❢♦✐ ♣❡♥s❛❞♦✱ ❡s❝r✐t♦ ❡ ♦r❣❛♥✐③❛❞♦ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❢❛❝✐❧✐t❛r ❛ ❛♣r❡♥❞✐③❛❣❡♠✱ ♣❛r❛ ❛❥✉❞❛r ♦s ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s ❝♦♠ r❡❧❛çã♦ à ●❡♦♠❡tr✐❛✱ ♥♦ ✐♥t✉✐t♦ ❞❡ ❜✉s❝❛r ♠❛✐♦r ❡♥t❡♥❞✐♠❡♥t♦ s♦❜r❡ ♦ t❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t✱ ❝✉❥♦ ❡♥✉♥❝✐❛❞♦ é✿

❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t✿ ❉❛❞❛s ❞✉❛s ❡❧✐♣s❡s ❞✐s❥✉♥t❛s ❡ ✉♠❛ ❞❡♥tr♦ ❞❛ ♦✉tr❛✱ s❡ ❡①✐st❡ ✉♠ ♣♦❧í❣♦♥♦ ❝✐r❝✉♥s❝r✐t♦ à ❡❧✐♣s❡ ✐♥t❡r♥❛ ❡ ✐♥s❝r✐t♦ à ❡①t❡r♥❛✱ ❞❡ n✲❧❛❞♦s✱

❡♥tã♦ q✉❛❧q✉❡r ♣♦♥t♦ ♣❡rt❡♥❝❡♥t❡ ❛ ❡❧✐♣s❡ ❡①t❡r♥❛ é ✈ért✐❝❡ ❞❡ ❛❧❣✉♠ ♣♦❧í❣♦♥♦ ❝♦♠ ❛s ♠❡s♠❛s ❝❛r❛❝t❡ríst✐❝❛s✳

❊①✐st❡♠ ✈ár✐❛s ♣r♦✈❛s ❞❡ss❡ ♥♦tá✈❡❧ t❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t✱ ❛ ♠❛✐♦r✐❛ ❞❛s q✉❛✐s ♥ã♦ sã♦ ❡❧❡♠❡♥t❛r❡s✳ P♦r ✐ss♦✱ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ❡♥❝♦♥tr❛r ✉♠❛ ♣r♦✈❛ ❡❧❡♠❡♥t❛r ♣❛r❛ ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ♥♦ ❝❛s♦ ❡♠ q✉❡ n = 3✱ ✐st♦ é✱ q✉❛♥❞♦ t❡♠♦s ❞✉❛s ❡❧✐♣s❡s ❡

✉♠ tr✐â♥❣✉❧♦✳

❖ t❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t é ✉♠ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ ❞❛ ❣❡♦♠❡tr✐❛ ❡♥✈♦❧✈❡♥❞♦ ❝ô♥✐❝❛s ✐♥s❝r✐t❛s ❡ ❝✐r❝✉♥s❝r✐t❛s ❛ ✉♠ ♣♦❧í❣♦♥♦✳ P♦r ✐ss♦ ✈❛♠♦s ♠♦str❛r ✉♠❛ ❛❜♦r❞❛❣❡♠ q✉❡ ♥❡❝❡ss✐t❛ ❛♣❡♥❛s ❞❡ ❢❡rr❛♠❡♥t❛s ❞♦s ♣r♦❣r❛♠❛s ❞❡ ❊♥s✐♥♦ ▼é❞✐♦✳

◆❡st❡ ❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ❝♦♥tr✐❜✉✐çõ❡s

❝✐❡♥tí✜❝❛s ❢✉♥❞❛♠❡♥t❛✐s ❞❡ ❏❡❛♥ ❱✐❝t♦r P♦♥❝❡❧❡t ✭✶✼✽✽✲✶✽✻✼✮ r❡❧❛❝✐♦♥❛❞❛s ❛ s❡✉s tr❛❜❛❧❤♦s ❝♦♠ ❛ ●❡♦♠❡tr✐❛✳

❆q✉✐ ✈❛♠♦s ❛♣r❡s❡♥t❛r ♣❛rt✐❝✉❧❛r✐❞❛❞❡s ❞♦ t❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ❡♠ s✐t✉❛❝õ❡s q✉❡ ♣❡r♠✐t❡♠ ♣❡r❝❡❜❡r ❛ ❢❛❝✐❧✐❞❛❞❡ ❝♦♠ q✉❡ t✉❞♦ ❛❝♦♥t❡❝❡✳ ▼♦str❛r❡♠♦s s✐t✉❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ tr✐â♥❣✉❧♦s ❡ ❝ír❝✉❧♦s✱ q✉❛❞r✐❧át❡r♦s ❡ ❝ír❝✉❧♦s ❡ ❡❧✐♣s❡s✳

❙❛❜❡♠♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ q✉❡ ❞❛❞♦ ✉♠ tr✐â♥❣✉❧♦ é s❡♠♣r❡ ♣♦ssí✈❡❧ ❝♦♥str✉✐r ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ❡ ♦✉tr❛ ❝✐r❝✉♥s❝r✐t❛ ❛♦ tr✐â♥❣✉❧♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ t❡♠♦s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s✱ ✉♠❛ ✐♥t❡r♥❛ ❛ ♦✉tr❛✳ ❙❡rá q✉❡ é ♣♦ssí✈❡❧ ❝♦♥str✉✐r ✉♠ tr✐â♥❣✉❧♦ ❛♦ ♠❡s♠♦ t❡♠♣♦ ✐♥s❝r✐t♦ ♥❛ ❡①t❡r✐♦r ❡ ❝✐r❝✉♥s❝r✐t♦ ♥❛ ✐♥t❡r✐♦r❄ ❆♣r❡s❡♥t❛♠♦s ❛q✉✐ ✉♠ r❡s✉❧t❛❞♦ ❜❡♠ ❝♦♥❤❡❝✐❞♦ q✉❡ ♥♦s ❞á ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ t❛❧ tr✐â♥❣✉❧♦ ❡①✐st❛✳

❈❛♣ít✉❧♦ ✶

❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t

◆❡st❡ ❝❛♣ít✉❧♦ ❡st✉❞❛r❡♠♦s ❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❡ P♦♥❝❡❧❡t ♣❛r❛ tr✐â♥❣✉❧♦s✱ q✉❛❞r✐❧át❡r♦s ❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s✳ ❖s r❡s✉❧t❛❞♦s q✉❡ ✐r❡♠♦s ❛♣r❡s❡♥t❛r ❛q✉✐ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✸❪✱ ❬✹❪✱ ❬✼❪✱ ❬✶✵❪ ❡ ❬✶✷❪✳ ❆♥t❡s ❞❡ ❢❛③❡r♠♦s ❛s ❞❡♠♦♥str❛çõ❡s ❞❡s❡❥á✈❡✐s ✈❛♠♦s ♠♦str❛r ❛❧❣✉♠❛s ♣r♦♣♦s✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ ❝❤❡❣❛r♠♦s ❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t✳

✶✳✶ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦

P❛r❛ ♣r♦✈❛r♠♦s ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❛ s❡çã♦ ♣r❡❝✐s❛♠♦s ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s✱ q✉❡ ♣r♦✈❛r❡♠♦s ❛q✉✐ ♣❛r❛ ❞❡✐①❛r ❡st❡ tr❛❜❛❧❤♦ ♠❛✐s ❝♦♠♣❧❡t♦✳

❉❡✜♥✐çã♦ ✶ ❉✐③❡♠♦s q✉❡ ✉♠❛ r❡t❛ t ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ Γ sã♦ t❛♥❣❡♥t❡s ♦✉✱

❛✐♥❞❛✱ q✉❡ ❛ r❡t❛ t é t❛♥❣❡♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ Γ✱ s❡ t ❡ Γ t✐✈❡r❡♠ ❡①❛t❛♠❡♥t❡ ✉♠

♣♦♥t♦ P ❡♠ ❝♦♠✉♠✳ ◆❡st❡ ❝❛s♦✱ P é ❞✐t♦ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛ ❞❡ t ❡ Γ✳

Pr♦♣♦s✐çã♦ ✶✳✶ ✭❬✶✱ Pr♦♣♦s✐çã♦ ✸✳✶✻❪✮ ❙❡❥❛♠ Γ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ P ✉♠

♣♦♥t♦ ❞❡ Γ✳ ❙❡ t é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ←→OP✱ ❡♥tã♦ t é t❛♥❣❡♥t❡

❛ Γ ❡♠ P✳

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✶✳ ♥♦ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦

Pr♦✈❛✿ ❚♦♠❡♠♦s ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ Γ ❞❡ ❝❡♥tr♦ ❡♠O ❡ r❛✐♦r ❡ s❡❥❛ P ✉♠ ♣♦♥t♦

q✉❛❧q✉❡r ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛Γ✳ ❖ s❡❣♠❡♥t♦OP é ✉♠ r❛✐♦✳ ❙❡❥❛t❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r

❛♦ s❡❣♠❡♥t♦ OP ♥♦ ♣♦♥t♦ P✳ ❚♦♠❡♠♦s ✉♠ ♦✉tr♦ ♣♦♥t♦ Q ♥❛ r❡t❛ t ❡ tr❛❝❡♠♦s

♦ s❡❣♠❡♥t♦ OQ✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✶✳ ❈♦♠♦ ♦ ❧❛❞♦ OQ é ♦♣♦st♦ ❛♦ ♠❛✐♦r

â♥❣✉❧♦ ❞♦ tr✐â♥❣✉❧♦∆OP Q✱ t❡♠♦s q✉❡OQ > OP =r✳ ❉❛í ♦ ♣♦♥t♦Q♥ã♦ ♣❡rt❡♥❝❡

❛ ❝✐r❝✉♥❢❡rê♥❝✐❛Γ❡ ♣♦rt❛♥t♦té ❛ r❡t❛ t❛♥❣❡♥t❡ q✉❡ ♣r♦❝✉r❛♠♦s✳ ❈♦♥❝❧✉í♠♦s ❛ss✐♠

❋✐❣✉r❛ ✶✳✶✿ ❘❡t❛ t❛♥❣❡♥t❡

q✉❡ t♦❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ✉♠ r❛✐♦ ♥♦ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛✳

❉❡✜♥✐çã♦ ✷ ✭❬✶✱ ❉❡✜♥✐çã♦ ✸✳✶❪✮ ❉❛❞❛ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ P r❡❧❛t✐✈❛ ❛ ♣♦♥t♦s ❞♦

♣❧❛♥♦✱ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✭❛❜r❡✈✐❛♥❞♦ ▲●✮ ❞♦s ♣♦♥t♦s q✉❡ ♣♦ss✉❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡

P é ♦ s✉❜❝♦♥❥✉♥t♦ £ ❞♦ ♣❧❛♥♦ q✉❡ s❛t✐s❢❛③ ❛s ❞✉❛s ❝♦♥❞✐çõ❡s ❛ s❡❣✉✐r✿

✭❛✮❚♦❞♦ ♣♦♥t♦ ❞❡ £ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ _P✳

✭❜✮❚♦❞♦ ♣♦♥t♦ ❞♦ ♣❧❛♥♦ q✉❡ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ P ♣❡rt❡♥❝❡ ❛ £✳

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ £ é ♦ _LG ❞❛ ♣r♦♣r✐❡❞❛❞❡ _P s❡ £ ❢♦r ❝♦♥st✐t✉í❞♦ ❡①❛t❛♠❡♥t❡

♣❡❧♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ tê♠ ❛ ♣r♦♣r✐❡❞❛❞❡ P✱ ♥❡♠ ♠❛✐s ♥❡♠ ♠❡♥♦s✳ ◆♦ q✉❡

s❡❣✉❡✱ ✈❡r❡♠♦s ❛❧❣✉♥s ❧✉❣❛r❡s ❣❡♦♠étr✐❝♦s ❡❧❡♠❡♥t❛r❡s✳

❉❡✜♥✐çã♦ ✸ ✭❬✶✱ ❊①❡♠♣❧♦ ✸✳✷❪✮ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ r ❡ ✉♠ ♣♦♥t♦ O ❞♦

♣❧❛♥♦✱ ♦ ▲● ❞♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ ❞✐st❛♠ r ❞♦ ♣♦♥t♦ O é ♦ ❝ír❝✉❧♦ Γ ❞❡ ❝❡♥tr♦

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✶✳ ♥♦ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦

O ❡ r❛✐♦ r✿

AO=r⇐⇒A∈Γ(O, r)

✳

❋✐❣✉r❛ ✶✳✷✿ ❈ír❝✉❧♦ ❝♦♠♦ ▲●✳

❉❡✜♥✐çã♦ ✹ ❉❛❞♦s ♦s ♣♦♥t♦s A ❡ B ♥♦ ♣❧❛♥♦✱ ❛ ♠❡❞✐❛tr✐③ ❞♦ s❡❣♠❡♥t♦ AB é ❛

r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ AB ❡ q✉❡ ♣❛ss❛ ♣♦r s❡✉ ♣♦♥t♦ ♠é❞✐♦✳

Pr♦♣♦s✐çã♦ ✶✳✷ ✭❬✶✱ Pr♦♣♦s✐çã♦ ✸✳✹❪✮ ❉❛❞♦s ♣♦♥t♦s A ❡ B ♥♦ ♣❧❛♥♦✱ ❛ ♠❡❞✐❛tr✐③

❞❡ AB ❡ ♦ LG ❞♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ ❡q✉✐❞✐st❛♠ ❞❡ A ❡ ❞❡ B✳

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

Pr♦✈❛✿ ❙❡❥❛♠M ♦ ♣♦♥t♦ ♠é❞✐♦ ❡ m ❛ ♠❡❞✐❛tr✐③ ❞❡ AB ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✸✳

❙❡ P ∈ m✱ ❡♥tã♦✱ ♥♦ tr✐â♥❣✉❧♦ ∆P AB✱ P M é ♠❡❞✐❛♥❛ ♣♦✐s t❡♠♦s q✉❡ M é ♣♦♥t♦

♠é❞✐♦ ❞❡ AB ❡ ❛❧t✉r❛ ❞❡✈✐❞♦ ❛♦ s❡❣♠❡♥t♦ P M s❡r ♣❡r♣❡♥❞✐❝✉❧❛r ❛ AB ❡ ❞❛í✱ ♦

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✶✳ ♥♦ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦

❋✐❣✉r❛ ✶✳✹✿ P A=P B =_⇒P _∈ ✭♠❡❞✐❛tr✐③ ❞❡ AB✮✳

tr✐â♥❣✉❧♦ ∆P AB é ✐sós❝❡❧❡s ❞❡ ❜❛s❡ AB✳ ▲♦❣♦ P A=P B✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛P

✉♠ ♣♦♥t♦ ♥♦ ♣❧❛♥♦ t❛❧ q✉❡ P A=P B ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✹✳ ❊♥tã♦✱ ♦ tr✐â♥❣✉❧♦

∆P AB é ✐sós❝❡❧❡s ❞❡ ❜❛s❡AB✱ ❞♦♥❞❡ s❡❣✉❡ q✉❡ ❛ ♠❡❞✐❛♥❛ ❡ ❛ ❛❧t✉r❛ r❡❧❛t✐✈❛s ❛ AB

❝♦✐♥❝✐❞❡♠ ❡ ❡q✉✐✈❛❧❡♠ ❛♦ s❡❣✉✐♠❡♥t♦ P M✱ ❧♦❣♦ P M _⊥AB✱ ❧♦❣♦P _∈m✱ ✭♠❡❞✐❛tr✐③

❞❡ AB✮✱ ♦ q✉❡ é ♦ ♠❡s♠♦ q✉❡ ❞✐③❡r q✉❡ P M é ♠❡❞✐❛tr✐③ ❞❡ AB✱ ❝♦♠♦ q✉❡rí❛♠♦s

♣r♦✈❛r✳

Pr♦♣♦s✐çã♦ ✶✳✸ ✭❬✶✱ Pr♦♣♦s✐çã♦ ✸✳✷✺❪✮ ❙❡❥❛♠ Γ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O ❡ P ✉♠ ♣♦♥t♦ ❡①t❡r✐♦r à ♠❡s♠❛✳ ❙❡ ♦s ♣♦♥t♦s A ❡ B✱ ♣❡rt❡♥❝❡♥t❡s ❛ Γ✱ sã♦ t❛✐s q✉❡ ←→

P A ❡ ←→P B sã♦ t❛♥❣❡♥t❡s ❛ Γ✱ ❡♥tã♦ P A = P B ❡ ←→P O é ❛ ♠❡❞✐❛tr✐③ ❞❡ AB✳ ❊♠

♣❛rt✐❝✉❧❛r t❡♠♦s ←→P O _⊥ ←→AB✳

❋✐❣✉r❛ ✶✳✺✿ ❘❡t❛s t❛♥❣❡♥t❡s

Pr♦✈❛✿ ❈♦♠♦ OA = OB✱ ❥á q✉❡ sã♦ r❛✐♦s ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡ ♦s â♥❣✉❧♦s PAOÒ = PBOÒ = 90◦_{✱ ♦s tr✐â♥❣✉❧♦s ∆P AO ❡ ∆P BO sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦}

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✶✳ ♥♦ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦

❡s♣❡❝✐❛❧ ❝❛t❡t♦ ❡ ❤✐♣♦t❡♥✉s❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❞❡ tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s✳ ❱❡❥❛ ❋✐❣✉r❛ ✶✳✺✳ ❊♠ ♣❛rt✐❝✉❧❛r t❡♠✲s❡ P A=P B✳ ❆❣♦r❛✱ ❝♦♠♦P ❡ O ❡q✉✐❞✐st❛♠ ❞❡A ❡ ❞❡ B✱

s❡❣✉❡ q✉❡ ←→P O é ❛ ♠❡❞✐❛tr✐③ ❞♦ s❡❣♠❡♥t♦ AB✱ ❝♦♠♦ q✉❡rí❛♠♦s✳

❖ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐r é ♠✉✐t❛s ✈❡③❡s ❝❤❛♠❛❞♦ ❞❡ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✱ ✈❡❥❛ ♣♦r ❡①❡♠♣❧♦ ❬✺❪✳

❚❡♦r❡♠❛ ✶✳✶ ❉❛❞❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ Γ ❞❡ r❛✐♦ r✱ ✐♥s❝r✐t❛ ❡♠ ✉♠ tr✐â♥❣✉❧♦

∆ABC r❡tâ♥❣✉❧♦ ❡♠ A✱ t❡♠✲s❡

AB+AC =BC+ 2r. ✭✶✳✶✮

Pr♦✈❛✿ ❈♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦∆ABC✱ r❡tâ♥❣✉❧♦ ❡♠A✱ ❝✐r❝✉♥s❝r✐t♦ à ❝✐r❝✉♥❢❡rê♥❝✐❛

Γ ❞❡ r❛✐♦ r ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✻✳ ❚♦♠❛♥❞♦ ♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛P✱ Q ❡ R

❋✐❣✉r❛ ✶✳✻✿ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦

❞♦s ❧❛❞♦s BC✱ AC ❡ AB r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❡♠♦s OP = OQ = OR = r✱ r❛✐♦ ❞❛

❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❈♦♠♦ ♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ sã♦ t❛♥❣❡♥t❡s à ❝✐r❝✉♥❢❡rê♥❝✐❛✱ s❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶ q✉❡ ♦s r❛✐♦s OP, OQ ❡ OR sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛♦ ❧❛❞♦sBC, AC ❡ AB✱ ♥♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ P, Q e R r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P♦rt❛♥t♦✱ ♦ q✉❛❞r✐❧át❡r♦

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❈✐r❝✉♥❢❡rê♥❝✐❛

ORAQ é ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ r✳ ❏á q✉❡✿

AC =AQ+QC ✭✶✳✷✮

❡

AB =AR+RB, ✭✶✳✸✮

s♦♠❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛s ❡q✉❛çõ❡s ✭✶✳✷✮ ❡ ✭✶✳✸✮ ♦❜t❡♠♦s

AB+AC =AQ+QC +AR+RB. ✭✶✳✹✮

❙❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✸ q✉❡ AQ = AR = r✱ QC = P C ❡ RB = BP✳ ❊♥tã♦ ❛

❡q✉❛çã♦ ❞❛❞❛ ❡♠ ✭✶✳✹✮ ❡q✉✐✈❛❧❡ ❛

AB +AC =BP +P C+r+r=BC+ 2r

❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

✶✳✷ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❈✐r❝✉♥❢❡rê♥❝✐❛

❆♥t❡s ❞❡ ♣r♦✈❛r♠♦s ✉♠ r❡s✉❧t❛❞♦ q✉❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♣r❡❝✐s❛♠♦s ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s✳

Pr♦♣♦s✐çã♦ ✶✳✹ ❙❡❥❛♠ Γ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ AB✱ CD ❞✉❛s ❝♦r❞❛s ❞❡ Γ✳ ❙❡

❛s ❝♦r❞❛s AB ❡ CD✱ s❡ ✐♥t❡rs❡❝t❛♠ ♥♦ ♣♦♥t♦ P ✐♥t❡r✐♦r ❛ Γ✱ ❡♥tã♦

P A.P B =P C.P D.

Pr♦✈❛✿ ❚r❛ç❛♥❞♦ ♦s s❡❣♠❡♥t♦s BC ❡AD✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✼✱ ♣❡❧♦ t❡♦r❡♠❛

❞♦ â♥❣✉❧♦ ✐♥s❝r✐t♦ ✭✈❡❥❛ ❬✶✱ Pr♦♣♦s✐çã♦ ✸✳✶✽❪✮✱ t❡♠♦s ❛s ✐❣✉❛❧❞❛❞❡s ♣❛r❛ ♦s â♥❣✉❧♦s

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❈✐r❝✉♥❢❡rê♥❝✐❛

❋✐❣✉r❛ ✶✳✼✿ P♦tê♥❝✐❛ ❞❡ P♦♥t♦

DABÒ =BCDÒ ❡ABCÒ =CDAc ✳ ❈♦♠♦ BP CÒ =DP AÒ ✭♣♦✐s sã♦ â♥❣✉❧♦s ♦♣♦st♦s ♣❡❧♦

✈ért✐❝❡✮✱ t❡♠♦s q✉❡ ♦s tr✐â♥❣✉❧♦s ∆CP B ❡∆AP D sã♦ s❡♠❡❧❤❛♥t❡s ✭❝❛s♦ ❆ ❆✮✳ ❉❛í

t❡♠♦s ♣♦r s❡♠❡❧❤❛♥ç❛ q✉❡

P A P C =

P D P B.

▲♦❣♦ P A.P B =P C.P D✱ ❝♦♠♦ ❞❡s❡❥❛❞♦✳

❈♦r♦❧ár✐♦ ✶ ❙ã♦ ❞❛❞♦s ♥♦ ♣❧❛♥♦ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ R✱ ❡ ✉♠ ♣♦♥t♦ P

✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦✳ ❙❡ ✉♠❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P ✐♥t❡rs❡❝t❛ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♥♦s

♣♦♥t♦s A ❡ B ❡♥tã♦✿

P A.P B =R2

−OP2. ✭✶✳✺✮

Pr♦✈❛✿ ❈♦♥s✐❞❡r❡♠♦s s♦♠❡♥t❡ ♦ ❝❛s♦ ❡♠ q✉❡P é ✐♥t❡r✐♦r ❛♦ ❞✐s❝♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛

❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❚r❛❝❡ ♣♦r P ♦ ❞✐â♠❡tr♦ CD ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❝♦♠ P ♣❡rt❡♥❝❡♥t❡

❛ OC✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✽✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✹ ✭t❡♦r❡♠❛ ❞❛s ❝♦r❞❛s✮ t❡♠♦s

q✉❡ P A.P B =P C.P D✳ ❈♦♠♦P C =R−OP ❡ P D =R+OP✱ t❡♠♦s

P A.P B =P C.P D= (R₋OP).(R+OP) =R2

−OP2.

❙❡AB é ✉♠ ❞✐â♠❡tr♦✱ ❜❛st❛ ❝♦♥s✐❞❡r❛r C =A ♦✉B ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣♦s✐çã♦ ❞❡P

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❈✐r❝✉♥❢❡rê♥❝✐❛

❋✐❣✉r❛ ✶✳✽✿ P♦tê♥❝✐❛ ❞❡ P♦♥t♦ ■♥t❡r✐♦r

Pr♦♣♦s✐çã♦ ✶✳✺ ✭❬✶✱ Pr♦♣♦s✐çã♦ ✸✳✺❪✮ ❙❡❥❛AOBÒ ✉♠ â♥❣✉❧♦ ❞❛❞♦✳ ❙❡P é ✉♠ ♣♦♥t♦

❞♦ ♠❡s♠♦✱ ❡♥tã♦

d(P,−→AO) = d(P,BO−−→)_⇐⇒P _∈(bissetriz de AOBÒ ).

❋✐❣✉r❛ ✶✳✾✿ P _∈ ✭❜✐ss❡tr✐③ ❞❡∠AOB✮

Pr♦✈❛✿ ❙✉♣♦♥❤❛ q✉❡ P ♣❡rt❡♥❝❡ à ❜✐ss❡tr✐③ ❞❡ AOBÒ ❡ s❡❥❛♠ M ❡ N✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ♣és ❞❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❜❛✐①❛❞❛s ❞❡ P às r❡t❛s ←→AO ❡ ←→BO

❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✾✳ ❈♦♠♦ MOPÒ = NOPÒ ✱ OM PÓ = ON Pc = 90◦ _{❡ OP}

é ❝♦♠✉♠✱ s❡❣✉❡ q✉❡ ♦s tr✐â♥❣✉❧♦s ∆OM P ❡ ∆ON P sã♦ ❝♦♥❣r✉❡♥t❡s ♣♦r LAAo✳

❉❛í✱ P M = P N✱ ♦✉ s❡❥❛✱ d(P,←→OA) ❂ d(P,←→OB)✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ P ✉♠

♣♦♥t♦ ♥♦ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦ AOBÒ ✱ t❛❧ q✉❡ P M ❂ P N✱ ♦♥❞❡ M ❡ N sã♦ ♦s ♣és

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❈✐r❝✉♥❢❡rê♥❝✐❛

❞❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❜❛✐①❛❞❛s ❞❡ P r❡s♣❡❝t✐✈❛♠❡♥t❡ às r❡t❛s ←→OA ❡ ←→OB. ❊♥tã♦✱ ♦s

tr✐â♥❣✉❧♦s ∆M OP ❡ ∆N OP sã♦ ♥♦✈❛♠❡♥t❡ ❝♦♥❣r✉❡♥t❡s✱ ❛❣♦r❛ ♣❡❧♦ ❝❛s♦ CH ❞❡

tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ✭P M =P N ❡ ❖P ❝♦♠✉♠✮✳ ▼❛s ❛í MOPÒ =NOPÒ ✱ ❞♦♥❞❡ P

❡stá s♦❜r❡ ❛ ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ AOBÒ ✱ ❝♦♠♦ q✉❡rí❛♠♦s ♣r♦✈❛r✳

❚❡♦r❡♠❛ ✶✳✷ ✭❬✶✱ ❚❡♦r❡♠❛ ✹✳✸✶❪✮✳ ❯♠ ❝ír❝✉❧♦ γ ❞❡ r❛✐♦ r ❡ ❝❡♥tr♦ I é ✐♥t❡r✐♦r ❛

✉♠ ❝✐r❝✉❧♦ Γ ❞❡ r❛✐♦R ❡ ❝❡♥tr♦ O✳ ❙❡ A _∈Γ ❡ AB ❡ AC sã♦ ❝♦r❞❛s ❞❡ Γt❛♥❣❡♥t❡s

❛ γ✱ ❡♥tã♦ γ é ♦ ❝ír❝✉❧♦ ✐♥s❝r✐t♦ ♥♦ tr✐â♥❣✉❧♦∆ABC s❡ ❡ só s❡

R2

−d2

= 2rR.

❝♦♠ d ❂ OI✳

Pr♦✈❛✿ ❙❡ P é ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❛ ❜✐ss❡tr✐③ AI ❞♦ â♥❣✉❧♦ BACÒ ❝♦♠ Γ✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✶✵✱ s❡❣✉❡ ❞♦ ❈♦r♦❧ár✐♦ ✶ q✉❡

AI.IP =R2

−OI2 =R2

−d2

. ✭✶✳✻✮

❆❣♦r❛✱ s❡♥❞♦X ❡Y r❡s♣❡❝t✐✈❛♠❡♥t❡ ♦s ♣és ❞❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s tr❛ç❛❞❛s ❞❡ O ❡I

❋✐❣✉r❛ ✶✳✶✵✿ ❆ ❞✐stâ♥❝✐❛OI

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❈✐r❝✉♥❢❡rê♥❝✐❛

❛ BP ❡AC✱ t❡♠♦s

BOXÒ = 1

2BOPÒ =BAPÒ =PACÒ =IAY.Ò

❈♦♠♦ ❛♠❜♦s ♦s tr✐â♥❣✉❧♦s ∆BOX ❡ ∆IAY tê♠ ✉♠ â♥❣✉❧♦ ❞❡ ✾✵◦✱ s❡❣✉❡ ❡♥tã♦ ❞♦

❝❛s♦ ❆✳❆✳ q✉❡ ∆BOX ❡∆IAY sã♦ s❡♠❡❧❤❛♥t❡s P♦rt❛♥t♦✱

BX IY =

BO AI

♦✉ ❛✐♥❞❛✱

BX.AI =BO.IY . ✭✶✳✼✮

❈♦♠♦ BO=R✱ IY =r ❡ BX = 1

2BP ,s❡❣✉❡ ❞❛s ❡q✉❛çõ❡s ✭✶✳✻✮ ❡ ✭✶✳✼✮ q✉❡

R2

−OI2 =R2

−d2

=AI.IP = 2Rr.IP BP,

❞❡ ♠❛♥❡✐r❛ q✉❡

OI2 =R2

−2Rr ⇐⇒BP =IP .

❖❜s❡r✈❡ q✉❡ BP =IP s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ tr✐â♥❣✉❧♦ ∆P BI é ✐sós❝❡❧❡s ❞❡ ❜❛s❡ BI✱

❞❡ ♠♦❞♦ q✉❡PBIÒ =PIBb ✳ ❈♦♠♦ PIBb é â♥❣✉❧♦ ❡①t❡r♥♦ ❛♦ tr✐â♥❣✉❧♦ ∆ABI✱ s❡❣✉❡

q✉❡

PIBb =IABÒ +IBA.Ò

P♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ♦ â♥❣✉❧♦ ✐♥s❝r✐t♦ t❡♠♦s

PBCÒ =PACÒ =PABÒ =IAB.Ò

P♦rt❛♥t♦

IABÒ +IBAÒ =PIBb =PBIÒ =PBCÒ +CBIÒ =IABÒ +CBIÒ

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❈✐r❝✉♥❢❡rê♥❝✐❛

♦ q✉❡ ♥♦s ❞á IBAÒ = CBIÒ ❞❡ ♠♦❞♦ q✉❡ I ♣❡rt❡♥❝❡ à ❜✐ss❡tr✐③ ❞❡ ABCÒ ❡ ❛ss✐♠

é ✐♥t❡rs❡çã♦ ❞❡ ❞✉❛s ❜✐ss❡tr✐③❡s ❞♦ tr✐â♥❣✉❧♦ ∆ABC✱ ♦ q✉❡ ♥♦s ❣❛r❛♥t❡ q✉❡ I é ♦

✐♥❝❡♥tr♦ ❞❡∆ABC ❡γ ❡stá ✐♥s❝r✐t❛ ❡♠∆ABC✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ ❧❛❞♦BC t❛♠❜é♠

é t❛♥❣❡♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ γ✳

❖ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐r é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❊st❡ ❣❛r❛♥t❡ q✉❡✱ ❞❛❞❛s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s✱ ✉♠❛ ✐♥t❡r♥❛ ❛ ♦✉tr❛✱ s❡ ❡①✐st❡ ✉♠ tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡①t❡r♥❛ ❡ ❝✐r❝✉♥s❝r✐t♦ ♥❛ ✐♥t❡r♥❛✱ ❡♥tã♦ ❝❛❞❛ ♣♦♥t♦ P ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡①t❡r♥❛ é ✈ért✐❝❡ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❝♦♠ ❛s ♠❡s♠❛s

❝❛r❛❝t❡ríst✐❝❛s✳

❚❡♦r❡♠❛ ✶✳✸ ✭P♦♥❝❡❧❡t✱ ❬✶❪✮ ❙❡❥❛♠ γ ❡ Γ r❡s♣❡❝t✐✈❛♠❡♥t❡ ♦s ❝ír❝✉❧♦s ✐♥s❝r✐t♦s

❡ ❝✐r❝✉♥s❝r✐t♦s ❛ ✉♠ tr✐â♥❣✉❧♦ ∆ABC✳ ❙❡ A′ _{6= A✱ B✱ C é ♦✉tr♦ ♣♦♥t♦ ❞❡ Γ✱ ❡}

A′_B′ _{❡ A}′_C′ _{sã♦ ❝♦r❞❛s ❞❡ Γt❛♥❣❡♥t❡s ❛ γ✱ ❡♥tã♦ γ é ♦ ❝ír❝✉❧♦ ✐♥s❝r✐t♦ ♥♦ tr✐â♥❣✉❧♦} ∆A′_B′_C′_.

❋✐❣✉r❛ ✶✳✶✶✿ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t

Pr♦✈❛✿ ❙❡❥❛♠ γ(I;r) ❡ Γ(O;R)✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✶✶✱ ❡♥tã♦ ♦ ❢❛t♦ ❞❡ γ s❡r

♦ ❝ír❝✉❧♦ ✐♥s❝r✐t♦ ❡♠ ∆ABC ❣❛r❛♥t❡✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ q✉❡OI2 =R2

−2Rr.

❉❡ ♣♦ss❡ ❞❡ss❛ ✐❣✉❛❧❞❛❞❡✱ ❛♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ r❡❢❡r✐❞♦ t❡♦r❡♠❛ ❛♦ tr✐â♥❣✉❧♦

∆A′_B′_C′_{✱ ❝♦♥❝❧✉í♠♦s q✉❡ B}′_C′ _{t❛♥❣❡♥❝✐❛ γ✱ ❝♦♥❢♦r♠❡ ❞❡s❡❥❛❞♦✳}

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✸✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦s ◗✉❛❞r✐❧át❡r♦s

✶✳✸ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦s ◗✉❛❞r✐❧át❡r♦s

◆❡st❛ s❡çã♦ ❡st✉❞❛r❡♠♦s ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ P♦♥❝❡❧❡t✱ ♥♦ ❡st✉❞♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡ q✉❛❞r✐❧át❡r♦s✳ ❖ r❡s✉❧t❛❞♦ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛q✉✐ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✹❪✳ ❋♦✐ ▲❡♦♥❤❛r❞ ❊✉❧❡r q✉❡ s❡ ✐♥t❡r❡ss♦✉ ♣♦r ❡ss❡ ♣r♦❜❧❡♠❛ ❡ r❡❛❧✐③♦✉ ❛❧❣✉♥s ❡s❢♦rç♦s ♣❛r❛ ❡♥❝♦♥tr❛r ❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡✱ ❞❛❞❛s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s λ

✐♥t❡r♥❛ ❛ Γ✱ ❡①✐st✐ss❡ ✉♠ q✉❛❞✐r❧át❡r♦ ✐♥s❝r✐t♦ ❡♠ Γ ❡ ❝✐r❝✉♥s❝r✐t♦ ❡♠ λ✱ ♣♦ré♠ ❢♦✐

s❡✉ ❛❧✉♥♦ ◆✐❦♦❧❛✐ ■✈❛♥♦✈✐❝❤ ❋✉ss q✉❡♠ ❞❡✉ ❝♦♠ ❡❧❛ ❡♠ ✶✼✾✼ ✭✈❡❥❛ ❬✾❪✮✳ ❚❛❧ ❝♦♥❞✐çã♦ é✿

2r2 (R2

+d2

) = (R2

−d2

)2

.

▼❛✐s ♣r❡❝✐s❛♠❡♥t❡ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

❚❡♦r❡♠❛ ✶✳✹ ❙❡❥❛♠ M, Z ♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s ✐♥s❝r✐t♦s ❡ ❝✐r❝✉♥s❝r✐t♦s ❛♦

q✉❛❞r✐❧át❡r♦ P QRS ❝♦♠ r❛✐♦s r, R✳ ❉❡♥♦t❛♥❞♦ ♣♦r d ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❝❡♥tr♦s M, Z ❞❡ss❡s ❝ír❝✉❧♦s d=M Z. t❡♠✲s❡ s❡♠♣r❡ q✉❡✿

2r2 (R2

+d2

) = (R2

−d2

)2

. ✭✶✳✽✮

❆ s❡❣✉✐r ♣r♦✈❛r❡♠♦s ❡st❡ r❡s✉❧t❛❞♦✱ q✉❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✹✱ Pá❣✳ ✶✽✽❪✳ ❉❛ ♠❡s♠❛ ♠❛♥❡✐r❛ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞♦s tr✐â♥❣✉❧♦s✱ s❡ ❛ ❡q✉❛çã♦ ✭✶✳✽✮ é s❛t✐s❢❡✐t❛ ❡♥tã♦ q✉❛❧q✉❡r ❝♦r❞❛ ❞❡ Γ t❛♥❣❡♥t❡ ❛ λ ♣♦❞❡ ❞❡s❝r❡✈❡r ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♠ ✈ért✐❝❡s ❡♠

Γ ❡ ❧❛❞♦s t❛♥❣❡♥t❡s ❛λ✳ P❛r❛ ♣r♦✈❛r♠♦s ❡st❡ r❡s✉❧t❛❞♦ ♣r❡❝✐s❛♠♦s ❞❡ ❛❧❣✉♥s ❧❡♠❛s

♣r❡❧✐♠✐♥❛r❡s✳

❉❡✜♥✐çã♦ ✺ ❉✐③❡♠♦s q✉❡ ✉♠ q✉❛❞r✐❧át❡r♦ é ❜✐❝ê♥tr✐❝♦ s❡ ❡❧❡ é s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ✐♥s❝r✐t♦ ❡♠ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ❝✐r❝✉♥s❝r✐t♦ ❡♠ ✉♠❛ ♦✉tr❛✳

Pr♦♣♦s✐çã♦ ✶✳✻ ❯♠ q✉❛❞r✐❧át❡r♦ P QRS ♣♦ss✉✐ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ❝♦♠

s❡✉s ❧❛❞♦s t❛♥❣❡♥t❡s ❛ ❡❧❛ ♥♦s ♣♦♥t♦s X, X′_{, Y, Y}′ _{❡ s❡❣♠❡♥t♦s XX}′ _{❡ Y Y}′

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✸✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦s ◗✉❛❞r✐❧át❡r♦s

Pr♦✈❛✿ ❱❛♠♦s s✉♣♦r q✉❡P QRS s❡❥❛ ✉♠ q✉❛❞r✐❧át❡r♦ ❜✐❝ê♥tr✐❝♦✳ ❙❡❥❛♠Γ♦ ❝ír❝✉❧♦

❝✐r❝✉♥s❝r✐t♦✱ ❝♦♠ ❝❡♥tr♦ ❡♠ O✱ ❡γ ♦ ❝ír❝✉❧♦ ✐♥s❝r✐t♦✱ ❝♦♠ ❝❡♥tr♦ ❡♠ I✳ ❉❡♥♦t❡♠♦s

♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ❞♦s ❧❛❞♦s ♦♣♦st♦s P Q❡ RS ❝♦♠ ♦ ❝ír❝✉❧♦ γ ♣♦r X ❡X′_{✱ ♦s}

♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ❞♦s ❧❛❞♦s ♦♣♦st♦s QR ❡ SP ♣♦r Y′ _{❡ Y✱ ❡ ❛✐♥❞❛ ♦ ♣♦♥t♦ ❞❡}

✐♥t❡rs❡çã♦ ❞❛s ❝♦r❞❛s XX′ _{❡ Y Y}′ _♣♦rO′_{✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✶✷✳}

❋✐❣✉r❛ ✶✳✶✷✿ ◗✉❛❞r✐❧át❡r♦ ❇✐❝ê♥tr✐❝♦

❈♦♠♦ ♦ q✉❛❞r✐❧át❡r♦ P QRS é ✐♥s❝r✐tí✈❡❧ s❡✉s â♥❣✉❧♦s ♦♣♦st♦s sã♦ s✉♣❧❡♠❡♥t❛r❡s✳

❊♥tã♦✱ PQRÒ +PSRÒ = 180◦_{✳ ❉♦s q✉❛❞r✐❧át❡r♦sIXQY}′ _{❡ IX}′_{SY ♦❜t❡♠♦s}

XIYb ′ _+IÓ_Y′_Q+Y′_QXÒ _{+QXIc = 360}◦ _{✭✶✳✾✮}

❡

X′IYb +IY SÒ +YSXÒ ′+SXÓ′_{I = 360}◦_. ✭✶✳✶✵✮

❙♦♠❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✶✳✾✮ ❡ ✭✶✳✶✵✮ ❡ ❧❡♠❜r❛♥❞♦ q✉❡

IÓY′_Q=QXIc _{=IY S}Ò _=SXÓ′_{I = 90}◦

✭♣♦✐s sã♦ â♥❣✉❧♦s ❞❡ t❛♥❣ê♥❝✐❛s✮ ❡ Y′_QXÒ _{+ YSX}Ò ′ _{= 180}◦ _{✭â♥❣✉❧♦s ♦♣♦st♦s ❞♦}

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✸✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦s ◗✉❛❞r✐❧át❡r♦s

q✉❛❞r✐❧át❡r♦ ✐♥s❝r✐t♦✮ ♦❜t❡♠♦s✱

XIYb ′+X′IYb + 360◦+ 180◦ = 720♦, ✭✶✳✶✶✮

❞❡ ♦♥❞❡ s❡ ❝♦♥❝❧✉✐ q✉❡

XIYb ′_+X′_{IYb = 180, ✭✶✳✶✷✮}

q✉❡ é ♦ ♠❡s♠♦ q✉❡

ù

XY′_{+Y X}ù′ _{= 180}◦_.

❆❣♦r❛✱ ❝♦♠♦ α é ✉♠ â♥❣✉❧♦ ❡①❝ê♥tr✐❝♦ ✐♥t❡r✐♦r ❛ γ✱ t❡♠♦s

α= XIYb

′_+X′_IYb

2 =

180♦ 2 = 90

◦_{, ✭✶✳✶✸✮}

❧♦❣♦ XX′_{⊥Y Y}′_. ■ss♦ ♠♦str❛ q✉❡ ❛s ❝♦r❞❛s ❞❡ t❛♥❣ê♥❝✐❛ ❞♦s ❞♦✐s ♣❛r❡s ❞❡ ❧❛❞♦s

♦♣♦st♦s ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ ❜✐❝ê♥tr✐❝♦ sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳

❱❡❥❛♠♦s q✉❡ ❡ss❛ ❝♦♥❞✐çã♦ t❛♠❜é♠ é s✉✜❝✐❡♥t❡✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❜✐❝ê♥tr✐❝♦

P QRS é ♦❜t✐❞♦ s❡ ❛s t❛♥❣❡♥t❡s P Q✱ RS✱ SP ❡ P Q✱ sã♦ tr❛ç❛❞❛s ♣❡❧♦s ♣♦♥t♦s ❞❡

t❛♥❣ê♥❝✐❛ X✱ X′_{✱ Y ❡ Y}′ _{❞❡ ❞✉❛s ❝♦r❞❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s XX}′ _{❡Y Y}′ _{❞❡ ✉♠ ❝ír❝✉❧♦}

❛r❜✐trár✐♦ γ✳ ❯♠❛ ✈❡③ α = XOc′_{Y = 90}◦ _{s❡❣✉❡ ❞❡ ✭✶✳✶✸✮ q✉❡ ❛ ❡q✉❛çã♦ ✭✶✳✶✷✮ é}

s❛t✐s❢❡✐t❛✳ ❆❧é♠ ❞✐ss♦✱ s❡♥❞♦

IXQc =IÓY′_{Q=IY S}Ò _=IXÓ′_{S = 90}♦

❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ♦♣♦st♦s XP YÒ ❡ X′_RYÒ ′ _{é ❞❡ 180}◦_{✱ ♦✉}

s❡❥❛✱ q✉❡ P QRS é t❛♠❜é♠ ✉♠ q✉❛❞r✐❧át❡r♦ ✐♥s❝r✐tí✈❡❧✱ ♣♦rt❛♥t♦ ❜✐❝ê♥tr✐❝♦✱ ❝♦♠♦

❞❡s❡❥á✈❛♠♦s✳

❯♠❛ ❞❛s ♠❛♥❡✐r❛s ❞❡ s❡ ♦❜t❡r ❛ ❞❡s❡❥❛❞❛ r❡❧❛çã♦ ❡♥tr❡ ♦s r❛✐♦s ❡ ♦ ❡✐①♦ ❞♦s ❝❡♥tr♦s ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❝✐r❝✉♥s❝r✐t❛ ❡ ✐♥s❝r✐t❛ é ♣♦r ♠❡✐♦ ❞♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ❧✉❣❛r ❣❡♦♠étr✐❝♦✳

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✸✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦s ◗✉❛❞r✐❧át❡r♦s

Pr♦❜❧❡♠❛ ✶ ❯♠ â♥❣✉❧♦ r❡t♦ é ❣✐r❛❞♦ s♦❜r❡ s❡✉ ✈ért✐❝❡ ✜①♦✱ q✉❡ ❡stá ❧♦❝❛❧✐③❛❞♦ ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠ ❝ír❝✉❧♦✱ s♦❜r❡ s❡✉ ❞✐â♠❡tr♦✳ ❊♥❝♦♥tr❛r ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s ❞✉❛s t❛♥❣❡♥t❡s ❞♦ ❝ír❝✉❧♦ q✉❡ ♣❛ss❛♠ ❛tr❛✈és ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❞♦s ❧❛❞♦s ❞♦ â♥❣✉❧♦ ❝♦♠ ♦ ❝ír❝✉❧♦✳

Pr♦✈❛✿ ❈♦♥s✐❞❡r❡♠♦s ♦ tr✐â♥❣✉❧♦∆OXY ✐♥t❡r♥♦ à ❝✐r❝✉♥❢❡rê♥❝✐❛Γ❞❡ r❛✐♦r✱ ❝♦♠♦

♥❛ ❋✐❣✉r❛ ✶✳✶✸ ❡ P ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s t❛♥❣❡♥t❡s à Γ ❡♠ X ❡ Y✳ ❯♠❛ ✈❡③

q✉❡ ♦ tr✐â♥❣✉❧♦XOY é r❡tâ♥❣✉❧♦ ❡♠O✱ t♦♠❡OF ❝♦♠♦ s❡♥❞♦ ❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦

❞❛❞♦ ❡♠ r❡❧❛çã♦ à ❤✐♣♦t❡♥✉s❛ XY ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡

❋✐❣✉r❛ ✶✳✶✸✿ ◗✉❛❞r✐❧át❡r♦ ❞❡ P♦♥❝❡❧❡t

OF2 =XF .F Y . ✭✶✳✶✹✮

❉❡♥♦t❡♠♦s ♣♦r M ♦ ❝❡♥tr♦ ❞❡ Γ✳ ❙❛❜❡♠♦s q✉❡ −−→P M é ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ XP YÒ

❡ é ♦rt♦❣♦♥❛❧ ❛ XY ❡♠ N✱ ♣♦♥t♦ ♠é❞✐♦ ❞❡st❡ s❡❣♠❡♥t♦ ❞❡ ♠♦❞♦ q✉❡ M P é

♣❛r❛❧❡❧♦ ❛ OF ✭✈❡❥❛ Pr♦♣♦s✐çã♦ ✶✳✺✮✳ ■♥tr♦❞✉③✐♥❞♦✱ r′ _{= M N ❡ r” = N X✱}

❡①tr❛í❞♦s ❞♦ tr✐â♥❣✉❧♦ ∆M N X r❡tâ♥❣✉❧♦ ❡♠ N✱ e = M O✱ e′ _{= M T = e.cos(ϕ)}

❡ e′′ _{= OT = e.s❡♥(ϕ) ❡①tr❛í❞♦s ❞♦ tr✐â♥❣✉❧♦ ∆M T O✱ r❡tâ♥❣✉❧♦ ❡♠ T✱ ♦♥❞❡ OT}

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✸✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦s ◗✉❛❞r✐❧át❡r♦s

é ✉♠❛ ♣❛r❛❧❡❧❛ ❛ N F ♣❛ss❛♥❞♦ ♣♦r O ❡ ✐♥t❡rs❡❝t❛♥❞♦ M P ❡♠ T✱ ❝♦♠ ♦ â♥❣✉❧♦ OM TÓ =ϕ✳ ❈♦♠♦

OF =N T =M N ₋T M =r′_−e′_,

XF =N X ₋N F =N X₋OT =r′′_−e′′

❡

F Y =F N+N Y =r′′+e′′

❞❡ ✭✶✳✶✹✮ ✈❡♠ q✉❡

(r′_−e′₎2

= (r′′_−e′′_).(r′′_+e′′₎

♦✉ ❛✐♥❞❛

r′2

−2.r′_.e′_+e′2 =r′′2

−e′′2

. ✭✶✳✶✺✮

❙♦♠❛♥❞♦ r′2 ❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❡ ✭✶✳✶✺✮ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦

2.r′2

−2.r′_.e′_+e′2 =r′′2

−e′′2 +r′2

,

❞❛í

2.r′2

−2.r′_.e′ _{= (r}′2 +r′′2

)₋(e′2 +e′′2

). ✭✶✳✶✻✮

❙❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ♥♦ tr✐â♥❣✉❧♦∆M T O q✉❡e2 =e′2

+e′′2

✳ ❆❧é♠ ❞✐ss♦✱ ❥á q✉❡ M X =r✱ ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ∆M N X t❡♠♦sr2

=r′2

+r′′2✳ ❆ss✐♠✱ s❡❣✉❡

❞❡ ✭✶✳✶✻✮ q✉❡

2.r′2

−2.r′.e′ = (r′2 +r′′2

)−(e′2 +e′′2

) =r2

−e2

. ✭✶✳✶✼✮

◆♦ ❡♥t❛♥t♦✱ e′ _{=e.cos(ϕ) ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡}

2.r′2

−2.r′.e.cos(ϕ) =r2

−e2

. ✭✶✳✶✽✮

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✸✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦s ◗✉❛❞r✐❧át❡r♦s

❈♦♠♦ ♦ tr✐â♥❣✉❧♦ ∆M XP é r❡tâ♥❣✉❧♦ ❡♠ X ❡N X é ✉♠❛ ❛❧t✉r❛ ❝♦♠ N s❡♥❞♦ ✉♠

♣♦♥t♦ ❞❛ ❜❛s❡ M P✱ ❤✐♣♦t❡♥✉s❛ ❞♦ tr✐â♥❣✉❧♦✱ ♦❜t❡♠♦s

M X2 =M N .M P .

❈♦♠♦ M X =r✱ M N =r′ _{❡ ❞❡♥♦t❛♥❞♦ M P =p✱ t❡r❡♠♦s r}2

=r′_{.p♦✉ ❛✐♥❞❛}

r′ ₌ r 2

p.

■♥tr♦❞✉③✐♥❞♦ ❡ss❡ ✈❛❧♦r ❛ ♣❛rt✐r ❞❡ ✭✶✳✶✽✮ ❝❤❡❣❛♠♦s ❛

2.r

4

p2 −2.

r2

p.e.cos(ϕ) = r

2

−e2

.

❉✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r r2

−e2

❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ♣♦r p2

✈❛♠♦s ♦❜t❡r ❛ ❡q✉❛çã♦❀

2.r4

r2

−e2 −

2.r2

p r2

−e2.e.cos(ϕ) =p

2

❞❡ ♦♥❞❡ s❡ ♦❜té♠ q✉❡

2.r4

r2

−e2 =p

2

+ 2.r 2

r2

−e2.p.e.cos(ϕ). ✭✶✳✶✾✮

❊s❝♦❧❤❡♥❞♦ d=r2

.e/(r2

−e2

)✱ ❡ s✉❜st✐t✉✐♥❞♦ ♥❛ ❡q✉❛çã♦ ✭✶✳✶✾✮ ♦❜t❡♠♦s 2.r4

r2

−e2 =p

2

+ 2.d.p.cos(ϕ). ✭✶✳✷✵✮

❆❣♦r❛✱ ❞❡♥♦t❛♠♦sR =ZP ❛ ❞✐stâ♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦Z s♦❜r❡ ❛ ❡①t❡♥sã♦ ❞❡OM ❛té P ♦♥❞❡ d = M Z✱ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ Z ❛♦ ❝❡♥tr♦ M ❞❡ Γ✳ ◆♦ tr✐â♥❣✉❧♦ ∆ZM P✱

♣❡❧❛ ❧❡✐ ❞♦s ❝♦ss❡♥♦s ♦❜t❡♠♦s

R2 =d2

+p2

+ 2.d.p.cos(180◦_{−ϕ) = d}2 +p2

+ 2.d.p.cos(ϕ). ✭✶✳✷✶✮

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✸✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦s ◗✉❛❞r✐❧át❡r♦s

❉❛í✱ ❝♦♠❜✐♥❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✶✳✷✵✮✱ ❡ ✭✶✳✷✶✮ t❡r❡♠♦s q✉❡❀

R2 =d2

+ 2.r 4

r2

−e2. ✭✶✳✷✷✮

❖✉ ❛✐♥❞❛

r2

−e2

= 2r

4

R2

−d2 ✭✶✳✷✸✮

❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱Rt❡♠ ✉♠ ✈❛❧♦r ❝♦♥st❛♥t❡✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ♣♦s✐çã♦ ❞♦s ♣♦♥t♦s X ❡Y✳ ❖ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞❡s❡❥❛❞♦ é ❛ss✐♠✱ ✉♠ ❝ír❝✉❧♦_C ❝✉❥♦ ❝❡♥tr♦ éZ✱ q✉❡ ❡stá

s✐t✉❛❞♦ s♦❜r❡ ❛ ❡①t❡♥sã♦ ❞❡OM✱ ❞❡t❡r♠✐♥❛❞♦ ♣♦rd=r2

.e/(r2

−e2

)❡ ❝✉❥♦ r❛✐♦Ré

❞❡t❡r♠✐♥❛❞♦ ♣♦rR2 =d2

+ 2.r4

/(r2

−e2

)✳ ◆❛t✉r❛❧♠❡♥t❡✱ t❛♠❜é♠ ♣❡rt❡♥❝❡♠ ❛ ❡st❡

❧✉❣❛r ❣❡♦♠étr✐❝♦ ♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ Q✱ R ❡ S ❞❛s t❛♥❣❡♥t❡s✱ q✉❡ sã♦ ♦❜t✐❞❛s

q✉❛♥❞♦ s❡ t✐r❛r ❛s t❛♥❣❡♥t❡s ❛tr❛✈és ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❞♦ ❝ír❝✉❧♦ ❝♦♠ ❛s ❡①t❡♥sõ❡s ❞♦s ❡✐①♦s XO ❡ Y O✳ ❖ q✉❛❞r✐❧át❡r♦ P QRS é ✉♠ q✉❛❞r✐❧át❡r♦ ❜✐❝ê♥tr✐❝♦

♥❛ ♠❡❞✐❞❛ ❡♠ q✉❡ ❡stá ✐♥s❝r✐t♦ ❡♠ ✉♠ ❝ír❝✉❧♦ ❡ ❝✐r❝✉♥s❝r✐t♦ ❡♠ ♦✉tr♦✳ ❙❡ ♦ â♥❣✉❧♦ r❡t♦ XOYÒ é ❣✐r❛❞♦ ❡♠ t♦r♥♦ ❞♦ ❝❡♥tr♦ O✱ ❞❡ ♠♦❞♦ q✉❡ ♦s ♣♦♥t♦sX ❡Y ❞❡s❝r❡✈❛♠

♦ ❝ír❝✉❧♦ Γ✱ ♦ q✉❛❞r✐❧át❡r♦ P QRS ❛ss✉♠❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ♣♦s✐çõ❡s ❞✐❢❡r❡♥t❡s✱ ♠❛s

s❡♠♣r❡ s❡rá ❜✐❝ê♥tr✐❝♦✳ ▲♦❣♦ ❛s ❢ór♠✉❧❛s ♦❜t✐❞❛s ♣❛r❛ d ❡ R ❝♦♥té♠ ❛ s♦❧✉çã♦

❞❡s❡❥❛❞❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✳ ❏á q✉❡ r2

−e2

= 2.r4

/(R2

−d2

)✱

s✉❜st✐t✉✐♥❞♦ ❡ss❡ ✈❛❧♦r ❡♠ d=r2

.e/(r2

−e2

) ♦❜t❡♠♦s q✉❡

r2

.e=d.2.r4

/(R2

−d2

)

❞❡ ♦♥❞❡ s❡ ♦❜té♠

e= 2.d.r2

/(R2

−d2

).

❈❛♣ít✉❧♦ ✶✳ ❚❡♦r❡♠❛s ❞❡ P♦♥❝❡❧❡t ✶✳✸✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥♦s ◗✉❛❞r✐❧át❡r♦s

❉✐st♦ s❡❣✉❡ q✉❡

r2

−e2

= r2

−(2.d.r2

/(R2

−d2

))2 = r

2 (R2

−d2

)2

−4d2

r4 (R2

−d2

)2

= r

2 [(R2

−d2

)2

−4r2

d2 ] (R2

−d2

)2 . ✭✶✳✷✹✮

❆❣♦r❛ ♣♦r ❝♦♠♣❛r❛çã♦ ❞❛s ❡q✉❛çõ❡s ✭✶✳✷✸✮✱ ✭✶✳✷✹✮✱ ❝❤❡❣❛♠♦s ❛ ❡q✉❛çã♦

2.r4

R2

−d2 =

r2 [(R2

−d2

)2

−4r2

d2 ] (R2

−d2

)2 .

▼✉❧t✐♣❧✐❝❛♥❞♦ ❡ss❛ ❡q✉❛çã♦ ♣♦r (R2

−d2

)2 ❡♥❝♦♥tr❛♠♦s ✉♠❛ ♦✉tr❛ ❡q✉❛çã♦

2.r2 (R2

−d2

) = (R2

−d2

)2

−4r2

d2

.

❆❞✐❝✐♦♥❛♥❞♦ ❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s 4r2

d2 ❡ ❝♦❧♦❝❛♥❞♦

2.r2 ❡♠ ❡✈✐❞ê♥❝✐❛ ♥♦ ♣r✐♠❡✐r♦

♠❡♠❜r♦ é q✉❡✱ ✜♥❛❧♠❡♥t❡✱ s❡ ♦❜té♠ ❛ r❡❧❛çã♦ ❡♥tr❡R✱r ❡d❡✐①♦ q✉❡ ✉♥❡ ♦s ❝❡♥tr♦s

❞♦s ❝ír❝✉❧♦s ❝✐r❝✉♥s❝r✐t♦ ❡ ✐♥s❝r✐t♦ ❞♦ q✉❛❞r✐❧át❡r♦ ❜✐❝ê♥tr✐❝♦

2r2 (R2

+d2

) = (R2

−d2

)2

,

❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

❈❛♣ít✉❧♦ ✷

❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❊❧✐♣s❡

◆❡st❡ ❝❛♣ít✉❧♦ ❛❜♦r❞❛r❡♠♦s ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ P♦♥❝❡❧❡t ♣❛r❛ tr✐â♥❣✉❧♦s ❡ ❡❧✐♣s❡s✳ ❆❧❣✉♥s ❞♦s r❡s✉❧t❛❞♦s q✉❡ tr❛r❡♠♦s ❛q✉✐ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✶✸❪✳ ❖ ❡♥✉♥❝✐❛❞♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t só r❡q✉❡r ❛ ❞❡✜♥✐çã♦ ❞❛ ❡❧✐♣s❡✳

❚❡♦r❡♠❛ ✷✳✶ ✭P♦r✐s♠❛ ❞❡ P♦♥❝❡❧❡t✮ ❉❛❞❛s ❞✉❛s ❡❧✐♣s❡s✱ ✉♠❛ ✐♥t❡r♥❛ à ♦✉tr❛✱ s❡ ❡①✐st❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ n✲❧❛❞♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡ ✐♥s❝r✐t♦ ♥❛ ❡❧✐♣s❡ ❡①t❡r♥❛ ❡

❝✐r❝✉♥s❝r✐t♦ à ❡❧✐♣s❡ ✐♥t❡r♥❛✱ ❡♥tã♦ q✉❛❧q✉❡r ♣♦♥t♦ ❞❛ ❡❧✐♣s❡ ❡①t❡r♥❛ é ✈ért✐❝❡ ❞❡ ❛❧❣✉♠ ♣♦❧í❣♦♥♦ ❝♦♠ ❛s ♠❡s♠❛s ❝❛r❛❝t❡ríst✐❝❛s✳

P❛r❛ ❛ ♣r♦✈❛ ❞❡st❡ t❡♦r❡♠❛ ❢❛r❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ q✉❡ ✉s❛ ❛♣❡♥❛s ❞❡ ❢❡rr❛♠❡♥t❛s ❞♦s ♣r♦❣r❛♠❛s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ♣❛❞rã♦✳ ■st♦ ♥ã♦ é ✉♠ ♣r♦❜❧❡♠❛ s✐♠♣❧❡s✱ ♣♦ré♠ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ❡♥❝♦♥tr❛r ✉♠❛ ♣r♦✈❛ ❡❧❡♠❡♥t❛r ❞❡ ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ♣❛r❛ ❛ s✐t✉❛çã♦ ❞♦ ❝❛s♦ n= 3 q✉❛♥❞♦ t❡♠♦s ❞✉❛s ❡❧✐♣s❡s✿

e : x 2

a2 +

y2

b2 = 1 ✭✷✳✶✮

❡

e1 :

x2

a2 1

+y 2

b2 1

= 1 ✭✷✳✷✮

❞❡ t❛❧ ♠♦❞♦ q✉❡ e1 s❡❥❛ ✐♥t❡r✐♦r ❛ e✳ P❛r❛ ❡st❡ ❝❛s♦ t❡r❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

❈❛♣ít✉❧♦ ✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❊❧✐♣s❡

❚❡♦r❡♠❛ ✷✳✷ ❙✉♣♦♥❤❛ q✉❡ ❛ ❡❧✐♣s❡e1 ❝♦♠ ❡q✉❛çã♦ ❞❛❞❛ ♣♦r ✭✷✳✷✮ ❡st❡❥❛ ♥♦ ✐♥t❡r✐♦r

❞❛ ❡❧✐♣s❡ e ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✶✮✱ ✈❡r ❋✐❣✉r❛ ✷✳✶✱ ❡ s❡❥❛♠ t❛✐s q✉❡✿

a > b >0, a1 > b1 >0, a > a1, b > b1.

❊♥tã♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

✐✮ ❡①✐st❡ ✉♠ tr✐â♥❣✉❧♦ ∆A0B0C0 ✐♥s❝r✐t♦ ❡♠ e✱ ❡ ❝✐r❝✉♥s❝r✐t♦ ❡♠ e1❀

✐✐✮ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦

a1

a + b1

b = 1;

✐✐✐✮ ♣❛r❛ q✉❛❧q✉❡r ♣♦♥t♦ A ♥❛ ❡❧✐♣s❡ e ♣♦❞❡✲s❡ ❡♥❝♦♥tr❛r ✉♠ ú♥✐❝♦ tr✐â♥❣✉❧♦ ∆ABC

✐♥s❝r✐t♦ ❡♠ e ❡ ❝✐r❝✉♥s❝r✐t♦ ❡♠ e1✳

❆♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ❛♥t❡s ❞❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛✳ ❱❡r❡♠♦s q✉❡ ♥ã♦ ❤á ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ❡♠ ❝♦♥s✐❞❡r❛r ❛ ❡❧✐♣s❡ ❡①t❡r♥❛ ❝♦♠♦ s❡♥❞♦ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❝♦♠♦ ♥❛ ❋✐❣✉r❛ ✷✳✶✳

❋✐❣✉r❛ ✷✳✶✿ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♣❛r❛ ❡❧✐♣s❡ ❡ ❝ír❝✉❧♦✳

❈❛♣ít✉❧♦ ✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❊❧✐♣s❡ ✷✳✶✳ ❘❡❞✉çã♦ ❛♦ ❝❛s♦ ❝ír❝✉❧♦ ❡ ❡❧✐♣s❡

✷✳✶ ❘❡❞✉çã♦ ❛♦ ❝❛s♦ ❝ír❝✉❧♦ ❡ ❡❧✐♣s❡

❈♦♥s✐❞❡r❡♠♦s ❛s ❞✉❛s ❡❧✐♣s❡s

e: x 2

a2 +

y2

b2 = 1. ✭✷✳✸✮

❡

e1 :

x2 a2 1 +y 2 b2 1

= 1 ✭✷✳✹✮

❝♦♠

a > b >0, a1 > b1 >0, a > a1, b > b1

❞❡ ♠♦❞♦ q✉❡ e1 é ✐♥t❡r✐♦r ❛ e✳ P♦❞❡♠♦s ✉s❛r ✉♠❛ s✐♠♣❧❡s ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s

♥♦ ♣❧❛♥♦✱ ❢❛③❡♥❞♦

X = x

a, ❡ Y = y

b ✭✷✳✺✮

❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❛ ❡❧✐♣s❡ e ❞❡ ❡q✉❛çã♦ ✭✷✳✸✮ ♥❛s ♥♦✈❛s ❝♦♦r❞❡♥❛❞❛sX✱Y✱ s❡❥❛ ❞❛❞❛

♣❡❧❛ ❡q✉❛çã♦

X2 +Y2

= 1, ✭✷✳✻✮

♦✉ s❡❥❛✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♦ ❝ír❝✉❧♦ ❝♦♠ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ O ❡ r❛✐♦ r = 1 ♥♦ ♥♦✈♦

s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳

❆ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡♠ q✉❡ x = aX e y = bY ♥♦s ❧❡✈❛ ❛ x2 = a2

X2

❡

y2 =b2

Y2 ❞❡ ♠♦❞♦ q✉❡

x2 a2 1 = a 2 X2 a2 1 = X 2 a2 1 a2 = X 2 A2 1 ❡ y 2 b2 1 = b 2 Y2 b2 1 = Y 2 b2 1 b2 = Y 2 B2 1 ,

♦♥❞❡ A1 =a1/a ❡ B1 =b1/b✳

▲♦❣♦ ❛ s❡❣✉♥❞❛ ❡❧✐♣s❡ e1 t♦r♥❛✲s❡

X2 A2 1 +Y 2 B2 1

= 1. ✭✷✳✼✮

❈❛♣ít✉❧♦ ✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❊❧✐♣s❡ ✷✳✶✳ ❘❡❞✉çã♦ ❛♦ ❝❛s♦ ❝ír❝✉❧♦ ❡ ❡❧✐♣s❡

➱ ❝❧❛r♦ q✉❡ ❡st❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣r❡s❡r✈❛ ❛s ♥♦çõ❡s ❞❡ ✐♥t❡rs❡çã♦✱ ❛ r❡t❛ s❡ tr❛♥s❢♦r♠❛ ❡♠ r❡t❛✱ ♦ ❝ír❝✉❧♦ ❡♠ ❝ír❝✉❧♦✱ ❛ ❡❧✐♣s❡ ❡♠ ❡❧✐♣s❡ ✭♦✉ ❝ír❝✉❧♦✮ ❡ s❡ ❛ r❡t❛ ❡ ❛ ❡❧✐♣s❡ sã♦ t❛♥❣❡♥t❡s ♣❡r♠❛♥❡❝❡♠ t❛♥❣❡♥t❡s ❛♣ós ❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✷✳✷✳

Pr♦♣♦s✐çã♦ ✷✳✶ ❙❡ ✉♠❛ r❡t❛r ❡ ❛ ❡❧✐♣s❡ e1 sã♦ t❛♥❣❡♥t❡s✱ ♣❡r♠❛♥❡❝❡rã♦ t❛♥❣❡♥t❡s

❛♣ós ❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞❛❞❛ ❡♠ ✭✷✳✺✮✳

❋✐❣✉r❛ ✷✳✷✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❛ ❊❧✐♣s❡ ♣♦r ▼✉❞❛♥ç❛ ❞❡ ❱❛r✐á✈❡❧✳

Pr♦✈❛✿ ❙❡❥❛♠ ❛s ❡❧✐♣s❡s e1 : x 2

a2

1 +

y2

b2

1 = 1 ❡ e

′ 1 : X2 A2 1 + Y2 B2

1 = 1 ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡ e 1

❛tr❛✈és ❞❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ X = x

a ❡ Y = y

b ♦♥❞❡ A1 = a1

a ❡ B1 = b1

b✳

❚♦♠❛♥❞♦ r : y = mx +n ❛ r❡t❛ t❛♥❣❡♥t❡ ❛ e1 ♥♦ ♣♦♥t♦ P0(x0;y0)✱ ❝♦♥s✐❞❡r❡

r′ _{: y =} amx b +

n

b ❛ r❡t❛ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡ P0✱ ♣❡❧❛ ♠❡s♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧✱ ❡♥tã♦ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡

i)P(x;y)_∈r _⇐⇒ P′₍x a,

y b)∈r′❀

❈♦♠♦P(x;y)∈r:y=mx+n t❡♠♦s q✉❡

y=mx+n _⇐⇒y = amx

a +n⇐⇒ y b = amx a b + n

b ⇐⇒Y = amX

b + n b

♣♦rt❛♥t♦ P′₍x a,

y b)∈r′✳

ii)P(x;y)_∈e1 ⇐⇒P′(x_a,y_b)∈e′1❀

❈❛♣ít✉❧♦ ✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❊❧✐♣s❡ ✷✳✶✳ ❘❡❞✉çã♦ ❛♦ ❝❛s♦ ❝ír❝✉❧♦ ❡ ❡❧✐♣s❡

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❛ ✐✮✱ ✈❡♠♦s q✉❡ P(x;y)_∈e1 : x 2

a2

1 +

y2

b2

1 = 1 s❡ ❡ só s❡

x2

a2 1.a2

a2

+ y

2

b2 1.b2

b2

= 1_⇐⇒ x

2 A2 1.a 2 + y2 B2 1.b

2 = 1⇐⇒

X2 A2 1 +Y 2 B2 1 = 1

▲♦❣♦ P′₍x a,

y

b)∈e′1✳

❉❛í i) ❡ ii) sã♦ s❛t✐s❢❡✐t❛s ❡ ♣♦rt❛♥t♦ iii) P =rTe1 ⇐⇒ P′ =r′Te′1❀ t❛♠❜é♠

é s❛t✐s❢❡✐t❛✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ss✐♠ ❛ ♣r♦✈❛✳

❖❜s❡r✈❛♥❞♦ ❛ tr❛♥s❢♦r♠❛çã♦ x= aX e y =bY✱ ❣r❛ç❛s ❛ ❡st❡ r❡s✉❧t❛❞♦✱ ❛ ♣❛rt✐r

❞❡ ❛❣♦r❛ ✈❛♠♦s tr❛❜❛❧❤❛r ❝♦♠ ♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ O ❡ r❛✐♦ ✶✱

x2 +y2

= 1 ✭✷✳✽✮

❡ ❛ ❡❧✐♣s❡

e1 :

x2 a2 1 +y 2 b2 1

= 1, 1> a1 > b1 >0 ✭✷✳✾✮

✐♥t❡r♥❛ à ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ ✶✱ ❝♦♠♦ é ♠♦str❛❞♦ ♥❛ ❋✐❣✉r❛ ✷✳✶✳

Pr❡♣❛r❛♠♦s ✉♠❛ ❧✐st❛ ❞❡ ♣❡r❣✉♥t❛s q✉❡ ♣r❡♣❛r❛ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭♦✉ ❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t✮ ♣❛r❛ ♦ ❝❛s♦n= 3✿ ❉❛❞♦ ✉♠❛ ❡❧✐♣s❡e1 : x

2

a2 1+

y2

b2

1 = 1❡ ♦ ♣♦♥t♦

A0(x0, y0)♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠O ❡ r❛✐♦ ✶✱ ✈❛♠♦s ❡♥❝♦♥tr❛r ❛s r❡t❛s ←−→

A0A1 ❡ ←−→A0A2 t❛♥❣❡♥t❡s ❛ e1 ❝♦♥t❡♥❞♦ ♦ ♣♦♥t♦ A0✱ ❡ ❡♥❝♦♥tr❛r t❛♠❜é♠ ♦s ♣♦♥t♦s

A1, A2 ❞❛ ✐♥t❡rs❡çã♦ ❞❡ss❛s r❡t❛s A0A1, A0A2 ❝♦♠ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ x 2

+y2 = 1✱

✭♣r❡❝✐s❛♠♦s ❡①♣r❡ss❛r ❛ ❢ór♠✉❧❛ ❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ A1, A2 ❡♠ t❡r♠♦s ❞❡ x0, y0 ❡

❞♦s ❝♦❡✜❝✐❡♥t❡s ❛♥❣✉❧❛r❡sk1, k2 ❞❛s r❡t❛s ←−→A0A1 ❡ ←−→A0A2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❯s❛♠♦s ❛

♣❛r❛♠❡tr✐③❛çã♦

xj = cos(ϕj), yj =s❡♥(ϕj), j = 0,1,2, ✭✷✳✶✵✮ ❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ϕj ❡ θj = arctang(kj) ♣❛r❛ j = 1, 2✳ P❛r❛ ✐st♦ ❢❛r❡♠♦s ❛❧❣✉♥s ❧❡♠❛s✳

▲❡♠❛ ✷✳✶ ❉❛❞❛ ❛ ❡❧✐♣s❡✱ e1 : x 2

a2 +

y2

b2 = 1✱ ❛ r❡t❛ t :y−y0 =k(x−x0) q✉❡ ❝♦♥té♠

❈❛♣ít✉❧♦ ✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❊❧✐♣s❡ ✷✳✶✳ ❘❡❞✉çã♦ ❛♦ ❝❛s♦ ❝ír❝✉❧♦ ❡ ❡❧✐♣s❡

❋✐❣✉r❛ ✷✳✸✿ ◗✉❛♥❞♦ A0A1 é t❛♥❣❡♥t❡ ❛e1❄

♦ ♣♦♥t♦ A0(x0, y0) é t❛♥❣❡♥t❡ ❛ ❡❧✐♣s❡ e1 s❡✱ ❡ s♦♠❡♥t❡ s❡

(y0−kx0) 2

=b2 +k2

a2

.

Pr♦✈❛✿ P❛r❛ q✉❡ ❛ r❡t❛ t:y₋y0 =k(x−x0)s❡❥❛ t❛♥❣❡♥t❡ ❛ ❡❧✐♣s❡e1 : x 2

a2

1 +

y2

b2

1 = 1

❞❡✈❡♠♦s ♠♦str❛r q✉❡ e1 ❡ t tê♠ ❛♣❡♥❛s ✉♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠✳ ■st♦ é✱ ❞❡✈❡♠♦s

r❡s♦❧✈❡r ♦ s✐st❡♠❛ ❢♦r♠❛❞♦ ♣♦r t ❡e1✳ ▲♦❣♦✿

      

x2

a2 +

y2

b2 = 1 =⇒b 2

x2 +a2

y2 =a2

b2 (1)

y−y0 =k(x−x0) =⇒y=kx−kx0+y0(2)

❙✉❜st✐t✉✐♥❞♦ ✭✷✮ ❡♠ ✭✶✮✱ ✈❛♠♦s ♦❜t❡r ❛ ❡q✉❛çã♦

b2

x2 +a2

(kx−kx0+y0) 2

=a2

b2

.

❉❡s❡♥✈♦❧✈❡♥❞♦ ♦ ♣r♦❞✉t♦ ♥♦tá✈❡❧ (kx₋ kx0 +y0)

2✱ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ♣❛ss❛ ❛ s❡r

❈❛♣ít✉❧♦ ✷✳ ❚❡♦r❡♠❛ ❞❡ P♦♥❝❡❧❡t ♥❛ ❊❧✐♣s❡ ✷✳✶✳ ❘❡❞✉çã♦ ❛♦ ❝❛s♦ ❝ír❝✉❧♦ ❡ ❡❧✐♣s❡

r❡♣r❡s❡♥t❛❞❛ ♣♦r

b2

x2 +a2

(k2

x2 +k2

x2 0+y

2 0 −2k

2

xx0+ 2kxy0 −2kx0y0) =a 2

b2

.

❱❛♠♦s ❛❣♦r❛ ❡❧✐♠✐♥❛r ♦ ♣❛rê♥t❡s❡ ❞❛ ❡q✉❛çã♦✱ ❞❛í ❡❧❛ ✜❝❛rá ♥♦ ❢♦r♠❛t♦

0 = b2

x2 +a2

k2

x2 +a2

k2

x2 0+a

2

y2 0 −2a

2

k2

xx0+ 2a 2

kxy0−2a 2

kx0y0−a 2

b2

= (b2 +a2

k2 )x2

+ (2a2

ky0−2a 2

k2

x0)x+a 2

(y2

0 −2kx0y0+k 2

x2 0)₋a2

b2

= (b2 +a2

k2 )x2

+ (2a2

ky0−2a 2

k2

x0)x+a 2

(y0−kx0) 2

−a2

b2

= (b2 +a2

k2 )x2

+ (2a2

ky0−2a 2

k2

x0)x+a 2

[(y0−kx0) 2

−b2

].

P❛r❛ q✉❡ ❡st❛ ❡q✉❛çã♦ ❞♦ ✷♦ _{❣r❛✉ ❡♠xt❡♥❤❛ ❛♣❡♥❛s ✉♠❛ s♦❧✉çã♦✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡}

❛ r❡t❛ t ❡ ❛ ❡❧✐♣s❡ e1 s❡❥❛♠ t❛♥❣❡♥t❡s✱ ❞❡✈❡♠♦s t❡r ∆ = 0✳ ❆ss✐♠

0 = 4a4

k2

(y0 −kx0) 2

−4a2

(b2 +a2

k2

)[(y0 −kx0) 2

−b2

]

= 4a4

k2

(y0 −kx0) 2

−4a2

(b2 +a2

k2

)(y0−kx0) 2

+ 4a2 (b2

+a2

k2 )b2

= [4a4

k2

−4a2

(b2 +a2

k2

)](y0−kx0) 2

+ 4a2

b2 (b2

+a2

k2 )

= (4a4

k2

−4a2

b2

−4a4

k2

)(y0−kx0) 2

+ 4a2

b2 (b2

+a2

k2 )

= −4a2

b2

(y0−kx0) 2

+ 4a2

b2 (b2

+a2

k2 )

❞❡ ♦♥❞❡ s❡ ♦❜té♠

4a2b2(y0−kx0) 2

= 4a2b2(b2+a2k2)

♦✉ ❛✐♥❞❛

(y0−kx0) 2

=b2 +a2

k2

❝♦♠♦ q✉❡rí❛♠♦s ♣r♦✈❛r✳

▲❡♠❛ ✷✳✷ ❉❛❞♦s ✉♠❛ ❡❧✐♣s❡ e1 : x 2

a2

1 +

y2

b2

1 = 1 ❡ ♦ ♣♦♥t♦ A

0(x0, y0) ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛