❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás
■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❊❧✐♣s❡ ❞❡ ❙t❡✐♥❡r
●❧❡✐ss♦♥ ❘♦❞r✐❣✉❡s ❙❛♥t❛♥❛
●♦✐â♥✐❛✲●♦✐ás✲❇r❛s✐❧
❊❧✐♣s❡ ❞❡ ❙t❡✐♥❡r
●❧❡✐ss♦♥ ❘♦❞r✐❣✉❡s ❙❛♥t❛♥❛
✶
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ P❛✉❧♦ ❍❡♥r✐q✉❡ ❞❡ ❆③❡✈❡❞♦ ❘♦❞r✐❣✉❡s
●♦✐â♥✐❛✲●♦✐ás✲❇r❛s✐❧
❆❣♦st♦✲✷✵✶✹
❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳ ➱ ♣r♦✐❜✐❞❛ ❛ r❡♣r♦❞✉çã♦ t♦t❛❧ ♦✉ ♣❛r❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ s❡♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❞♦ ❛✉t♦r ❡ ❞♦ ♦r✐❡♥t❛❞♦r✳
❆●❘❆❉❊❈■▼❊◆❚❖❙
➱ ❝♦♠ ♠✉✐t❛ ❛❧❡❣r✐❛ q✉❡ ❛❣r❛❞❡ç♦✿
• ❆ ❉❡✉s✱ q✉❡ ♠❡ ♣r♦♣♦r❝✐♦♥♦✉ ♠❛✐s ❡ss❛ ❝♦♥q✉✐st❛✱ ♣♦✐s s❡♠ ❉❡✉s✱ ♥❛ ❞✐ss♦ t❡r✐❛
❛❝♦♥t❡❝✐❞♦❀
• ❆ ♠✐♥❤❛ ❡s♣♦s❛ ❏✉❧❧✐❛♥❛ q✉❡ s❡♠♣r❡ ❡st❡✈❡ ❞♦ ♠❡✉ ❧❛❞♦ ❞✉r❛♥t❡ t♦❞♦s ❡ss❡s ❞✐❛s✱
♠❡ ❝♦♠♣r❡❡♥❞❡♥❞♦ ❡ ❞❛♥❞♦ ♦ s✉♣♦rt❡ ♥❡❝❡ssár✐♦ ♣❛r❛ ❝✉rs❛r ♦ ♠❡str❛❞♦❀
• ❆ ♠✐❤❛ ✜❧❤❛ ●❛❜r✐❡❧❧❛ ❙♦♣❤✐❛✱ q✉❡ ❛♣❡s❛r ❞❡ ♣❡q✉❡♥❛✱ ❛♦ ♦❜s❡r✈❛r ♦ s❡✉ s♦rr✐s♦
❡ s❡✉ ♦❧❤❛r ♠❡ ♠♦t✐✈❛✈❛ ❛✐♥❞❛ ♠❛✐s ❛ ♣❛ss❛r ♣♦r ❝✐♠❛ ❞❛s ❞✐✜❝✉❧❞❛❞❡s ❡ ❢❛③❡r ♦ ❝✉rs♦❀
• ❆♦s ♠❡✉s ♣❛✐s✱ ❉✐✈✐♥♦ ❡ ❆❧✐❝✐❛✱ q✉❡ sã♦ ♠❡✉s ✐♥❝❡♥t✐✈❛❞♦r❡s ❡ ♣r♦✈❡❞♦r❡s ❞❡
♠❡✐♦s ♣❛r❛ ♠❡✉s ❡st✉❞♦s ❞❡s❞❡ ❛ ✐♥❢â♥❝✐❛✱ ❞❡✈❡r✐❛ t❡r ✉♠ ❞✐♣❧♦♠❛ ♣❛r❛ ♦s ♣❛✐s ♣♦✐s ♦s ♠❡✉s sã♦ ❞✐♣❧♦♠❛❞♦s ❥✉♥t♦ ❝♦♠✐❣♦❀
• ❆♦s ♠❡✉s ✐r♠ã♦s ❈❧❡✐t♦♥ ❡ ❋❡r♥❛♥❞♦✱ q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈❛r❛♠ ❛ ✐r ❛❞✐❛♥t❡
❡♠ ♠❡✉s ❡st✉❞♦s❀
• ❆♦s ♠❡✉s ❛♠✐❣♦s ❱✐t♦r ❇r❛❣❛✱ q✉❡ ❞❡s❞❡ ❛ ❣r❛❞✉❛çã♦ ♥♦s ❛❥✉❞❛♠♦s ✉♠ ❛♦ ♦✉tr♦
♥♦s ❡st✉❞♦s✱ ▼ár✐♦ ❏♦♥❛s q✉❡ ❝♦♥❤❡❝❡♠♦s ♥❡ss❡ ♠❡str❛❞♦ ❡ ♣❛r❡❝❡ q✉❡ ❥á ♥♦s ❝♦♥❤❡❝❡♠♦s ❛ ♠✉✐t♦ ♠❛✐s t❡♠♣♦ ❡ ▼♦r❛❡s✱ q✉❡ ❢♦✐ ✉♠ ✐♥❝❡t✐✈❛❞♦r ♦ t❡♠♣♦ t♦❞♦❀
• ❆♦ ♦r✐❡♥t❛❞♦r ❉r✳ P❛✉❧♦ ❍❡♥r✐q✉❡✱ q✉❡ ❞❡s❞❡ ♦ ❝♦♠❡ç♦ ♥♦s r❡❝❡❜❡✉ ❞❡ ❜r❛ç♦s
❛❜❡rt♦✱ ❡ ♥♦s ❞✐s♣❡♥s♦✉ ❞✉r❛♥t❡ t♦❞♦ ♦ t❡♠♣♦ ❛ ❛t❡♥ç❛♦ ❡ t♦❞❛s ❛s ✐♥❢♦r♠❛çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ♦ s✉❝❡ss♦ ❞❡ss❡ tr❛❜❛❧❤♦❀
• ❊ ❡♠ ❣❡r❛❧ ❛ t♦❞♦s ♦s ❢❛♠✐❧✐❛r❡s ❡ ❛♠✐❣♦s q✉❡ ❞❡ ✉♠❛ ❢♦r♠❛ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛
♠❡ ❛❥✉❞❛r❛♠ ❝♦♠ ✐♥❝❡♥t✐✈♦s ✈❡r❜❛✐s ♦✉ ❝♦♠ ❛t✐t✉❞❡s ❛ ❝♦♥❝❧✉✐r ♦ ❝✉rs♦❀
❘❊❙❯▼❖
P❛❧❛✈r❛s✲❝❤❛✈❡
❆❜str❛❝t
❚❤✐s ✇♦r❦ ✐s ❢♦❝✉s❡❞ ♦♥ ❞❡♠♦♥str❛t✐♥❣✱ ✉s✐♥❣ ❣❡♦♠❡tr✐❝ t♦♦❧s✱ t❤❡ ❙t❡✐♥❡r✬s ❊❧❧✐♣s❡ t❤❡♦r❡♠ ✇❤✐❝❤ s❛②s t❤❛t ❛❧❧ ❡❧❧✐♣s❡s ❡♥t❡r❡❞ ✐♥ ❛ ❣✐✈❡♥ tr✐❛♥❣❧❡✱ t❤❡ ❧❛r❣❡r ❛r❡❛ ✐s t♦✉❝❤✐♥❣ t❤❡ tr✐❛♥❣❧❡ s✐❞❡s ✐♥ t❤❡✐r r❡s♣❡❝t✐✈❡ ♠✐❞❞❧❡ ♣♦✐♥ts✳
❋♦r t❤✐s ✇❡ ❞✐❞✱ ✐♥ ❝❤❛♣t❡r ♦♥❡✱ ❛♥ ❛♣♣r♦❛❝❤ t♦ t❤❡ st✉❞② ♦❢ tr✐❛♥❣❧❡s✱ t❤❡ ❡❧❧✐♣s❡s✱ t❤❡ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥✱ ❛♥❞ ✇❡ ♣r❡s❡♥t s♦♠❡ r❡s✉❧ts ♦❢ ❛r✐t❤♠❡t✐❝ ❛♥❞ ❣❡♦♠❡tr②✱ t❤✐s ✇♦r❦ t✐t❧❡❞ ❛s ♣r❡❧✐♠✐♥❛r② ❧❡♠♠❛s✱ t♦ ❜❡ t❤❡ t♦♦❧s ✭♣r❡r❡q✉✐s✐t❡s✮ ❢♦r ♣r♦♦❢ t❤❡ t❤❡♦r❡♠ t❤❛t t❤❡ ♠❛✐♥ ✐s ♠❛❞❡ ✐♥ ❝❤❛♣t❡r t✇♦✳
✶✷
❑❡②✇♦r❞s
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✶✸
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶ ❚r✐â♥❣✉❧♦ ❞❡ ❉❡s❝❛r❣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷ ❈❛r❛✈❡❧❛s P♦rt✉❣✉❡s❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✸ ❊stá❞✐♦ ❞❡ ❋✉t❡❜♦❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✹ P♦♥t❡ ❞❡ ❋❡rr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✺ P❛ss❛r❡❧❛ ❞❡ P❡❞❡str❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✻ ❈♦♥str✉çã♦ ❈✐✈✐❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✼ P♦❧✐❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✽ P♦❧í❣♦♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✾ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✶✵ ✶♦✲❈❛s♦ ❞❡ ❈♦♥❣r✉ê♥❝✐❛✿ ▲▲▲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✶✶ ✷♦❈❛s♦ ❞❡ ❈♦♥❣r✉ê♥❝✐❛✿ ▲❆▲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✶✷ ✸♦ ❈❛s♦ ❞❡ ❈♦♥❣r✉ê♥❝✐❛✿ ❆▲❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✶✸ ✶♦ ❈❛s♦ ❞❡ ❙❡♠❡❧❤❛♥ç❛✿ ❆❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✶✹ ✷♦ ❈❛s♦ ❞❡ ❙❡♠❡❧❤❛♥ç❛✿ ▲▲▲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✶✺ ✸♦ ❈❛s♦ ❞❡ ❙❡♠❡❧❤❛♥ç❛✿ ▲❆▲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✶✹
✸✹ Pr♦❥❡çã♦ ❞❡ ✉♠❛ ❘❡t❛ ❡♠ ✉♠ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✺ Pr♦❥❡çã♦ ❞❡ ✉♠ ❙❡❣♠❡♥t♦ ❡♠ ✉♠ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✻ Pr♦❥❡çã♦ ❞❡ ❙❡❣♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✼ Pr♦❥❡çã♦ ❞❡ ❙❡❣♠❡♥t♦s Pr♦♣♦r❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✽ Pr♦❥❡çã♦ ❞❡ ❘❡t❛s P❛r❛❧❡❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✾ Pr♦❥❡çã♦ ❞❡ ❊❧✐♣s❡ ❡♠ ❈✐r❝✉♥❢❡r❡♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✵ Pr♦❥❡çã♦ ❞❡ ➪r❡❛ ❞❡ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✶ ❚r✐â♥❣✉❧♦ ■♥❝r✐t♦ ❡♠ ✉♠❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✷ ❚r✐â♥❣✉❧♦ ❈✐r❝✉♥s❝r✐t♦ ❡♠ ✉♠❛ ❈✐r❝✉♥❢❡r❡♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✸ ❊❧✐♣s❡ ■♥❝r✐t❛ ❡♠ ✉♠ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✹✹ Pr♦❥❡çã♦ ❞❛ ❊❧✐♣s❡ ❡♠ ✉♠ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✺ ❊①❡♠♣❧♦ ❞❡ ❚r❡❧✐ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✻ ❆♣❧✐❝❛çõ❡s ❞❡ ❚r❡❧✐ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✼ ❙❡çã♦ ❞❡ ✉♠❛ ❚r❡❧✐ç❛ ❝♦♠ ❊❧❡tr♦❞✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✽ ❚r❛ç❛❞♦ ❞❛ ❊❧✐♣s❡ P❡❧♦ ▼ét♦❞♦ ❞♦ ❏❛r❞✐♥❡✐r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✹✾ ❊❧✐♣s❡ ❆tr❛✈és ❞♦s P♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✺✵ ❈♦♥str✉✐♥❞♦ ❊❧✐♣s❡ ❝♦♠ ❉♦❜r❛❞✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✺✶ ❈♦♥str✉✐♥❞♦ ❊❧✐♣s❡ ❝♦♠ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✺✷ ▼❛r❝❛♥❞♦ ❚rês P♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
✺✸ ❘♦t❛❝✐♦♥❛♥❞♦ ♦ P♦♥t♦Q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
✺✹ P♦♥t♦ ▼é❞✐♦M ❞❡Q′Q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
✺✺ ❈✐r❝✉♥❢❡rê♠❝✐❛ ❞❡ ❈❡♥tr♦ M ❡ ❘❛✐♦ M O ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
✺✻ ❉❡t❡r♠✐♥❛♥❞♦ ♦s ❊✐①♦s ❞❛ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✺✼ ❉✐r❡çã♦ ❞♦s ❊✐①♦s ❞❛ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✺✽ ❉❡t❡r♠✐♥❛♥❞♦ ♦s ❋♦❝♦s ❞❛ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✺✾ ❚r❛ç❛♥❞♦ ❛ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✻✵ ❊❧✐♣s❡ ❞❡ ❏♦s❡♣❤ ❈❛r♥♦② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
✻✶ ❚r❛ç❛❞♦ ❞♦ ❚r✐â♥❣✉❧♦ ABC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸
✻✷ P♦♥t♦F1 ♥♦ ■♥t❡r✐♦r ❞♦ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸
✻✸ ❙✐♠étr✐❝♦s ❞❡ F1 ❡♠ ❘❡❧❛çã♦ ❛♦s ▲❛❞♦s ❞♦ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸
✻✹ ❈❡♥tr♦F2 ❞❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ P❛ss❛ ♣❡❧♦s ❙✐♠étr✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹
✻✺ ■♥t❡rs❡❝çã♦ ❊♥tr❡ ♦s ❙❡❣♠❡♥t♦s q✉❡ ▲✐❣❛♠ F2❡ ♦s ❙✐♠étr✐❝♦s ✳ ✳ ✳ ✳ ✳ ✼✹
❙✉♠ár✐♦
✶ ■◆❚❘❖❉❯➬➹❖ ✶✼
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✶✽
✷✳✶ ❊st✉❞♦ ❞♦s ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✳✶ ❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛ ❞♦s ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✳✷ ❉❡✜♥✐çõ❡s ❡ Pr♦♣r✐❡❞❛❞❡s ❞♦s tr✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳✸ Pr♦♣r✐❡❞❛❞❡s ❞♦s ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷ ❊st✉❞♦ ❞❛s ❊❧✐♣s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷✳✶ ❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷✳✷ ❉❡✜♥✐çõ❡s ❡ Pr♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✸ Pr♦❥❡çã♦ ❖rt♦❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✸✳✶ Pr♦❥❡çã♦ ❖rt♦❣♦♥❛❧ ❞❡ ✉♠ P♦♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✸✳✷ Pr♦❥❡çã♦ ❖rt♦❣♦♥❛❧ ❞❡ ✉♠❛ ❋✐❣✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✸✳✸ Pr♦❥❡çã♦ ❖rt♦❣♦♥❛❧ ❞❡ ✉♠❛ ❘❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✸✳✹ Pr♦❥❡çã♦ ❖rt♦❣♦♥❛❧ ❞❡ ✉♠ ❙❡❣♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✸✳✺ Pr♦♣r✐❡❞❛❞❡s ❞❛ Pr♦❥❡çã♦ ❖rt♦❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸ ❆ ❊▲■P❙❊ ❉❊ ❙❚❊■◆❊❘ ✹✻
✸✳✶ ❘❡❧❛t♦ ❍✐stór✐❝♦ ❙♦❜r❡ ♦ ❆ss✉♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✷ ▲❡♠❛s Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✸ ❊❧✐♣s❡ ❞❡ ❙t❡✐♥❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✹ ❆P▲■❈❆➬Õ❊❙ ❊ ❘❖❚❊■❘❖❙ ❉❊ ❆❯▲❆ ✻✵
❙❯▼➪❘■❖ ✶✻
✶ ■◆❚❘❖❉❯➬➹❖ ✶✼
✶ ■◆❚❘❖❉❯➬➹❖
❖ ❡st✉❞♦ ❞❛s ❡❧✐♣s❡s✱ ❝♦♠♦ ❞❛s ❝ô♥✐❝❛s ❞❡ ✉♠ ♠♦❞♦ ❣❡r❛❧✱ ♥ã♦ é ❛❧❣♦ ❝♦♠✉♠ ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦✳ ●❡r❛❧♠❡♥t❡✱ ❡♠❜♦r❛ ♠✉✐t♦s ❧✐✈r♦s ❛té ❝❤❡❣❛♠ ❛ ❛❜♦r❞❛r ♦ ❛ss✉♥t♦✱ ♠❛s ♥❛ ♣rát✐❝❛ ❞❛s ❛✉❧❛s ♥ã♦ s❡ ❢❛❧❛ ❞❛s ❝ô♥✐❝❛s✱ ♥♦ ♠á①✐♠♦ sã♦ ❛❜♦r❞❛❞♦s ♦s ❛ss✉♥t♦s r❡❢❡r❡♥t❡s à ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡ ❣❡r❛❧♠❡♥t❡ ❛s ❝ô♥✐❝❛s ♥ã♦ sã♦ ♠❡♥❝✐♦♥❛❞❛s ♥❡♠ ♥♦s ♣❧❛♥♦s ❞❡ ❝✉rs♦s ❞♦s ♣r♦❢❡ss♦r❡s✳
❆ ✐♠♣♦rtâ♥❝✐❛ ❞❛s ❝ô♥✐❝❛s✱ ❡ ❡♠ ♣❛rt✐❝✉❧❛r ❞❛ ❡❧✐♣s❡✱ ♣❛r❛ ❛ ❢♦r♠❛çã♦✱ ♥ã♦ só ❞❡ ✉♠ ♣r♦✜ss✐♦♥❛❧ ❞❛ ár❡❛ ❞❡ ❡①❛t❛s ♠❛s ♣❛r❛ ✉♠ ❝✐❞❛❞ã♦ ❛t✐✈♦ ❡ ♣❛rt✐❝✐♣❛t✐✈♦ s♦❝✐❛❧✲ ♠❡♥t❡✱ é ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛✳ P♦✐s✱ t❛❧ ❡st✉❞♦ ❛❣r❡❣❛ ❝♦♥❤❡❝✐♠❡♥t♦s s✐♥❣✉❧❛r❡s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ ♠✉♥❞♦ ❡♠ q✉❡ ❡st❛♠♦s ✐♥s❡r✐❞♦s✱ ♥♦ ❝❛s♦ ❞❛ ❡❧✐♣s❡ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♥❤❡❝❡r s❡✉s ❡❧❡♠❡♥t♦s sã♦ ✐♠♣r❡s❝✐♥❞í✈❡✐s ♣❛r❛ ❡♥t❡♥❞❡r ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ s✐st❡♠❛ s♦❧❛r ♥♦ q✉❛❧ ♦ ♣❧❛♥❡t❛ t❡rr❛✱ ❡♠ q✉❡ ✈✐✈❡♠♦s✱ ❢❛③ ♣❛rt❡✳
◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ❛r❣✉♠❡♥t♦s ❡ ❡❧❡♠❡♥t♦s ❝♦❣♥✐t✐✈♦s ❝♦♠ ❛ ✐♥t❡♥sã♦ ❞❡ ❞❡s♣❡rt❛r ♥♦s ❛❧✉♥♦s ❡ ♥♦s ♣r♦❢❡ss♦r❡s ♦ ❞❡s❡❥♦ ❞❡ ❜✉s❝❛ ♣♦r ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛✐s ❛♣r♦❢✉♥❞❛❞♦ s♦❜r❡ ❛ ❡❧✐♣s❡✳ ❆❜♦r❞❛♠♦s ✉♠ t❡♠❛✿ ✧❆ ❊❧✐♣s❡ ❞❡ ❙t❡✐♥❡r✧q✉❡ ♣r♦♣♦r❝✐✲ ♦♥❛ ✉♠❛ ❛♥á❧✐s❡ ♠❛✐s ❛♣r♦❢✉♥❞❛❞❛ ❞♦ ❡st✉❞♦ ❞❛ ❡❧✐♣s❡✱ ♣♦ré♠ ✉s❛♥❞♦ ❢❡rr❛♠❡♥t❛s ❞❛ ♠❛t❡♠át✐❝❛ ❜ás✐❝❛✱ ♦ q✉❡ ❢❛❝✐❧✐t❛ ♦ ❡st✉❞♦ ❡ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ ❛ss✉♥t♦✳
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✶✽
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙
◆❡ss❛ s❡çã♦ ❢❛r❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞❛s ❞❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❞♦s tr✐â♥❣✉❧♦s ❡ q✉❛♥❞♦ ♥❡❝❡ssár✐♦ ❛♣r❡s❡♥t❛♥❞♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❝♦♠♦ ❥✉st✐✜❝❛t✐✈❛✳ ❆♣r❡s❡♥t❛r❡♠♦s ✉♠ ❜r❡✈❡ ❡st✉❞♦ s♦❜r❡ ❛s ❡❧✐♣s❡ r❡ss❛❧t❛♥❞♦ s❡✉s ❡❧❡♠❡♥t♦s ♣r✐♥❝✐♣❛✐s ❡ s✉❛s ♣r♦♣r✐❡✲ ❞❛❞❡s✳ ❋❛r❡♠♦s t❛♠❜é♠ ✉♠❛ ❛❜♦r❞❛❣❡♠ s♦❜r❡ ❛ t❡♦r✐❛ ❞❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧✳ ❚♦❞❛s ❡ss❛s ❛❜♦r❞❛❣❡♥s t❡♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ s❡r ✉♠ s✉♣♦rt❡ t❡ór✐❝♦ ♣❛r❛ ❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ ♣❛rt❡ ♣r✐♥❝✐♣❛❧ ❞♦ tr❛❜❛❧❤♦ q✉❡ é ♦ ❡st✉❞♦ ❞❛ ❊❧✐♣s❡ ❞❡ ❙t❡✐♥❡r✳
✷✳✶ ❊st✉❞♦ ❞♦s ❚r✐â♥❣✉❧♦s
✷✳✶✳✶ ❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛ ❞♦s ❚r✐â♥❣✉❧♦s
❖s tr✐â♥❣✉❧♦s sã♦ ❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s✱ ❥á q✉❡ q✉❛❧q✉❡r ♣♦❧í❣♦♥♦ ❝♦♠ ✉♠ ♥ú♠❡r♦ ♠❛✐♦r ❞❡ ❧❛❞♦s ♣♦❞❡ r❡❞✉③✐r✲s❡ ❛ ✉♠❛ s✉❝❡ssã♦ ❞❡ tr✐â♥❣✉❧♦s✱ ❛♦ tr❛ç❛r t♦❞❛s ❛s s✉❛s ❞✐❛❣♦♥❛✐s ❛ ♣❛rt✐r ❞❡ ✉♠ ✈ért✐❝❡✳ ❆ ❣❡♦♠❡tr✐❛ ❞♦ tr✐â♥❣✉❧♦ é ❞❡ ✉♠❛ r✐q✉❡③❛ ✐♥❝rí✈❡❧ ❡ t❡♠ ❛♣❛✐①♦♥❛❞♦✱ ❞✉r❛♥t❡ sé❝✉❧♦s✱ ♦s ♠❛t❡♠át✐❝♦s ❡ ❡st✉❞✐♦s♦s ❛♠❛❞♦r❡s✳ ❆ ❣❡♦♠❡tr✐❛ ❞❡ ❊✉❝❧✐❞❡s r❡s❡r✈❛ ✉♠ ❧✉❣❛r ♣r❡♣♦♥❞❡r❛♥t❡ ❛♦ tr✐â♥❣✉❧♦✳ ❈♦♠ ❡❢❡✐t♦✱ três ♣♦♥t♦s ♥ã♦ ❛❧✐♥❤❛❞♦s ✭♥ã♦ ❝♦❧✐♥❡❛r❡s✮ ❞❡t❡r♠✐♥❛♠ ✉♠ ❡ ✉♠ só ♣❧❛♥♦❀ ❡ ♦ tr✐â♥❣✉❧♦ é ♥❡st❡ ♣❧❛♥♦ ♦ ♣♦❧í❣♦♥♦ ♠❛✐s s✐♠♣❧❡s✳
◆ã♦ ❡①✐st❡ q✉❛❧q✉❡r r❡❢❡rê♥❝✐❛ ♣♦♥t✉❛❧ ❛ q✉❡♠ ♦✉ ❝♦♠♦ t❡rá s✐❞♦ ✐♥✈❡♥t❛❞♦ ♦✉ ❞❡s❝♦❜❡rt♦ ♦ tr✐â♥❣✉❧♦✳ ❚❡rá s✐❞♦✱ t❛❧✈❡③✱ ♦ ❤♦♠❡♠ q✉❡ ❛♦ ❧♦♥❣♦ ❞❛ s✉❛ ❡✈♦❧✉çã♦ t❡rá s❡♥t✐❞♦ ♥❡❝❡ss✐❞❛❞❡ ♥❛ s✉❛ ✈✐❞❛ ♣rát✐❝❛ ❞❡ t♦r♥❛r rí❣✐❞❛s ❡ s❡❣✉r❛s ❛❧❣✉♠❛s ❞❛s s✉❛s ❝♦♥str✉çõ❡s✳ P♦r ❡①❡♠♣❧♦✱ ♥♦s t❡♠♣♦s ♣r✐♠✐t✐✈♦s ❞❛ ❝✐✈✐❧✐③❛çã♦ ●r❡❣❛✱ ❢♦✐ ✉s❛❞♦ ♣❡❧♦s ❣r❡❣♦s ♦ tr✐â♥❣✉❧♦ ❞❡ ❞❡s❝❛r❣❛✳
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✶✾
❖ tr✐â♥❣✉❧♦ ❞❡ ❞❡s❝❛r❣❛ ❡r❛ ✉♠❛ ❝♦♥str✉çã♦ q✉❡ ♣❡r♠✐t✐❛ ❞❡s❝❛rr❡❣❛r ❛s ♣r❡ssõ❡s ❡①❡r❝✐❞❛s ♣♦r ❣r❛♥❞❡s ♣❡s♦s q✉❡ s❡ ❡♥❝♦♥tr❛✈❛♠ ♣♦r ❝✐♠❛ ❞❛s ♣♦rt❛s ❞♦s tú♠✉❧♦s ❡ ❞❛s ❝✐❞❛❞❡s✳ ❉❡✈✐❞♦ ❛♦ ♣❡s♦✱ ❛s ♣♦rt❛s ♣♦❞✐❛♠ r♦♠♣❡r✲s❡✱ ♠❛s ❝♦♠ ♦ tr✐â♥❣✉❧♦✱ ❡ss❡ ♣❡s♦ ❡r❛ s✉♣♦rt❛❞♦ ♣♦r ♣✐❧❛r❡s ❧❛t❡r❛✐s q✉❡ ❡r❛♠ ❢❡✐t♦s ❝♦♠ ♠❛t❡r✐❛❧ rí❣✐❞♦✳
◆♦ ♣r✐♥❝í♣✐♦ ❞❛ ■❞❛❞❡ ▼é❞✐❛✱ ❛♣❛r❡❝❡✉ ♥♦ ▼❡❞✐t❡rrâ♥❡♦✱ ✉♠❛ ✈❡❧❛ tr✐❛♥❣✉❧❛r✱ ❛❧✐✲ ♥❤❛❞❛ ❝♦♠ ♦ ❡✐①♦ ❧♦♥❣✐t✉❞✐♥❛❧ ❞♦ ❝❛s❝♦ ❞♦ ♥❛✈✐♦✱ ❝♦♥tr❛r✐❛♥❞♦ ❛ ❛té ❡♥tã♦ ✉t✐❧✐③❛❞❛✱ q✉❡ ❡r❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♠❡s♠♦ ❡✐①♦ ❡ ❞❡ ❝♦♥✜❣✉r❛çã♦ q✉❛❞r❛❞❛✱ ❝❤❛♠❛❞❛ ❘❡❞♦♥❞❛✱ ♣❡❧♦ ❢❛t♦ ❞❡ ✈✐st❛ ❞❡ ❧♦♥❣❡ ♣❛r❡❝❡r r❡❞♦♥❞❛✳ ◆ã♦ s❡ s❛❜❡ q✉❡♠ ❛ ✉t✐❧✐③♦✉ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③✳ ➪r❛❜❡s✱ ■♥❞✐❛♥♦s ♦✉ ❛té ■♥❞♦♥és✐♦s✱ sã♦ ❛♣♦♥t❛❞♦s ❝♦♠♦ ♦s ♣❡r❝✉rs♦r❡s ❞❡ t❛❧ s✐st❡♠❛✱ q✉❡ ♣❡r♠✐t❡ à ❡♠❜❛r❝❛çã♦ ♥❛✈❡❣❛r ❝♦♥tr❛ ♦ ✈❡♥t♦ ❛ ✉♥s ✺✵ ♦✉ ✻✵ ❣r❛✉s✳
❖s ➪r❛❜❡s ✉s❛✈❛♠ ♥❛ ♣❡s❝❛ ❡ ♥♦ tr❛♥s♣♦rt❡ ❞❡ ❣ê♥❡r♦s✱ ✉♠❛ ❡♠❜❛r❝❛çã♦ r♦❜✉st❛✱ ❞❡ ❢♦r♠❛s ✜♥❛s✱ ❞❡ ♣❡r✜❧ ❜❛✐①♦✱ ❝❤❛♠❛❞❛ ✧❈❛r❛✈♦✧q✉❡ ❡r❛ ❛r♠❛❞❛ ❝♦♠ ✉♠❛ ✈❡❧❛ tr✐❛♥❣✉❧❛r✳
❈♦♠ ❛ ♦❝✉♣❛çã♦ ❞❛ P❡♥í♥s✉❧❛ ■❜ér✐❝❛✱ é ❞❡ ♣r❡✈❡r q✉❡ ❡st❡ t✐♣♦ ❞❡ ❡♠❜❛r❝❛çã♦ t❡♥❤❛ ✈✐♥❞♦ ❝♦♠ ♦s ✐♥✈❛s♦r❡s ❡ t❡♥❤❛ ❝❤❛♠❛❞♦ ❛ ❛t❡♥çã♦ ❞♦s ❝♦♥str✉t♦r❡s ❞❡ ❡♠❜❛r❝❛✲ çõ❡s ❞❛ ❝♦st❛ ❞♦ ❆t❧â♥t✐❝♦✱ ❞❡✈✐❞♦ às s✉❛s q✉❛❧✐❞❛❞❡s ♥á✉t✐❝❛s✳ ❖s P♦rt✉❣✉❡s❡s ❡♥tã♦ ✐♥tr♦❞✉③✐r❛♠ ❣r❛♥❞❡s ♠❡❧❤♦r❛♠❡♥t♦s ❡ ♥❛s❝❡✉ ❛ ❈❛r❛✈❡❧❛ P♦rt✉❣✉❡s❛✳
❋✐❣✉r❛ ✷✿ ❈❛r❛✈❡❧❛s P♦rt✉❣✉❡s❛s
❆ ❈❛r❛✈❡❧❛ P♦rt✉❣✉❡s❛ ❢♦✐ ♦ ♥❛✈✐♦ ❡s❝♦❧❤✐❞♦ ♣❛r❛ ❛ ❞❡♠❛♥❞❛ ❞♦s ❞❡s❝♦❜r✐♠❡♥t♦s s✉❜st✐t✉✐♥❞♦ ❛s ❡♠❜❛r❝❛çõ❡s ❞❡ ✈❡❧❛ r❡t❛♥❣✉❧❛r✳ ❆ ✈❡❧❛ tr✐❛♥❣✉❧❛r ♦✉ ❧❛t✐♥❛ ✭❝♦♠♦ ♠✉✐t♦s ❛ ❝❤❛♠❛♠✮ ♣❡r♠✐t✐✉ ❛♦s ♣♦rt✉❣✉❡s❡s ♥❛✈❡❣❛r ❝♦♥tr❛ ♦ ✈❡♥t♦✳ ❉✉r❛♥t❡ ♠❛✐s ❞❡ ✹✺✵ ❛♥♦s ❛ ❈❛r❛✈❡❧❛ t♦r♥♦✉✲s❡ ❝é❧❡❜r❡ ♣❡❧♦ ▼✉♥❞♦✳ ◆❛ ❜❛s❡ ❞❡st❛ ✐♥✈❡♥çã♦ ❡stá ♦ tr✐â♥❣✉❧♦✳
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✵
◆❛ ❝♦❜❡rt✉r❛ ❞❡ ❡stá❞✐♦s ❝♦♠♦ ♦s ❞❛ ❝♦♣❛✳
❋✐❣✉r❛ ✸✿ ❊stá❞✐♦ ❞❡ ❋✉t❡❜♦❧
◆❛ ❝♦♥str✉çã♦ ❞❡ ♣♦♥t❡s ❞❡ ❢❡rr♦✳
❋✐❣✉r❛ ✹✿ P♦♥t❡ ❞❡ ❋❡rr♦
◆❛s ♣❛ss❛r❡❧❛s ♣❛r❛ ♣❛ss❛❣❡♥s ❞❡ ♣❡❞❡str❡s
❋✐❣✉r❛ ✺✿ P❛ss❛r❡❧❛ ❞❡ P❡❞❡str❡
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✶
❋✐❣✉r❛ ✻✿ ❈♦♥str✉çã♦ ❈✐✈✐❧
❊ ❡♠ ♠✉✐t❛s ♦✉tr❛s s✐t✉❛çõ❡s ✐♠♣♦rt❛♥t❡s✳
❈♦♠♦ ❱✐♠♦s✱ t❡r ♦ ❞♦♠í♥✐♦ ❞❛s ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ ❛♣❧✐❝❛çõ❡s ❞♦s tr✐â♥❣✉❧♦s é ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❞❡s❡♠♣❡♥❤♦ ❞❡ ♠✉✐t❛s ♣r♦✜ssõ❡s✱ ❞❛í ❛ ✐♠♣♦r✲ tâ♥❝✐❛ ❞♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠átr✐❝❛ t❡r ♦ ❝✉✐❞❛❞♦ ❞❡ ❢♦r♥❡❝❡r ✉♠ ❜♦♠ ❝✉rs♦ s♦❜r❡ ❡ss❡ ❛ss✉♥t♦ ❛ s❡✉s ❛❧✉♥♦s✳
✷✳✶✳✷ ❉❡✜♥✐çõ❡s ❡ Pr♦♣r✐❡❞❛❞❡s ❞♦s tr✐â♥❣✉❧♦s
❈❤❛♠❛♠♦s ❞❡ ♣♦❧✐❣♦♥❛❧ ❛ ✉♠❛ ✜❣✉r❛ ❢♦r♠❛❞❛ ♣♦r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s
A1, A2, A3, A4, ..., An−1, An
❡ ♣❡❧♦s s❡❣♠❡♥t♦s
A1A2, A2A3, A3A4, ..., An−1An.
❖s ♣♦♥t♦s sã♦ ♦s ✈ért✐❝❡s ❞❛ ♣♦❧✐❣♦♥❛❧ ❡ ♦s s❡❣♠❡♥t♦s sã♦ ♦s s❡✉s ❧❛❞♦s✳
❋✐❣✉r❛ ✼✿ P♦❧✐❣♦♥❛❧
❯♠ ♣♦❧í❣♦♥♦ é ✉♠❛ ♣♦❧✐❣♦♥❛❧ ❡♠ q✉❡ ❛s três ❝♦♥❞✐çõ❡s ❛❜❛✐①♦ sã♦ s❛t✐s❢❡✐t❛s✿
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✷
❜✮ ♦s ❧❛❞♦s ❞❛ ♣♦❧✐❣♦♥❛❧ s❡ ✐♥t❡r❝❡♣t❛♠ s♦♠❡♥t❡ ❡♠ s✉❛s ❡①tr❡♠✐❞❛❞❡s ❡
❝✮ ❞♦✐s ❧❛❞♦s ❝♦♠ ♠❡s♠❛ ❡①tr❡♠✐❞❛❞❡ ♥ã♦ ♣❡rt❡♥❝❡♠ ❛ ✉♠❛ ♠❡s♠❛ r❡t❛✳
❆❜❛✐①♦ t❡♠♦s ❡①❡♠♣❧♦s ❞❡ ♣♦❧í❣♦♥♦s✳
❋✐❣✉r❛ ✽✿ P♦❧í❣♦♥♦
❉❡✜♥✐çã♦ ✷✳✶✳ ❈❤❛♠❛♠♦s ❞❡ tr✐â♥❣✉❧♦ ❛ t♦❞❛ ♣♦❧✐❣♦♥❛❧ ❝♦♠ três ✈ért✐❝❡s ❡ três ❧❛❞♦s✳
❋✐❣✉r❛ ✾✿ ❚r✐â♥❣✉❧♦s
❉❡✜♥✐çã♦ ✷✳✷✳ ❉♦✐s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s s❡ ❢♦r ♣♦ssí✈❡❧ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦rr❡s✲ ♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ s❡✉s ✈ért✐❝❡s ❞❡ ♠♦❞♦ q✉❡ ❧❛❞♦s ❡ â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s s❡❥❛♠ ❝♦♥❣r✉❡♥t❡s✳
❆ ✈❡r✐✜❝❛çã♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ ❞♦✐s tr✐â♥❣✉❧♦s ♣♦❞❡ s❡r ❢❡✐t❛✱ ❞❡ ♠❛♥❡✐r❛ ❛❜r❡✲ ✈✐❛❞❛✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦s três ❝❛s♦s r❡❧❛❝✐♦♥❛❞♦s ❛❜❛✐①♦✿
• ✶♦ ❈❛s♦ ▲▲▲ ✭▲❛❞♦✱ ▲❛❞♦✱ ▲❛❞♦✮
❙❡ t♦❞♦s ♦s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❢♦r❡♠ ❝♦♥❣r✉❡♥t❡s ❛♦s ❧❛❞♦s ❞❡ ♦✉tr♦✱ ♦s ❞♦✐s
tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✳ P♦r ❡①❡♠♣❧♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦ t❡♠♦s a ∼=a′✱ b ∼=b′ ❡
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✸
❋✐❣✉r❛ ✶✵✿ ✶♦✲❈❛s♦ ❞❡ ❈♦♥❣r✉ê♥❝✐❛✿ ▲▲▲
• ✷♦ ❈❛s♦ ▲❆▲ ✭▲❛❞♦✱ ➶♥❣✉❧♦✱ ▲❛❞♦✮
❙❡ ❞♦✐s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❢♦r❡♠ ❝♦♥❣r✉❡♥t❡s ❛ ❞♦✐s ❧❛❞♦s ❞❡ ♦✉tr♦ tr✐â♥❣✉❧♦✱ ❡ ♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣♦r ❡ss❡s ❧❛❞♦s ❝♦♥❣r✉❡♥t❡s t❛♠❜é♠ ❢♦r❡♠ ❝♦♥❣r✉❡♥t❡s✱ ♦s
❞♦✐s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✳ P♦r ❡①❡♠♣❧♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦ t❡♠♦s Aˆ ∼= ˆA′✱
b ∼=b′ ❡ c∼=c′ ❧♦❣♦ △ABC ∼=△A′B′C′✳
❋✐❣✉r❛ ✶✶✿ ✷♦❈❛s♦ ❞❡ ❈♦♥❣r✉ê♥❝✐❛✿ ▲❆▲
• ✸♦ ❈❛s♦ ❆▲❆ ✭➶♥❣✉❧♦✱ ▲❛❞♦✱ ➶♥❣✉❧♦✮
❙❡ ✉♠ ❧❛❞♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❢♦r ❝♦♥❣r✉❡♥t❡ ❛ ✉♠ ❧❛❞♦ ❞❡ ♦✉tr♦ tr✐â♥❣✉❧♦✱ ❡ ♦s â♥❣✉❧♦s ❛❞❥❛❝❡♥t❡s ❛ ❡ss❡s ❧❛❞♦s t❛♠❜é♠ ❢♦r❡♠ ❝♦♥❣r✉❡♥t❡s✱ ♦s ❞♦✐s tr✐â♥❣✉❧♦s
sã♦ ❝♦♥❣r✉❡♥t❡s✳ P♦r ❡①❡♠♣❧♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦ t❡♠♦s c∼=c′ ❡ Aˆ∼= ˆA′ ❡ Bˆ ∼= ˆB′
❧♦❣♦ △ABC ∼=△A′B′C′✳
❋✐❣✉r❛ ✶✷✿ ✸♦ ❈❛s♦ ❞❡ ❈♦♥❣r✉ê♥❝✐❛✿ ❆▲❆
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✹
✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ s❡✉s ✈ért✐❝❡s ❞❡ ♠♦❞♦ q✉❡ â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s s❡❥❛♠ ✐❣✉❛✐s ❡ ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s s❡❥❛♠ ♣r♦♣♦r❝✐♦♥❛✐s✳
◆ã♦ é ♥❡❝❡ssár✐♦ ❝♦♥❢❡r✐r s❡ t♦❞♦s ♦s â♥❣✉❧♦s ❞❡ ❞♦✐s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s ❡ s❡ t♦❞♦s ♦s ❧❛❞♦s ❞♦s ♠❡s♠♦s sã♦ ♣r♦♣♦r❝✐♦♥❛✐s ♣❛r❛ s❛❜❡r s❡ ❛♠❜♦s sã♦ s❡♠❡❧❤❛♥t❡s✱ ❜❛st❛ q✉❡ ❡❧❡s ❛♣r❡s❡♥t❡♠ ❛❧❣✉♠❛s ❞❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s✳ ▼♦str❛r❡♠♦s✱ ❛ s❡❣✉✐r✱ três ❝❛s♦s q✉❡ ❢❛❝✐❧✐t❛♠ ❞❡t❡r♠✐♥❛r q✉❛♥❞♦ tr✐â♥❣✉❧♦s sã♦ s❡♠❡❧❤❛♥t❡s✳
• ✶♦ ❈❛s♦ ❆❆ ✭➶♥❣✉❧♦✱ ➶♥❣✉❧♦✮
❙❡ ❞♦✐s â♥❣✉❧♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ sã♦ ❝♦♥❣r✉❡♥t❡s ❛ ❞♦✐s â♥❣✉❧♦s ❞❡ ♦✉tr♦ tr✐â♥❣✉❧♦✱
♦s ❞♦✐s tr✐â♥❣✉❧♦s sã♦ s❡♠❡❧❤❛♥t❡s✳ P♦r ❡①❡♠♣❧♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦ t❡♠♦s Aˆ∼= ˆA′
❡ Bˆ ∼= ˆB′ ❧♦❣♦ △ABC ∼ △A′B′C′✳
❋✐❣✉r❛ ✶✸✿ ✶♦ ❈❛s♦ ❞❡ ❙❡♠❡❧❤❛♥ç❛✿ ❆❆
• ✷♦ ❈❛s♦ ▲▲▲ ✭▲❛❞♦✱ ▲❛❞♦✱ ▲❛❞♦✮
❙❡ t♦❞♦s ♦s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❢♦r❡♠ ♣r♦♣♦r❝✐♦♥❛✐s ❛♦s ❧❛❞♦s ❞❡ ♦✉tr♦✱ ♦s ❞♦✐s
tr✐â♥❣✉❧♦s sã♦ s❡♠❡❧❤❛♥t❡s✳ P♦r ❡①❡♠♣❧♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦ t❡♠♦s a
a′ =
b b′ =
c c′
❧♦❣♦ △ABC ∼ △A′B′C′✳
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✺
• ✸♦ ❈❛s♦ ▲❆▲ ✭▲❛❞♦✱ ➶♥❣✉❧♦✱ ▲❛❞♦✮
❙❡ ❞♦✐s tr✐â♥❣✉❧♦s ♣♦ss✉ír❡♠ ✉♠ â♥❣✉❧♦ ❝♦♥❣r✉❡♥t❡ ❢♦r♠❛❞♦ ❡♥tr❡ ❞♦✐s ❧❛❞♦s ❞❡ ♠❡❞✐❞❛s ♣r♦♣♦r❝✐♦♥❛✐s✱ ♦s ❞♦✐s tr✐â♥❣✉❧♦s sã♦ s❡♠❡❧❤❛♥t❡s✳ P♦r ❡①❡♠♣❧♦ ♥❛ ✜❣✉r❛
❛❜❛✐①♦ t❡♠♦s Bˆ ∼= ˆB′ ❡ a
a′ =
c
c′ ❧♦❣♦ △ABC ∼ △A ′B′C′✳
❋✐❣✉r❛ ✶✺✿ ✸♦ ❈❛s♦ ❞❡ ❙❡♠❡❧❤❛♥ç❛✿ ▲❆▲
❉❡✜♥✐çã♦ ✷✳✹✳ ❙❡❥❛ABC ✉♠ tr✐â♥❣✉❧♦ ❡ s❡❥❛ D✉♠ ♣♦♥t♦ ❞❛ r❡t❛ q✉❡ ❝♦♥té♠B ❡C✳
❖ s❡❣♠❡♥t♦ AD ❝❤❛♠❛✲s❡ ♠❡❞✐❛♥❛ ❞♦ tr✐â♥❣✉❧♦ r❡❧❛t✐✈❛♠❡♥t❡ ❛♦ ❧❛❞♦ BC✱ s❡ D ❢♦r
♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡BC✳ ❖ s❡❣♠❡♥t♦ AD ❝❤❛♠❛✲s❡ ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ Aˆ s❡ ❛ s❡♠✐rr❡t❛
SAD ❞✐✈✐❞❡ ♦ â♥❣✉❧♦ CABˆ ❡♠ ❞♦✐s â♥❣✉❧♦s ✐❣✉❛✐s✳ ❖ s❡❣♠❡♥t♦ AD ❝❤❛♠❛✲s❡ ❛❧t✉r❛
❞♦ tr✐â♥❣✉❧♦ r❡❢❡r❡♥t❡ ❛♦ ❧❛❞♦ BC✱ s❡ AD ❢♦r ♣❡r♣❡♥❞✐❝✉❧❛r ❛ r❡t❛ q✉❡ ❝♦♥té♠ BC✳ ❊
❝❤❛♠❛✲s❡ ♠❡❞✐❛tr✐③ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦rD ❡ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ BC✳
P♦r ❡①❡♠♣❧♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ❡♠ ✭❛✮ AD é ♠❡❞✐❛♥❛✱ ❡♠ ✭❜✮ AD é ❜✐ss❡tr✐③ ❡ ❡♠
✭❝✮ AD é ❛❧t✉r❛✳
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✻
❉❡✜♥✐çã♦ ✷✳✺✳ ❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✳ ◆❡ss❡ tr✐â♥❣✉❧♦ t❡♠♦s✿
❛✮ ❆s ♠❡❞✐❛tr✐③❡s ❞♦s três ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ s❡ ✐♥t❡r❝❡♣t❛♠ ❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦O✱
❝❤❛♠❛❞♦ ❝✐r❝✉♥❝❡♥tr♦✳ ❊st❡ ♣♦♥t♦ ❡stá ❡q✉✐❞✐st❛♥t❡ ✭à ♠❡s♠❛ ❞✐stâ♥❝✐❛✮ ❞♦s três ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ ❡ é ♦ ❝❡♥tr♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝✐r❝✉♥s❝r✐t❛ ❛♦ tr✐â♥❣✉❧♦✳
❜✮ ❆s ❜✐ss❡tr✐③❡s ❞♦s três â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ tr✐â♥❣✉❧♦ s❡ ✐♥t❡r❝❡♣t❛♠ ❡♠ ✉♠ ú♥✐❝♦
♣♦♥t♦ I ❝❤❛♠❛❞♦ ✐♥❝❡♥tr♦✱ q✉❡ ❡stá ❡q✉✐❞✐st❛♥t❡ ✭à ♠❡s♠❛ ❞✐stâ♥❝✐❛✮ ❞♦s ❧❛❞♦s
❞♦ tr✐â♥❣✉❧♦ ❡ é ♦ ❝❡♥tr♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ♥♦ tr✐â♥❣✉❧♦✳
❝✮ ❆s ❛❧t✉r❛s r❡❢❡r❡♥t❡s ❛♦s três ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ s❡ ✐♥t❡rs❡♣t❛♠ ❡♠ ✉♠ ú♥✐❝♦
♣♦♥t♦ H✱ ❝❤❛♠❛❞♦ ♦rt♦❝❡♥tr♦✳
❞✮ ❆s ♠❡❞✐❛♥❛s ❞♦ tr✐â♥❣✉❧♦ s❡ ✐♥t❡r❝❡♣t❛♠ ❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦ G ❝❤❛♠❛❞♦ ❜❛r✐✲
❝❡♥tr♦✱ q✉❡ ❞✐st❛ ❞♦✐s t❡rç♦s ❞♦ ✈ért✐❝❡ ❞❛ ♠❡❞✐❛♥❛ ❝♦rr❡s♣♦♥❞❡♥t❡✳ ❖ ❜❛r✐❝❡♥tr♦ é ♦ ❝❡♥tr♦ ❞❡ ❣r❛✈✐❞❛❞❡ ❞♦ tr✐â♥❣✉❧♦✳ ■st♦ q✉❡r ❞✐③❡r q✉❡✱ s❡ s✉s♣❡♥❞❡r♠♦s ✉♠ tr✐â♥❣✉❧♦ ❞❡ ♠❛t❡r✐❛❧ ❤♦♠♦❣ê♥❡♦ ♣❡❧♦ s❡✉ ❜❛r✐❝❡♥tr♦✱ ❡❧❡ ✜❝❛ ❡♠ ❡q✉✐❧í❜r✐♦✳
❱❡❥❛ ❛s ✐❧✉str❛çõ❡s ♥❛s ✜❣✉r❛s ❛❜❛✐①♦
❋✐❣✉r❛ ✶✼✿ ❇❛r✐❝❡♥tr♦✱ ❈✐r❝✉♥❝❡♥tr♦✱ ❍♦rt♦❝❡♥tr♦ ❡ ■♥❝❡♥tr♦✱ ❘❡s♣❡❝t✐✈❛♠❡♥t❡
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✼
❉❡✜♥✐çã♦ ✷✳✻✳ ❆ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ T✱ ❞❡ ✈ért✐❝❡s A✱ B ❡ C é ❞❛❞❛ ♣♦r
A[T] = AB ·hc 2
❖♥❞❡ AB é ❛ ❜❛s❡ ❞♦ tr✐â♥❣✉❧♦ ❡ hc é ❛ ❛❧t✉r❛ r❡❢❡r❡♥t❡ ❛♦ ✈ért✐❝❡ C ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦
❧❛❞♦ AB✳
❆ ❢ór♠✉❧❛ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ ❞❡✜♥✐çã♦ ♥❡ss❡ tr❛❜❛❧❤♦✱ ♠❛✐s s✉❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥❛ r❡❢❡rê♥❝✐❛ ❜✐❜❧✐♦❣rá✜❝❛ ❬✸❪✳
✷✳✶✳✸ Pr♦♣r✐❡❞❛❞❡s ❞♦s ❚r✐â♥❣✉❧♦s
❆♣r❡s❡♥t❛r❡♠♦s ❛ s❡❣✉✐r ❛❧❣✉♥s ❢❛t♦s ❞❡❝♦rr❡t❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ tr✐â♥❣✉❧♦✱ ♥❡st❡ tr❛✲ ❜❛❧❤♦ ❞❡♥♦♠✐♥❛❞♦s ❞❡ ♣r♦♣r✐❡❞❛❞❡s✳ ❆s ♣r♦♣r✐❡❞❛❞❡s ❞♦s tr✐â♥❣✉❧♦s s❡rã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ❞❡♠♦s♥tr❛çã♦ ❞♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦✳ ◗✉❡r❡♠♦s r❡ss❛❧t❛r q✉❡ ♦ ♦❜❥❡t✐✈♦ ♥ã♦ é ❛♣r❡s❡♥t❛r ✉♠ ❡st✉❞♦ ❝♦♠♣❧❡t♦ s♦❜r❡ ♦s tr✐â♥❣✉❧♦s✱ ♠❛✐s s✐♠ ❛❧❣✉♠❛s ❢❡rr❛♠❡♥t❛s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❥✉st✐✜❝❛r ♦ ❡st✉❞♦ ♣r♦♣♦st♦✳
Pr♦♣r✐❡❞❛❞❡ ✷✳✶✳ ❚♦❞♦ tr✐â♥❣✉❧♦ ♣♦❞❡ s❡r ❝❧❛ss✐✜❝❛❞♦ q✉❛♥t♦ ❛♦s ❧❛❞♦s ❝♦♠♦ ❡q✉✐✲ ❧át❡r♦✱ q✉❛♥❞♦ ♣♦ss✉✐ t♦❞♦s ♦s ❧❛❞♦s ❝♦♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛✱ ✐sós❝❡❧❡s✱ q✉❛♥❞♦ ♣♦ss✉✐ ❞♦✐s ❧❛❞♦s ❝♦♠ ♠❡s♠❛ ♠❡❞✐❞❛✱ ♦✉ ❡s❝❛❧❡♥♦✱ q✉❛♥❞♦ t♦❞♦s ♦s ❧❛❞♦s ♣♦ss✉❡♠ ♠❡❞✐❞❛s ❞✐❢❡r❡♥t❡s
◆❛ ✜❣✉r❛ ❛❜❛✐①♦ t❡♠♦s ❛ ✐❧✉str❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡
❋✐❣✉r❛ ✶✽✿ ❈❧❛ss✐✜❝❛çã♦ ❞♦s ❚r✐â♥❣✉❧♦s ◗✉❛♥t♦ ❛♦s ▲❛❞♦s
Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳ ❚♦❞♦ tr✐â♥❣✉❧♦ ♣♦❞❡ s❡r ❝❧❛ss✐✜❝❛❞♦ q✉❛♥t♦ ❛♦s â♥❣✉❧♦s ❝♦♠♦ ❛❝✉✲ tâ♥❣✉❧♦✱ q✉❛♥❞♦ ♣♦ss✉✐ t♦❞♦s ♦s â♥❣✉❧♦s ❛❣✉❞♦s✱ ♦❜t✉sâ♥❣✉❧♦✱ q✉❛♥❞♦ ♣♦ss✉✐ ✉♠ â♥❣✉❧♦ ♦❜t✉s♦✱ ♦✉ r❡tâ♥❣✉❧♦✱ q✉❛♥❞♦ ♣♦ss✉✐ ✉♠ â♥❣✉❧♦ r❡t♦✳
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✽
❋✐❣✉r❛ ✶✾✿ ❈❧❛ss✐✜❝❛çã♦ ❞♦s ❚r✐â♥❣✉❧♦s ◗✉❛♥t♦ ❛♦s ➶♥❣✉❧♦s
❉❡✜♥✐çã♦ ✷✳✼✳ ❯♠ ❝ír❝✉❧♦ ❡stá ✐♥s❝r✐t♦ ❡♠ ✉♠ ♣♦❧í❣♦♥♦✱ ❡♠ ♣❛rt✐❝✉❧❛r ❡♠ ✉♠ tr✐â♥✲ ❣✉❧♦✱ s❡ t♦❞♦s ♦s ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦ sã♦ t❛♥❣❡♥t❡s ❛♦ ❝ír❝✉❧♦✳ q✉❛♥❞♦ ✐st♦ ♦❝♦rr❡ ❞✐③✲s❡ q✉❡ ♦ ♣♦❧í❣♦♥♦ ❝✐r❝✉♥s❝r❡✈❡ ♦ ❝ír❝✉❧♦✳
❱❡❥❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✳
❋✐❣✉r❛ ✷✵✿ ❈✐r❝✉♥❢❡rê♥❝✐❛ ■♥s❝r✐t❛ ❡♠ ✉♠ ❚r✐â♥❣✉❧♦
Pr♦♣r✐❡❞❛❞❡ ✷✳✸✳ ❆ s♦♠❛ ❞❡ t♦❞♦s ♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ✐❣✉❛❧ ❛
180♦✳
Pr♦♣♦s✐çã♦ ✷✳✶✳ ❚♦❞♦ tr✐â♥❣✉❧♦ ♣♦ss✉✐ ✉♠ ❝ír❝✉❧♦ ✐♥s❝r✐t♦✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦✳ ❚r❛❝❡ ❛s ❜✐ss❡tr✐③❡s ❞♦s â♥❣✉❧♦s Aˆ ❡ Bˆ✳ ❊s✲
t❛s s❡ ❡♥❝♦♥tr❛♠ ❡♠ ✉♠ ♣♦♥t♦ P✳ ❞❡st❡ ♣♦♥t♦✱ ❜❛✐①❡ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛♦s ❧❛❞♦s ❞♦
tr✐â♥❣✉❧♦✳ ❙❡❥❛♠ E✱ F ❡ G ♦s ♣és ❞❡st❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ♥♦s ❧❛❞♦s AB✱ BC ❡ CA✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❱❛♠♦s ♣r♦✈❛r q✉❡
P E=P F =P G.
❆ss✐♠ ♦ ♣♦♥t♦ P é ♦ ❝❡♥tr♦ ❞❡ ✉♠ ❝ír❝✉❧♦ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦sE✱F ❡G✱ ❛❧é♠ ❞✐st♦✱
❝♦♠♦ ♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ABC sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛♦s r❛✐♦sP E✱ P F ❡P G❡❧❡s sã♦
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✾
❋✐❣✉r❛ ✷✶✿ ❈ír❝✉❧♦ ■♥s❝r✐t♦ ❡♠ ✉♠ ❚r✐â♥❣✉❧♦
P❛r❛ ♣r♦✈❛r q✉❡
P E =P F =P G
✈❛♠♦s ❝♦♠♣❛r❛r ♦s tr✐â♥❣✉❧♦s P GA ❡ P EA✱ ❡ ♦s tr✐â♥❣✉❧♦s P EB ❡ P EF✱ t♦❞♦s ❡❧❡s
sã♦ tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s✳ ◆♦s ❞♦✐s ♣r✐♠❡✐r♦s t❡♠♦s
PAGˆ =PAEˆ ✭P❆ é ❜✐ss❡tr✐③✮ ❡ P❊✱ P❋ ❡ P● sã♦ r❛✐♦s✳
◆♦s ❞♦✐s ú❧t✐♠♦s t❡♠♦s
PBEˆ =PBFˆ ✭P❇ é ❜✐ss❡tr✐③✮ ❡ P❊✱ P❋ ❡ P● sã♦ r❛✐♦s✳
P♦rt❛♥t♦ ♦s ❞♦✐s ♣❛r❡s ❞❡ tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✱ ♣❡❧♦ ✸♦ ❝❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✳
❉❛ ❝♦♥❣r✉ê♥❝✐❛ ❞♦s ❞♦✐s ♣r✐♠❡✐r♦s ❝♦♥❝❧✉í♠♦s q✉❡P G=P E✳ ❉❛ ❝♦♥❣r✉ê♥❝✐❛ ❞♦s ❞♦✐s
ú❧t✐♠♦s ♦❜t❡♠♦sP E =P F✳ ❊ ❝♦♥❝❧✉í♠♦s ❛ ❞❡♠♦str❛çã♦✳
✷✳✷ ❊st✉❞♦ ❞❛s ❊❧✐♣s❡s
✷✳✷✳✶ ❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✵
❋✐❣✉r❛ ✷✷✿ ▼♦❞❡❧♦ ❞❡ Pt♦❧♦♠❡✉
P❧❛tã♦ ✭✹✷✼✲✸✹✼ ❛✳❈✮ é ✐♠♣♦rt❛♥t❡ ♥❛ ❤✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦r s❡✉ ♣❛♣❡❧ ❝♦♠♦ ✐♥s♣✐r❛❞♦r ❡ ❣✉✐❛ ❞❡ ♦✉tr♦s ✜❧ós♦❢♦s ❡ ♠❛t❡♠át✐❝♦s✳ ❚❛❧✈❡③ ❛ P❧❛tã♦ s❡ ❞❡✈❛ ❛ ❞✐st✐♥çã♦ ❝❧❛r❛ q✉❡ s❡ ❢❛③✐❛ ♥❛ ●ré❝✐❛ ❛♥t✐❣❛ ❡♥tr❡ ❆r✐t♠ét✐❝❛ ✭♥♦ s❡♥t✐❞♦ ❞❡ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ♦ ❡st✉❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✮ ❡ ▲♦❣íst✐❝❛ ✭❛ té❝♥✐❝❛ ❞❡ ❝♦♠♣✉t❛çã♦✱ ❤♦❥❡ ❝❤❛♠❛❞❛ ❞❡ ❆r✐t♠ét✐❝❛✮✳ ❍á r❡❣✐str♦s ❞❡ q✉❡ ❛ ❆❝❛❞❡♠✐❛ P❧❛tô♥✐❝❛ ❞❡ ❆t❡♥❛s t♦r♥♦✉✲s❡ ♦ ❝❡♥tr♦ ♠❛t❡♠át✐❝♦ ❞♦ ♠✉♥❞♦ ❡ ❞❡ss❛ ❡s❝♦❧❛ ♣r♦✈✐❡r❛♠ ♦s ♣r✐♥❝✐♣❛✐s ♠❡str❡s ❡ ♣❡sq✉✐s❛❞♦r❡s ♥♦s ♠❡❛❞♦s ❞♦ sé❝✉❧♦ ■❱ ❛✳❈✳ ❉❡ss❡s✱ ♦ ♠❛✐♦r ❢♦✐ ❊✉❞♦①♦ ❞❡ ❈♥✐❞♦ ✭✹✵✽✲✸✺✺ ❛✳❈✳ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✮✱ ❞✐s❝í♣✉❧♦ ❞❡ P❧❛tã♦ q✉❡ s❡ t♦r♥♦✉ ♦ ♠❛✐s ❝é❧❡❜r❡ ♠❛t❡♠át✐❝♦ ❡ ❛strô♥♦♠♦ ❞❡ s❡✉ t❡♠♣♦✳
❋✐❣✉r❛ ✷✸✿ ▼♦❞❡❧♦ ❞❡ ❈♦♣ér♥✐♦
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✶
❞❡ss❡s r❛✐♦s ❣✐r❛✈❛ ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ t♦r♥♦ ❞❡ ✉♠ ❡✐①♦ ✜①♦ ❡♠ r❡❧❛çã♦ à s✉♣❡r❢í❝✐❡ ❞❛ ❡s❢❡r❛✳ ❊ss❡ ♠♦❞❡❧♦ t♦r♥❛✈❛ ❡①tr❡♠❛♠❡♥t❡ ❝♦♠♣❧✐❝❛❞❛ ❛ ❞❡s❝r✐çã♦ ❞♦s ♠♦✈✐♠❡♥t♦s ♣❧❛♥❡tár✐♦s✳ ◆♦ sé❝✉❧♦ ❳❱■✱ ◆✐❝♦❧❛✉ ❈♦♣ér♥✐❝♦ r❡t♦♠♦✉ ❛ t❡♦r✐❛ ❞❡ ❆r✐st❛r❝♦ ❞❡ ❙❛♠♦s ✭sé❝✉❧♦ ■■■ ❛✳❈✳✮ ❡ ♣r♦♣ôs ♦ s✐st❡♠❛ ❤❡❧✐♦❝ê♥tr✐❝♦✳ ❙✉❣❡r✐✉ q✉❡ ♦s ♣❧❛♥❡t❛s✱ ✐♥❝❧✉s✐✈❡ ❛ ❚❡rr❛✱ ❞❡s❝r❡✈❡♠ ór❜✐t❛s ❝✐r❝✉❧❛r❡s ❡♠ t♦r♥♦ ❞♦ ❙♦❧✳ ❯♠ sé❝✉❧♦ ❞❡♣♦✐s✱ ❛ t❡♦r✐❛ ❤❡❧✐♦✲ ❝ê♥tr✐❝❛ ❝♦♠❡ç❛ ❛ s❡r ❛♣❡r❢❡✐ç♦❛❞❛ ❡ ♠❛✐s ❜❡♠ ❝♦♠♣r❡❡♥❞✐❞❛✱ ❛ ♣❛rt✐r ❞♦s ❡st✉❞♦s ❞❡ ●❛❧✐❧❡✉ ❡ ❞♦ ❛strô♥♦♠♦ ❞✐♥❛♠❛rq✉ês ❚②❝❤♦ ❇r❛❤❡✱ ♥❛s❝✐❞♦ ❡♠ ✶✺✹✻✳ ❊♠ ✶✻✵✾✱ ●❛❧✐❧❡✉ ♣❛ss♦✉ ❛ ♦❜s❡r✈❛r ♦ ♣❧❛♥❡t❛ ❏ú♣✐t❡r ❛tr❛✈és ❞❡ ✉♠ t❡❧❡s❝ó♣✐♦✳ ❉❡s❝♦❜r✐✉ ♦❜❥❡t♦s q✉❡ s❡ ♠♦✈✐❛♠ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ♣❧❛♥❡t❛ ❡ q✉❡ ❝❛❞❛ ✉♠ ❞❡❧❡s ❛♣r❡s❡♥t❛✈❛ ór❜✐t❛ ❡ ♣❡rí♦❞♦ ❞✐❢❡r❡♥t❡s ❡♠ s✉❛ tr❛❥❡tór✐❛ ❛♦ r❡❞♦r ❞❡ ❏ú♣✐t❡r✳ ❉✉r❛♥t❡ ✷✵ ❛♥♦s✱ ❚②❝❤♦ ❝❛t❛❧♦❣♦✉ ♠✐✲ ❧❤❛r❡s ❞❡ ❡str❡❧❛s ❞❡ ♠❛♥❡✐r❛ ♣r❡❝✐s❛✳ ❊❧❛❜♦r♦✉ ♠❛♣❛s ❝❡❧❡st❡s q✉❡ t♦r♥❛r❛♠ ♣♦ssí✈❡❧ ✉♠❛ ❞❡s❝r✐çã♦ ♠❛✐s r❡❛❧✐st❛ ❞❛s ór❜✐t❛s ♣❧❛♥❡tár✐❛s✳ ❊ss❛ ❞❡s❝r✐çã♦ só ✜❝♦✉ ❝♦♥❤❡❝✐❞❛ ❛♣ós s✉❛ ♠♦rt❡✱ ♣♦r ✉♠ ❞❡ s❡✉s ❛❧✉♥♦s✱ ♦ ❛❧❡♠ã♦ ❏♦❤❛♥♥❡s ❑❡♣❧❡r ✭✶✺✼✶✲✶✻✸✵✮ q✉❡✱ ♣♦r ♠❡✐♦ ❞❡ ❝á❧❝✉❧♦s ♠❛t❡♠át✐❝♦s✱ ❝❤❡❣♦✉ às ór❜✐t❛s ❡❧í♣t✐❝❛s ♣❛r❛ ♦s ♣❧❛♥❡t❛s ❡ às ❧❡✐s q✉❡ ✜③❡r❛♠ ❛✈❛♥ç❛r ❛ ❆str♦♥♦♠✐❛✳ ❆ ♣r✐♠❡✐r❛ ❧❡✐ ❞❡ ❑❡♣❧❡r✱ ❝❤❛♠❛❞❛ ▲❡✐ ❞❛s Ór❜✐t❛s✱ ❞✐③ ❡①❛t❛♠❡♥t❡ q✉❡ ❝❛❞❛ ♣❧❛♥❡t❛ ❞❡s❝r❡✈❡ ✉♠❛ ór❜✐t❛ ❡❧í♣t✐❝❛ ❡♠ t♦r♥♦ ❞♦ ❙♦❧✱ q✉❡ ♦❝✉♣❛ ✉♠ ❞♦s ❢♦❝♦s ❞❛ ❡❧✐♣s❡✳
❋✐❣✉r❛ ✷✹✿ ▼♦❞❡❧♦ ❞❡ ❑❡♣❧❡r
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✷
❋✐❣✉r❛ ✷✺✿ ❈ô♥✐❝❛s ❆♣❛rt✐r ❞♦ ❈♦♥❡
❙❡ ❜❡♠ q✉❡ ♥❡ss❛ é♣♦❝❛✱ ♥ã♦ s❡ ❞✐s♣✉♥❤❛ ❞❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛❀ ❆♣♦❧ô♥✐♦ ❢❛③ ✉♠ tr❛t❛♠❡♥t♦ ❞❛s ❝ô♥✐❝❛s ❛♣❡♥❛s ❣❡♦♠❡tr✐❝❛♠❡♥t❡✱ ♠❛s q✉❡ s❡ ❛♣r♦①✐♠❛ ♠✉✐t♦ ❞♦ ♠♦❞❡❧♦ ❛♥❛❧ít✐❝♦ ❝♦♥❤❡❝✐❞♦ ❤♦❥❡✳ ❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣♦r ❡❧❡ ❢♦r❛♠ ♦s ú♥✐❝♦s q✉❡ ❡①✐st✐r❛♠ ❛té q✉❡ ❋❡r♠❛t ✭✶✻✵✶✲✶✻✻✺✮ ❡ ❉❡s❝❛rt❡s ✭✶✺✾✻✲✶✻✺✵✮✱ ❡♠ ✉♠❛ ❞❛s ♣r✐♠❡✐r❛s ❛♣❧✐❝❛çõ❡s ❞❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✱ r❡t♦♠❛r❛♠ ♦ ♣r♦❜❧❡♠❛ ❡st✉❞❛♥❞♦✲♦ q✉❛s❡ q✉❡ ❝♦♠♣❧❡t❛♠❡♥t❡✱ ♠❡s♠♦ ♥ã♦ ♠❛♥❡❥❛♥❞♦ ❝♦♦r❞❡♥❛❞❛s ♥❡❣❛t✐✈❛s✱ ❝♦♠ ❛s r❡str✐çõ❡s q✉❡ ✐st♦ ✐♠♣õ❡✳
✷✳✷✳✷ ❉❡✜♥✐çõ❡s ❡ Pr♦♣r✐❡❞❛❞❡s
❉❡✜♥✐çã♦ ✷✳✽✳ ❈❤❛♠❛♠♦s ❞❡ ❡❧✐♣s❡ ❛♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦sP t❛✐s q✉❡ ❛ s♦♠❛
❞❡ s✉❛s ❞✐stâ♥❝✐❛s ❛ ❞♦✐s ♣♦♥t♦s ✜①♦s F1 ❡ F2 é ❝♦♥st❛♥t❡✱ ✐❣✉❛❧ ❛ ✷❛✱ ❡ ♠❛✐♦r ❞♦ q✉❡
❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s ✜①♦s✳
❱❡❥❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦
❖❜s❡r✈❛♥❞♦ ❛ ❡❧✐♣s❡ ❞❛ ✜❣✉r❛ ❛❝✐♠❛ ✈❛♠♦s ✐❞❡♥t✐✜❝❛r ❛❧❣✉♥s ❡❧❡♠❡♥t♦s ✐♠♣♦rt❛♥t❡s✳
• ❖s ♣♦♥t♦s ✜①♦s F1 ❡ F2 sã♦ ♦s ❢♦❝♦s ❞❛ ❡❧✐♣s❡ ❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡❧❡s é 2c✱
❞❡♥♦♠✐♥❛❞❛ ❞✐stâ♥❝✐❛ ❢♦❝❛❧✳
• A1A2 é ♦ ❡✐①♦ ♠❛✐♦r ❞❛ ❡❧✐♣s❡ ❡ s✉❛ ♠❡❞✐❞❛ é ✐❣✉❛❧ ❛ 2a✳
• B1B2 é ♦ ❡✐①♦ ♠❡♥♦r ❞❛ ❡❧✐♣s❡ ❡ s✉❛ ♠❡❞✐❞❛ é ✐❣✉❛❧ ❛2b✳
• ❖ é ♦ ❝❡♥tr♦ ❞❛ ❡❧✐♣s❡ ✭✐♥t❡rs❡çã♦ ❞♦ ❡✐①♦ ♠❛✐♦r A1A2 ❝♦♠ ♦ ❡✐①♦ ♠❡♥♦r B1B2✮✳
• ❖s ♣♦♥t♦s A1A2 sã♦ ❞❡♥♦♠✐♥❛❞♦s ✈ért✐❝❡s ❞❛ ❡❧✐♣s❡✳
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✸
❋✐❣✉r❛ ✷✻✿ ❊❧❡♠❡♥t♦s ❞❛ ❊❧✐♣s❡
❆♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦B1OF2✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
❛ r❡❧❛çã♦ ♥♦tá✈❡❧ ❞❛ ❡❧✐♣s❡✿
a2
=b2
+c2
.
❋✐❣✉r❛ ✷✼✿ ❊❝❡♥tr✐❝✐❞❛❞❡ ❞❛ ❊❧✐♣s❡
❆ ✜❣✉r❛ ❛❝✐♠❛ ♠♦str❛ ♦ q✉❡ ❛ ❡①❝❡♥tr✐❝✐❞❛❞❡ ❛❢❡t❛ ♥❛ ❡❧✐♣s❡✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡✿
■✮ ◗✉❛♥❞♦ ❛ ❡①❝❡♥tr✐❝✐❞❛❞❡ s❡ ❛♣r♦①✐♠❛ ❞❡ 1✱ ❛ ❡❧✐♣s❡ é ♠❛✐s ❛❝❤❛t❛❞❛✳
■■✮ ◗✉❛♥❞♦ ❛ ❡①❝❡♥tr✐❝✐❞❛❞❡ s❡ ❛♣r♦①✐♠❛ ❞❡ ③❡r♦✱ ❛ ❡❧✐♣s❡ é ♠❛✐s ❛rr❡❞♦♥❞❛❞❛✳
❖ q✉❡ t♦r♥❛ ❛ ❡①❝❡♥tr✐❝✐❞❛❞❡e ✉♠❛ ♠❡❞✐❞❛ ✐♠♣♦rt❛♥t❡ ♥♦ ❡st✉❞♦ ❞❛s ❡❧✐♣s❡s✳ ❱❡❥❛
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✹
❋✐❣✉r❛ ✷✽✿ ❊❝❡♥tr✐❝✐❞❛❞❡ ❞❛ ❊❧✐♣s❡
❉❡✜♥✐çã♦ ✷✳✾✳ ❆ ár❡❛ ❞❡ ✉♠❛ ❡❧✐♣s❡ ε ❞❡ ❡✐①♦ ♠❛✐♦r ♠❡❞✐♥❞♦ 2a✱ ❡ ❞❡ ❡✐①♦ ♠❡♥♦r
♠❡❞✐♥❞♦2b é ❞❛❞♦ ♣♦r
A[ε] =πab
❆ ❢ór♠✉❧❛ ❞❛ ár❡❛ ❞❛ ❡❧✐♣s❡ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ ❞❡✜♥✐çã♦ ♥❡ss❡ tr❛❜❛❧❤♦✱ ♠❛✐s s✉❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥❛ r❡❢❡rê♥❝✐❛ ❜✐❜❧✐♦❣rá✜❝❛ ❬✸❪✳
❉❡✜♥✐çã♦ ✷✳✶✵✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❡❧✐♣s❡ ❡stá ✐♥❝r✐t❛ ❡♠ ✉♠ tr✐â♥❣✉❧♦ s❡ ❛ ❡❧✐♣s❡ é t❛♥❣❡♥t❡ ❛♦s três ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦✳
❆ ❞❡✜♥✐çã♦ ❛❝✐♠❛ é ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♣♦❞❡r♠♦s ❢❛③❡r ♦s ❡st✉❞♦ ❞❛ ❡❧✐♣s❡ ❞❡ ❙t❡✐♥❡r✱ ❛❣♦r❛ ✉♠❛ q✉❡stã♦ ♦♣♦rt✉♥❛ ♥❡ss❡ ♠♦♠❡♥t♦ é s♦❜r❡ ❛ ❡①✐stê♥❝✐❛ ❞❛ ❡❧✐♣s❡ ✐♥s❝r✐t❛ ♥♦ tr✐â♥❣✉❧♦✱ ♦ q✉❡ ❛s ❞✉❛s ♣r♦♣♦s✐çõ❡s s❡❣✉✐♥t❡s r❡s♣♦♥❞❡♠✳
Pr♦♣♦s✐çã♦ ✷✳✷✳ ❉❛❞❛ ✉♠❛ ❡❧✐♣s❡ε❞❡ ❢♦❝♦sF ❡F′✱ ❡ ✉♠ ♣♦♥t♦P ∈ε✱ ❛ r❡t❛ t❛♥❣❡♥t❡
❛ ε ❡♠ P é ❛ ♠❡❞✐❛tr✐③ ❞♦ s❡❣♠❡♥t♦ F Q ♦♥❞❡ Q é ✉♠ ♣♦♥t♦ ♣❡rt❡♥❝❡♥t❡ à s❡♠✐rr❡t❛
F′P t❛❧ q✉❡ F′Q= 2a✳
❉❡♠♦♥str❛çã♦✳ ❙❛❜❡♠♦s✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❊❧✐♣s❡✱ q✉❡ ❞❛❞♦s ❞♦✐s ♣♦♥t♦sF ❡F′ ❡ s❡❥❛
a✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ ❞❡ ♠♦❞♦ q✉❡ ✷❛ ❃F F′✳ ❆ ❡❧✐♣s❡ε❞❡ ❢♦❝♦sF ❡F′❡ s❡♠✐❡✐①♦
♠❛✐♦ra é ♦ ❝♦♥❥✉♥t♦ ε ❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦s P ❞♦ ♣❧❛♥♦ t❛✐s q✉❡
P F +P F′ = 2a.
❆❣♦r❛✱ ❉❛❞♦ P ∈ ε✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ♣♦♥t♦ Q ♣❡rt❡♥❝❡♥t❡ à s❡♠✐rr❡t❛ F′P t❛❧ q✉❡
F′Q= 2a✱ ✐st♦ ✐♠♣❧✐❝❛ q✉❡
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✺
❋✐❣✉r❛ ✷✾✿ ❘❡t❛ ❚❛♥❣❡♥t❡ à ❊❧✐♣s❡
❈♦♥s✐❞❡r❡♠♦s ❛ ♠❡❞✐❛tr✐③ t ❞♦ s❡❣♠❡♥t♦ F Q✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ❛❝✐♠❛✳
❖ q✉❡ q✉❡r❡♠♦s ♣r♦✈❛r é q✉❡ ❛ ♠❡❞✐❛tr✐③ t ❞♦ s❡❣♠❡♥t♦ F Q é ❛ r❡t❛ t❛♥❣❡♥t❡ à
❡❧✐♣s❡ε❡♠P✳ ❖❜s❡r✈❛♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡P é ♦ ♣♦♥t♦ ❞❛ ❡❧✐♣s❡ q✉❡ ❡stá ❡♥tr❡ ♦s ♣♦♥t♦s
F′ ❡ Q ❡ ❛s ❞✉❛s ❡q✉❛çõ❡s ❛❝✐♠❛✱ s❡❣✉❡ q✉❡
P Q = F Q−P F′
= 2a−P F′ ✭✶✮
= P F
❧♦❣♦ ♦ tr✐â♥❣✉❧♦ △F P Q é ✐sós❝❡❧❡s ❡ ❛ ♠❡❞✐❛tr✐③ t ❞♦ s❡❣♠❡♥t♦ F Q ♣❛ss❛ ♣❡❧♦
♣♦♥t♦P✳ ❆❣♦r❛ t❡♠♦s q✉❡ ♠♦str❛r q✉❡ t❡ t❛♥❣❡♥t❡ à ❡❧✐♣s❡ ε✱ ♦✉ s❡❥❛ q✉❡P é ♦ ú♥✐❝♦
♣♦♥t♦ ❞❛ ❡❧✐♣s❡ ♣♦r ♦♥❞❡ ❛ r❡t❛ t ♣❛ss❛✳ P❛r❛ ✐st♦ ❝♦♥s✐❞❡r❡♠♦s ✉♠ ❞❛❞♦ ♣♦♥t♦ A
♣❡rt❡♥❝❡♥t❡ ❛ r❡t❛ t ✱ ❆ ❞✐st✐♥t♦ ❞❡ P ✱ t❡♠♦s q✉❡
AF +AF′ = AQ+AF′ ♣♦✐s ❝♦♠♦ t é ♠❡✐❛tr✐③ ❞❡ F Q✱ AF =AQ
> F′Q ✭✷✮
= 2a.
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✻
q✉❡t∩ε=P✱ ♦✉ s❡❥❛ P é ♦ ú♥✐❝♦ ♣♦♥t♦ ❞❛ ❡❧✐♣s❡ q✉❡ ❡stá ♥❛ ♠❡❞✐❛tr✐③t ❞♦ s❡❣♠❡♥t♦
AQ✳ P♦rt❛♥t♦ t é t❛♥❣❡♥t❡ à ❡❧✐♣s❡ ε✳
❖❜s❡r✈❡ q✉❡ ♦ ♣♦♥t♦ Q ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ♦ ♣♦♥t♦ ❡♠ q✉❡ ❛ s❡♠✐rr❡t❛ F′P
✐♥t❡r❝❡♣t❛ ✉♠❛ ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❞✐r❡tr✐③❡s ❞❛ ❡❧✐♣s❡ε✱ ❛q✉❡❧❛ ❞❡ ❝❡♥tr♦ F′ ❡ r❛✐♦ 2a✳
❆❧é♠ ❞✐ss♦✱ ❛ ❝♦♥str✉ç❛♦ ❡❢❡t✉❛❞❛ r❡✈❡❧❛ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛ ✈❛❧✐❞❛❞❡ ❞❡ ✉♠❛ ❝♦♥❤❡❝✐❞❛
♣r♦♣r✐❡❞❛❞❡ r❡✢❡t♦r❛ ❞❛ ❡❧✐♣s❡✿ ♦s â♥❣✉❧♦s q✉❡ ♦ r❛✐♦s ❢♦❝❛✐sP F ❡P F′ ❢♦r♠❛♠ ❝♦♠ ❛
r❡t❛ t❛♥❣❡♥t❡ ❛ε ❡♠ P s❛♦ ❝♦♥❣r✉❡♥t❡s✳
Pr♦♣♦s✐çã♦ ✷✳✸✳ P❛r❛ ❝❛❞❛ ♣♦♥t♦ ❋ ♣❡rt❡♥❝❡♥t❡ ❛♦ ✐♥t❡r✐♦r ❞♦ tr✐â♥❣✉❧♦✱ ❡①✐st❡ ✉♠❛ ❡❧✐♣s❡ ✐♥s❝r✐t❛ ♥❡ss❡ tr✐â♥❣✉❧♦✱ t❡♥❞♦ ❋ ❝♦♠♦ ✉♠ ❞❡ s❡✉s ❢♦❝♦s✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦ ❛r❜✐trár✐♦ ❡ F ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ♥♦ s❡✉ ✐♥✲
t❡r✐♦r✳ ❱❛♠♦s ❞❡t❡r♠✐♥❛r ✉♠❛ ❡❧✐♣s❡ ε t❡♥❞♦ F ❝♦♠♦ ✉♠ ❞❡ s❡✉s ❢♦❝♦s ❞❡ ♠♦❞♦ q✉❡
❛s r❡t❛sAB✱ BC ❡ CA s❡❥❛♠ t❛♥❣❡♥t❡s ❛ε✳
▼♦t✐✈❛❞♦s ♣❡❧♦ ❡①♣♦st♦ ♥❛ ♣r♦♣♦s✐çã♦ ✷✳✷✱ ❝♦♥s✐❞❡r❡ ✉♠ tr✐â♥❣✉❧♦ ABC✳ ❙❡❥❛ F
✉♠ ♣♦♥t♦ ♥♦ ✐♥t❡r✐♦r ❞♦ tr✐â♥❣✉❧♦ ❡ s❡❥❛♠ Q✱ R ❡ S ♣♦♥t♦s s✐♠étr✐❝♦s ❛♦ ♣♦♥t♦ F
❞❛❞♦ ❡♠ r❡❧❛çã♦ às r❡t❛sAB✱BC ❡CAr❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❝❡♥tr♦F′ ❞❛
❝✐r❝✉♥❢❡rê♥❝✐❛ ❝✐r❝✉♥s❝r✐t❛ ♥♦ tr✐â♥❣✉❧♦QRS ❡ s❡❥❛♠P1✱P2❡P3❛s ✐♥t❡r❝❡❝sõ❡s ❡♥tr❡
♦s r❛✐♦s F′Q✱ F′R ❡ F′S ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ❝♦♥té♠ ♦s ♣♦♥t♦s Q✱ R ❡ S ❡ ♦s ❧❛❞♦s
AB✱ BC ❡CA ❞♦ tr✐â♥❣✉❧♦ ABC✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✳
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✼
❆ ❡❧✐♣s❡ ε❞❡ ❢♦❝♦s F ❡ F′ ❡ q✉❡ ♣❛ss❛ ♣♦r P1 é t❛❧ q✉❡ ❛s r❡t❛s AB✱ BC ❡CA sã♦
t❛♥❣❡♥t❡s ❛ ❡❧✐♣s❡ ε ❡♠ P1✱ P2 ❡ P3✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
◆♦t❡ q✉❡ ♦ ♣♦♥t♦ F′✱ ❛ss✐♠ ❝♦♠♦ F✱ t❛♠❜é♠ ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✐♦r ❞♦ tr✐â♥❣✉❧♦
ABC✳ ◆❛ r❡❛❧✐❞❛❞❡✱ F ❡F′ sã♦ ❝♦♥❥✉❣❛❞♦s ✐s♦❣♦♥❛✐s ❝♦♠ r❡❧❛çã♦ ❛♦ tr✐â♥❣✉❧♦ABC✳
■ss♦ s✐❣♥✐✜❝❛ q✉❡ ❛s s❡♠✐rr❡t❛sAF ❡ AF′ sã♦ s✐♠étr✐❝❛s ❡♠ r❡❧❛çã♦ à r❡t❛ q✉❡ ❝♦♥té♠
❛ ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ ❞❡ ✈ért✐❝❡ A ❞♦ tr✐â♥❣✉❧♦ ABC ❡ ♣r♦♣r✐❡❞❛❞❡s ❛♥á❧♦❣❛s ✈❛❧❡♠
♣❛r❛ ♦s ♣❛r❡s ❞❡ s❡♠✐rr❡t❛sBF✱BF′ ❡ CF✱CF′✳ ❱❡❥❛♠♦s ✉♠❛ ♣r♦✈❛✳
❙❡❥❛
x=m(BAFˆ )✶ =m(BAQˆ ) ❡ y=m(CAFˆ ) =m(CASˆ ),
t❡♠♦s q✉❡
m(QASˆ ) = 2(x+y).
P♦r ♦✉tr♦ ❧❛❞♦✱AQS é ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s✱ ♣♦✐s
AQ=AF =AS
❡✱ ❝♦♠♦ F′ ♣❡rt❡♥❝❡ ❛ ♠❡❞✐❛tr✐③ ❞♦ s❡❣♠❡♥t♦ QS✱ ♣♦✐s F′ é ♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛
q✉❡ ♣❛ss❛ ♣♦rQ ❡S✱ s❡❣✉❡ q✉❡
m(SAFˆ ′) =x+y.
▼❛sm(SACˆ ) = y✱ ❞♦♥❞❡
m(CAFˆ ′) = x=m(BAFˆ ).
▲♦❣♦✱AF ❡ AF′ sã♦ s❡♠✐rr❡t❛s s✐♠étr✐❝❛s ❡♠ r❡❧❛çã♦ à r❡t❛ q✉❡ ❝♦♥té♠ ❛ ❜✐ss❡tr✐③ ❞♦
â♥❣✉❧♦ ❞♦ ✈ért✐❝❡ ❆ ❞♦ tr✐â♥❣✉❧♦ABC✳ P♦rt❛♥t♦ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ♦ s❡❣♠❡♥t♦
AB é t❛♥❣❡♥t❡ à ❡❧✐♣s❡ ε ❡♠ P1✳ ❆♥❛❧♦❣❛♠❡♥t❡ ♣r♦✈❛♠♦s ❛s t❛♥❣ê♥❝✐❛ ❞❛ ❡❧✐♣s❡ ❡♠
AC ❡ BC✳
✷✳✸ Pr♦❥❡çã♦ ❖rt♦❣♦♥❛❧
✷✳✸✳✶ Pr♦❥❡çã♦ ❖rt♦❣♦♥❛❧ ❞❡ ✉♠ P♦♥t♦
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✽
❉❡✜♥✐çã♦ ✷✳✶✶✳ ❈❤❛♠❛✲s❡ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡ ✉♠ ♣♦♥t♦P s♦❜r❡ ✉♠ ♣❧❛♥♦α ♦ ♣é
❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r P′ ❛♦ ♣❧❛♥♦ ❝♦♥❞✉③✐❞❛ ♣❡❧♦ ♣♦♥t♦ P✳ ❖ ♣❧❛♥♦ α é ❝❤❛♠❛❞♦ ❞❡ ♣❧❛♥♦
❞❡ ♣r♦❥❡çã♦ ❡ ❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ α ♣❛ss❛♥❞♦ ♣♦r P ❡ P′ é ❝❤❛♠❛❞❛ ❞❡ ♣r♦❥❡t❛♥t❡
❞♦ ♣♦♥t♦ P✳
◆❛ ✜❣✉r❛ ❛❜❛✐①♦ ❡①❡♠♣❧✐✜❝❛♠♦s ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✳
❋✐❣✉r❛ ✸✶✿ Pr♦❥❡çã♦ ❞❡ ✉♠ P♦♥t♦ ❡♠ ✉♠ P❧❛♥♦
P❛r❛ r❡♣r❡s❡♥t❛r ❛ ♣r♦❥❡çã♦ P′ ❞♦ ♣♦♥t♦ P ♥♦ ♣❧❛♥♦ α ❞❡ ♣r♦❥❡çã♦✱ ✉s❛r❡♠♦s ❛
♥♦t❛çã♦P′ =projαP✳
✷✳✸✳✷ Pr♦❥❡çã♦ ❖rt♦❣♦♥❛❧ ❞❡ ✉♠❛ ❋✐❣✉r❛
❉❡✜♥✐çã♦ ✷✳✶✷✳ ❈❤❛♠❛✲s❡ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡ ✉♠❛ ✜❣✉r❛ F s♦❜r❡ ✉♠ ♣❧❛♥♦ α ❛♦
❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♣r♦❥❡çõ❡s ♦rt♦❣♦♥❛✐s ❞♦s ♣♦♥t♦s q✉❡ ❝♦♠♣õ❡♠ ❛ ✜❣✉r❛ F✳
❈♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦
❋✐❣✉r❛ ✸✷✿ Pr♦❥❡çã♦ ❞❡ ✉♠❛ ❋✐❣✉r❛ ❡♠ ✉♠ P❧❛♥♦
P❛r❛ r❡♣r❡s❡♥t❛r ❛ ♣r♦❥❡çã♦ F′ ❞❛ ✜❣✉r❛ F ♥♦ ♣❧❛♥♦ α ❞❡ ♣r♦❥❡çã♦✱ ✉s❛r❡♠♦s ❛
✷ ❊❙❚❯❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✾
✷✳✸✳✸ Pr♦❥❡çã♦ ❖rt♦❣♦♥❛❧ ❞❡ ✉♠❛ ❘❡t❛
❆ ♣r♦❥❡çã♦ ❞❡ ✉♠❛ r❡t❛ ❡♠ ✉♠ ♣❧❛♥♦ ❞❡♣❡♥❞❡ ❞❛ ♣♦s✐çã♦ ❞❛ r❡t❛ ❡♠ r❡❧❛çã♦ ❛♦ ♣❧❛♥♦✳
❙❡ ✉♠❛ r❡t❛ r é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ α✱ ❝♦♠♦ ❛ ✐♥t❡r❝❡sã♦ ❡♥tr❡ ❛ r❡t❛ ❡ ♦ ♣❧❛♥♦ é
✉♠ ♣♦♥t♦ P✱ ❡♥tã♦ ♣❡❧❛s ❞❡✜♥✐çõ❡s ❛♥t❡r✐♦r❡s ❝❤❡❣❛♠♦s q✉❡ ❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❛
r❡t❛ r ♥♦ ♣❧❛♥♦ α é ❡①❛t❛♠❛♥t❡ ♦ ♣♦♥t♦ P✳ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✳
❋✐❣✉r❛ ✸✸✿ Pr♦❥❡çã♦ ❞❡ ✉♠❛ ❘❡t❛ P❡r♣❡♥❞✐❝✉❧❛r ❡♠ ✉♠ P❧❛♥♦
❆❣♦r❛ s❡ ❛ r❡t❛ r ♥ã♦ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ α ❛ Pr♦❥❡çã♦ é ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛
❞❡✜♥✐çã♦ ❛❜❛✐①♦
❉❡✜♥✐çã♦ ✷✳✶✸✳ ❈❤❛♠❛✲s❡ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡ ✉♠❛ r❡t❛ r ♥ã♦✲♣❡r♣❡♥❞✐❝✉❧❛r ❛♦
♣❧❛♥♦ α✱ ♦ tr❛ç♦ ❡♠ α ♣r♦✈♦❝❛❞♦ ♣♦r ✉♠ ♣❧❛♥♦ β ♣❡r♣❡♥❞✐❝✉❧❛r ❛ α q✉❡ ❝♦♥té♠ ❛ r❡t❛
r✳
❆ ✜❣✉r❛ ❛❜❛✐①♦ ✐❧✉str❛ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✳
❋✐❣✉r❛ ✸✹✿ Pr♦❥❡çã♦ ❞❡ ✉♠❛ ❘❡t❛ ❡♠ ✉♠ P❧❛♥♦
P❛r❛ r❡♣r❡s❡♥t❛r ❛ ♣r♦❥❡çã♦r′ ❞❛ r❡t❛r♥♦ ♣❧❛♥♦α❞❡ ♣r♦❥❡çã♦✱ ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦
