❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚
❆ ❘❡t❛ ❞❡ ❊✉❧❡r ❡ ❛
❈✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ◆♦✈❡ P♦♥t♦s✿
❯♠ ❖❧❤❛r ❆❧❣é❜r✐❝♦
†♣♦r
❆♥t♦♥✐♦ ▼❛r❝♦s ❞❛ ❙✐❧✈❛ ❙♦✉t♦
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦Pr♦❢✳ ❉r✳ P❡❞r♦ ❆♥t♦♥✐♦ ●ó♠❡③ ❱❡♥❡❣❛s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❆❣♦st♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
S728r Souto, Antonio Marcos da Silva.
A reta de Euler e a circunferência dos nove pontos: um olhar algébrico / Antonio Marcos da Silva Souto.-- João Pessoa, 2013.
72f.
Orientador: Pedro Antonio Gómez Venegas Dissertação (Mestrado) - UFPB/CCEN
1. Geometria. 2. Álgebra Linear. 3. Geogebra. 4. Pontos notáveis do triângulo. 5. Reta de Euler. 6. Circunferência dos nove pontos.
❆♦ ♠❡✉ ❙❡♥❤♦r ❏❡s✉s ❈r✐st♦✱ ♣❡❧❛ ❢♦rç❛ s✉♣r✐❞❛ ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✳
❆♦s ♠❡✉s ♣❛✐s✱ ▼❛rt✐♥❤♦ ❙♦✉t♦ ❞❡ ❆❧♠❡✐❞❛ ❡ ❱❡rô♥✐❝❛ ▼❛r✐❛ ❞❛ ❙✐❧✈❛ ❙♦✉t♦✱ ♣♦r ♠❡ ❡❞✉❝❛r❡♠ ❝♦♠ ❛♠♦r ❡ ❤✉♠✐❧❞❛❞❡✱ s✉❛s ✈✐❞❛s sã♦ ♠❡✉s ❡①❡♠♣❧♦s✳
➚ ♠✐♥❤❛ q✉❡r✐❞❛ ❡s♣♦s❛ ❘♦sâ♥❣❡❧❛ ❑át✐❛ ❡ ♠❡✉s ❛♠❛❞♦s ✜❧❤♦s ❆♥✐♥❤❛✱ ❆♥❞ré ❡ ❆❧í❝✐❛✱ ♣♦r s✉♣♦rt❛r❡♠ ❝♦♠ ♣❛❝✐ê♥❝✐❛ ❛ ♠✐♥❤❛ ❛✉sê♥❝✐❛ ❞✉r❛♥t❡ ❛s ✈✐❛❣❡♥s ❡ ♥♦s ♠♦♠❡♥t♦s ❞❡ r❡❝❧✉sã♦ ❢♦rç❛❞❛ ♣❡❧❛s r✐❣♦r♦s❛s ❛t✐✈✐❞❛❞❡s ❞♦ ❝✉rs♦✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❉r✳ P❡❞r♦ ❆♥t♦♥✐♦ ●ó♠❡③ ❱❡♥❡❣❛s✱ ♣♦r ♠❡ ❣✉✐❛r ♥❡ss❛ ❞✐❢í❝✐❧ ❡ ár❞✉❛ t❛r❡❢❛ ❡♠ ✉♠ ♣r❛③♦ ❞❡ t❡♠♣♦ tã♦ ❝✉rt♦ ❡ ❡♠ ✉♠ ♣❡rí♦❞♦ tã♦ ❞♦❧♦r♦s♦ ❞❛ s✉❛ ✈✐❞❛✳
❆♦s ♣r♦❢❡ss♦r❡s ❉r❛✳ ❏❛❝q✉❡❧✐♥❡ ❘♦❥❛s ❡ ❉r✳ ❘❛♠ó♥ ▼❡♥❞♦③❛✱ ♣♦r ❛❝❡✐t❛r❡♠ ♣❛rt✐❝✐♣❛r ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✳
❆♦s ♠❡✉s ❛♠✐❣♦s✴❝♦❧❡❣❛s ❞❡ t✉r♠❛ q✉❡ tr❛♥s❢♦r♠❛r❛♠ ♦s ♠♦♠❡♥t♦s ❞❡ t❡♥sã♦ ❡♠ s❛❜♦r♦s❛s ❡①♣❡r✐ê♥❝✐❛s ❞❡ ✉♥✐ã♦ ❡ ❛♠✐③❛❞❡✱ ❥á ❡st♦✉ ❝♦♠ s❛✉❞❛❞❡✳
❆♦s ❝♦♦r❞❡♥❛❞♦r❡s ❉r✱ ❏♦ã♦ ▼❛r❝♦s ❡ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡✱ ♣♦r ❛❝❡✐t❛r❡♠ ❛ss✉✲ ♠✐r ♦ ♣✐♦♥❡✐r♦ ❞❡s❛✜♦ ❞❡ ✐♥✐❝✐❛r ❡ ♣r♦ss❡❣✉✐r ♦ P❘❖❋▼❆❚ ♥❛ ❯❋P❇✳
❆♦s ♠❡✉s ❚✐♦s✴♣❛❞r✐♥❤♦s ❉r✳ ❆♥tô♥✐♦ ❙♦✉t♦ ❈♦✉t✐♥❤♦ ❡ ▲✐③❡t❡ ❞❡ ❋❛r✐❛s ❈♦✉✲ t✐♥❤♦✱ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ♥❡st❛ ❡ ❡♠ ♦✉tr❛s ❥♦r♥❛❞❛s✱ ♣♦r ♠❡ ❛❝♦❧❤❡r❡♠ ❡ ♠❡ ❤♦s♣❡❞❛r❡♠ ❞✉r❛♥t❡ ♦s ✐♥t❡r♠✐♥á✈❡✐s ✜♥s ❞❡ s❡♠❛♥❛ ❝♦♠ ♠✉✐t♦ ❝❛r✐♥❤♦❀
❆♦ ♠❡✉ ❛♠✐❣♦✱ ❝♦❧❡❣❛✱ ❡①✲♣r♦❢❡ss♦r ❡ ❝♦♠♣❛♥❤❡✐r♦ ❞❡ ✈✐❛❣❡♠✱ ❏♦sé ❊❞♠❛r ❇❡✲ ③❡rr❛ ❏ú♥✐♦r✱ q✉❡ ♠❡ ❡♥❝♦r❛❥♦✉ ❛ ♣r♦ss❡❣✉✐r ❝♦♥s✐❣♦ ❛té ♦ ✜♠✱ ♣♦r ♥ã♦ ♠❡ ❞❡✐①❛r só ♥♦s ✶✵✵✵ q✉✐❧ô♠❡tr♦s s❡♠❛♥❛✐s ❞❡ ❥♦r♥❛❞❛❀
➚ ❙❇▼✱ ❛♦ ■▼P❆ ❡ ❛♦ ▼❊❈ ♣♦r ❞✐s♣♦♥✐❜✐❧✐③❛r ❛♦ ❞♦❝❡♥t❡ ❞♦ ❊♥s✐♥♦ ❇ás✐❝❛ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ r❡❛❧✐③❛r ♦ q✉❛s❡ ✉tó♣✐❝♦ s♦♥❤♦ ❞❛ ❢♦r♠❛çã♦ ❡♠ ♥í✈❡❧ ❞❡ ♠❡str❛❞♦❀
❊st❡ tr❛❜❛❧❤♦ é ♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠❛ ♣❡sq✉✐s❛ s♦❜r❡ ❛ r❡t❛ ❞❡ ❊✉❧❡r ❡ ❛ ❝✐r❝✉♥❢❡rê♥✲ ❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s✳ ❋♦✐ ✉t✐❧✐③❛❞♦ ♦ s♦❢t✇❛r❡ ❣❡♦❣❡❜r❛ ♣❛r❛ ✐❧✉str❛r ❛s ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s ❡ ❛♣r❡s❡♥t❛r ❛❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s ♣rát✐❝❛s ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ♣♦♥t♦s ♥♦tá✲ ✈❡✐s ❞♦ tr✐â♥❣✉❧♦✱ ❞❛ r❡t❛ ❞❡ ❊✉❧❡r ❡ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s ❛♦s ❡st✉❞❛♥t❡s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❚♦❞❛✈✐❛✱ ♦ tr❛❜❛❧❤♦ s❡ ❜❛s❡♦✉ ♥❛s ❞❡♠♦♥str❛çõ❡s✱ ❝♦♠ ♦ ✉s♦ ❞❛ ➪❧❣❡❜r❛ ▼♦❞❡r♥❛ ❡ ❞❛ ➪❧❣❡❜r❛ ▲✐♥❡❛r✱ ❞❛ ❡①✐stê♥❝✐❛ ❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♦❜❥❡t♦ ❞❡st❛ ♣❡sq✉✐s❛✱ s♦❜r❡t✉❞♦ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦s ♣♦♥t♦s ♥♦ ♣❧❛♥♦✱ ❢✉♥❞❛♠❡♥✲ t❛❧ ♥❡st❛s ❞❡♠♦♥str❛çõ❡s✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ●❡♦♠❡tr✐❛✱ ➪❧❣❡❜r❛ ▲✐♥❡❛r✱ ●❡♦❣❡❜r❛✱ P♦♥t♦s ◆♦tá✈❡✐s ♥♦ ❚r✐✲ â♥❣✉❧♦✱ ❘❡t❛ ❞❡ ❊✉❧❡r✱ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ◆♦✈❡ P♦♥t♦s✳
❚❤✐s ✇♦r❦ ✐s t❤❡ r❡s✉❧t ♦❢ ❛ r❡s❡❛r❝❤ ♦♥ t❤❡ ❊✉❧❡r ❧✐♥❡ ❛♥❞ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ t❤❡ ♥✐♥❡ ♣♦✐♥ts✳ ❚❤❡ s♦❢t✇❛r❡ ❣❡♦❣❡❜r❛ ✇❛s ✉s❡❞ t♦ ✐❧❧✉str❛t❡ ❣❡♦♠❡tr✐❝ ❝♦♥str✉❝t✐♦♥s ❛♥❞ ♣r❡s❡♥t s♦♠❡ ♣r❛❝t✐❝❛❧ ❛❝t✐✈✐t✐❡s ❢♦r t❤❡ st✉❞② ♦❢ ♥♦t❛❜❧❡ ♣♦✐♥ts ♦❢ t❤❡ tr✐❛♥✲ ❣❧❡✱ t❤❡ ❊✉❧❡r ❧✐♥❡ ❛♥❞ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ t❤❡ ♥✐♥❡ ♣♦✐♥ts t♦ ❤✐❣❤ s❝❤♦♦❧ st✉❞❡♥ts✳ ❍♦✇❡✈❡r✱ t❤❡ ✇♦r❦ ✇❛s ❜❛s❡❞ ♦♥ t❤❡ ♣r♦♦❢✱ ✇✐t❤ t❤❡ ✉s❡ ♦❢ ▼♦❞❡r♥ ❆❧❣❡❜r❛ ❛♥❞ ▲✐♥❡❛r ❆❧❣❡❜r❛✱ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♦❜❥❡❝t ♦❢ t❤✐s r❡s❡❛r❝❤✱ ❡s♣❡❝✐❛❧❧② t❤❡ ✉♥✐✈❡rs❛❧ ♣r♦♣❡rt② ♦❢ ♣♦✐♥ts ✐♥ t❤❡ ♣❧❛♥❡✱ ❝r✐t✐❝❛❧ ✐♥ t❤❡s❡ ❞❡♠♦♥str❛t✐♦♥s✳
❑❡②✇♦r❞s✿ ●❡♦♠❡tr②✱ ▲✐♥❡❛r ❆❧❣❡❜r❛✱ ●❡♦❣❡❜r❛✱ ◆♦t❛❜❧❡ P♦✐♥ts ♦❢ ❛ ❚r✐❛♥❣❧❡✱ ❊✉❧❡r ▲✐♥❡✱ ◆✐♥❡✲♣♦✐♥ts ❈✐r❝❧❡✳
✶ ❆ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛ ♥❛ ❈♦♥str✉çã♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❛ ❈✐r❝✉♥❢❡✲
rê♥❝✐❛ ❞♦s ◆♦✈❡ P♦♥t♦s ✶
✶✳✶ ❘❡❝♦rt❡s ❍✐stór✐❝♦s ❞❛s ❉❡s❝♦❜❡rt❛s ●❡♦♠étr✐❝❛s ❞♦ ❙é❝✉❧♦ ❳■❳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ ❆s ✧♣ré✲❞❡s❝♦❜❡rt❛s✧❞❡ ❊✉❧❡r ♥♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳❱■■■ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✷ ❙é❝✉❧♦ ❳■❳✿ ❆ r❡✲❞❡s❝♦❜❡rt❛ ❞❛ ❣❡♦♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✸ ❯♠ ❜❡❧♦ ❞❡s❢❡❝❤❡ ♣❛r❛ ✉♠ sé❝✉❧♦ ❜r✐❧❤❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❈♦♥str✉çã♦ ❞❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ◆♦✈❡ P♦♥t♦s ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷✳✶ ❖s ♣♦♥t♦s ♥♦tá✈❡✐s ❞♦ tr✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳✷ ❆ ❝♦♥str✉çã♦ ❞❛ r❡t❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷✳✸ ❆ ❝♦♥str✉çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷ ❉❡✜♥✐çõ❡s ❆❧❣é❜r✐❝❛s ✶✼
✷✳✶ ❊s♣❛ç♦ ❆✜♠ ❡ ❊s♣❛ç♦ ❱❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✶✳✶ ❊s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✳✷ ❊s♣❛ç♦ ❛✜♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✶✳✸ ❖❜s❡r✈❛çõ❡s ✐♠♣♦rt❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷ ❘❡t❛s✱ P❧❛♥♦s ❡ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷✳✶ ❘❡t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷✳✷ P❧❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷✳✸ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ Pr♦♣r✐❡❞❛❞❡ ❯♥✐✈❡rs❛❧ ❞❡ P♦♥t♦s ♥✉♠ P❧❛♥♦P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✷✳✸✳✶ Pr♦♣r✐❡❞❛❞❡ ❯♥✐✈❡rs❛❧ ❞❡ ♣♦♥t♦s ❡♠ ✉♠ ♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸✳✷ Pr♦♣r✐❡❞❛❞❡ ❯♥✐✈❡rs❛❧ ❞❡ ♣♦♥t♦s ♥✉♠ ♣❧❛♥♦ P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✷✳✸✳✸ P♦♥t♦ ♠é❞✐♦ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹ ❖rt♦❣♦♥❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹✳✶ ❉❡✜♥✐çã♦ ❞❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹✳✷ ❉✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞♦ ❡s♣❛ç♦ ❛✜♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✹✳✸ Pr♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ✈❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✹✳✹ Pr♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❛✜♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✹✳✺ ❘❡❧❛çã♦ ❡♥tr❡ ❛s ♣r♦❥❡çõ❡s ✈❡t♦r✐❛❧ ❡ ❛✜♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✹✳✻ ❆❧❣✉♠❛s ✐♠♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✸ ❖ ❈✐r❝✉♥❝❡♥tr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✹ ❖ ❖rt♦❝❡♥tr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✺ ❆ ❘❡t❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✻ ❆ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ◆♦✈❡ P♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✻✳✶ ❖s P♦♥t♦s ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✻✳✷ ❈✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✻✳✸ ❆ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✻✳✹ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✶
❈♦♠♦ ♦ ♣ró♣r✐♦ ♥♦♠❡ s✉❣❡r❡✱ ❛ ●❡♦♠❡tr✐❛✱ ❞♦ ❣r❡❣♦ ●❡♦✭t❡rr❛✮✱ ♠❡tr✐❛✭♠❡❞✐❞❛✮✱ ❛♣r❡s❡♥t❛✲s❡ ❝♦♠♦ r❛♠♦ ❞❛ ▼❛t❡♠át✐❝❛ út✐❧ ♣❛r❛ r❡s♦❧✈❡r q✉❡stõ❡s ❞♦ ❝♦t✐❞✐❛♥♦ ❝♦♠♦ ♠❡❞✐r ♣♦rçõ❡s ❞❡ t❡rr❛s ❡♥tr❡ ✐♥✜♥✐t❛s ♦✉tr❛s ❛♣❧✐❝❛çõ❡s✳ ❯♠❛ ✐♥t❡r✈❡♥çã♦ ♣❡❞❛❣ó✲ ❣✐❝❛ q✉❡ ❛♣r❡s❡♥t❛ss❡ ❛♦ ❡st✉❞❛♥t❡ ❞❡ ❢♦r♠❛ ♣rát✐❝❛ ❛ ●❡♦♠❡tr✐❛ q✉❡ ❢❛s❝✐♥♦✉ ❡ ❛té ❞✐✈❡rt✐✉ ♥♦ss♦s ♣r❡❝✉rs♦r❡s ❞❛ ●ré❝✐❛ ❛♥t✐❣❛ ❢♦✐ ♦ q✉❡ ♥♦s ♠♦t✐✈♦✉ ❛ ❡s❝♦❧❤❡r ❛ ❘❡t❛ ❞❡ ❊✉❧❧❡r ❡ ❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ◆♦✈❡ P♦♥t♦s ❝♦♠♦ ♦ t❡♠❛ ❞❡ss❡ tr❛❜❛❧❤♦✱ ❛♣❡s❛r ❞❡ ♥ã♦ t❡r ❛❧❣✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ré✈✐♦✱ ♦ q✉❡ ♥♦s ❢♦rç♦✉ ❛ ♣❡sq✉✐s❛r ♠✉✐t♦✳ ◆❡st❡ ♣❡rí♦❞♦ ♣✉❞❡♠♦s ✈❛❧♦r✐③❛r ✉♠❛ ❢❡rr❛♠❡♥t❛ ♣❡❞❛❣ó❣✐❝❛ ♠✉✐t♦ út✐❧ q✉❡ ❢♦✐ ♥♦s ❛♣r❡s❡♥t❛❞❛ q✉❛♥❞♦ ❝✉rsá✈❛♠♦s ❛ ❞✐s❝✐♣❧✐♥❛ ▼❆✸✻ ✲ ❘❡❝✉rs♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ♥♦ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❛ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛✳ ◆❡st❛✱ ❝♦♥❤❡❝❡♠♦s ❞❡ ❢♦r♠❛ ♠❛✐s ♣r♦❢✉♥❞❛✱ ♦ s♦❢t✇❛r❡ ❧✐✈r❡ ●❡♦❣❡❜r❛ q✉❡ ♥♦s ❛❥✉❞♦✉ ♥❛ ✈✐s✉❛❧✐③❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ t✉❞♦ q✉❡ í❛♠♦s ❝♦♥❤❡❝❡♥❞♦✳
❆♣ós ❛❧❣✉♠❛s ❝♦♥str✉çõ❡s ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ❝♦♠ ❛✉①í❧✐♦ ❞♦ s♦❢t✇❛r❡✱ ♣❛ss❛♠♦s ❛ ❞❡s❡♥✈♦❧✈❡r ❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ◆♦✈❡ P♦♥t♦s ❡ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ P♦st❡r✐♦r♠❡♥t❡✱ ❞✐❛♥t❡ ❞❡ ✉♠ ❛rt✐❣♦ q✉❡ ❛♣r❡s❡♥t❛✈❛ ❛ ❈ô♥✐❝❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s✱ ❛s ❝♦♥str✉çõ❡s ❞✐♥â♠✐❝❛s ♥♦s ❛tr❛✐❛♠ ♠❛✐s ❛✐♥❞❛ ❡ s✉r❣✐✉ ✉♠❛ ♣❡r❣✉♥t❛✱ ♣♦r q✉❡ ❛ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ♣r✐✈❛ s❡✉s ❡st✉❞❛♥t❡s ❞❡ ❝♦♥❤❡❝❡r ❡st❛s ♠❛r❛✈✐❧❤❛s ♦❢❡r❡❝❡r❡❝❡♥❞♦ ✉♠❛ ●❡♦♠❡tr✐❛ ❜✉r♦❝rát✐❝❛ ❡ s✉♣❡r✜❝✐❛❧✱ ❧✐♠✐t❛♥❞♦✲s❡ ❛ ❛♣r❡s❡♥t❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ♣♦♥t♦s ◆♦tá✈❡✐s ❞♦ ❚r✐â♥❣✉❧♦✱ ♦ ❜❛r✐❝❡♥tr♦✱ ♦ ❝✐r❝✉♥❝❡♥tr♦✱ ♦ ✐♥❝❡♥tr♦ ❡ ♦ ♦rt♦❝❡♥tr♦✱ q✉❡✱ ♥❛ ♥♦ss❛ ♣rát✐❝❛✱ ♥ã♦ s❡r✈✐❛ ♣❛r❛ ♥❛❞❛✱ ❥á q✉❡ s❡ s❛❜❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ❜❛r✐❝❡♥tr♦ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦ ♣♦♥t♦ ❞❡ ❝❡♥tr♦ ❞❡ ♠❛ss❛ ❞♦ tr✐✲ â♥❣✉❧♦✱ ❞♦ ❝✐r❝✉♥❝❡♥tr♦ q✉❡ é ♦ ♣♦♥t♦ ❝❡♥tr❛❧ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ❝✐r❝✉♥s❝r❡✈❡ ♦ tr✐â♥❣✉❧♦ ❡ ❞♦ ✐♥❝❡♥tr♦✱ q✉❡ é ♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ♥♦ tr✐â♥❣✉❧♦ ❡ ♥ã♦ ❤❛✈✐❛✱ ❛té ❡♥tã♦✱ ✉♠❛ ✉t✐❧✐③❛çã♦ ♣❛r❛ ♦ ♦rt♦❝❡♥tr♦✱ q✉❛♥❞♦ ❞❡s❝♦❜r✐♠♦s q✉❡ ❡st❡ é ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ❡♥❝♦♥tr♦ ❞♦ ♣♦♥t♦ ❝❡♥tr❛❧ ❞❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ◆♦✈❡ P♦♥t♦s✳
❚♦❞❛✈✐❛✱ ❛♦ ❝♦♥❤❡❝❡r ♦ ❛rt✐❣♦ ❆ ❘❡t❛ ❞❡ ❊✉❧❡r ❡ ❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ◆♦✈❡ P♦♥✲ t♦s✱ ❞♦s ❛✉t♦r❡s ❘♦❥❛s ❡ ▼❡♥❞♦③❛✱ ✈❡❥❛ ❬✷✵❪✱ ♣✉❞❡ t❡r ✉♠ ♥♦✈♦ ♦❧❤❛r q✉❡✱ ❛té ❡♥tã♦✱ ♥ã♦ ❤❛✈✐❛ ❡①♣❡r✐♠❡♥t❛❞♦✱ ❛ ✉t✐❧✐③❛çã♦ ❞❛ á❧❣❡❜r❛ ❡ s❡✉s ❝♦♥❝❡✐t♦s ♣❛r❛ ❞❡t❡r♠✐✲ ♥❛r ❡ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❝❛❞❛ ♣♦♥t♦ ♥♦tá✈❡❧ ❞♦ tr✐â♥❣✉❧♦✱ ❛ ❝♦❧✐♥❡❛r✐❞❛❞❡ ❞♦ ❜❛r✐❝❡♥tr♦✱ ❝✐r❝✉♥❝❡♥tr♦ ❡ ♦rt♦❝❡♥tr♦✱ q✉❡ ❞❡✜♥❡♠ ✉♠❛ r❡t❛✱ ❛ r❡t❛ ❞❡ ❊✉❧❡r ❡ ❛
♣❛r❛ ❛q✉❡❧❡s ❡st✉❞❛♥t❡s ❛ q✉❡♠ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦s ❝♦♥❝❡✐t♦s ✐♥✐❝✐❛✐s ❞❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛ tã♦ ❜r✉s❝❛♠❡♥t❡ ♥♦ ✐♥✐❝✐♦ ❞♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❉❡❞✐❝❛♠♦s ♦ ❝❛♣ít✉❧♦ ✷ ♣❛r❛ ❛♣r❡s❡♥t❛çã♦ ❞❛s ❞❡✜♥✐çõ❡s✱ t❡♦r❡♠❛s ❡ ♣r♦♣♦s✐çõ❡s q✉❡ s❡rã♦ út❡✐s ♥♦ ♦❜❥❡t✐✈♦ ❞❡ss❡ tr❛❜❛❧❤♦✱ q✉❡ é✱ ❞❡♠♦♥str❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❡①✐s✲ tê♥❝✐❛s ❞♦s ♣♦♥t♦s ♥♦tá✈❡✐s✱ r❡t❛ ❞❡ ❊✉❧❡r ❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s✱ s♦❜r❡ ❛ ót✐❝❛ ❞❛ á❧❣❡❜r❛✱ q✉❡ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ♥♦ ❝❛♣ít✉❧♦ ✸✳
◆♦ ❝❛♣ít✉❧♦ ✐♥✐❝✐❛❧✱ ❢❛r❡♠♦s ✉♠ ❜r❡✈❡ r❡❝♦rt❡ ❤✐stór✐❝♦ ❞❛s ❞❡s❝♦❜❡rt❛s ❡st✉❞❛❞❛s ♥❡ss❡ tr❛❜❛❧❤♦ ❡ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❞✐❞át✐❝♦✲ ♣❡❞❛❣ó❣✐❝❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s q✉❡ ♣r❡t❡♥✲ ❞❡♠ ✐❧✉str❛r ❡ ❢❛❝✐❧✐t❛r ♦ ♣r♦❝❡ss♦ ❊♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠✱ s♦❜r❡t✉❞♦ ❞♦ ❡st✉❞❛♥t❡ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ❡ ❆♥❛❧ít✐❝❛✳
❆ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛ ♥❛
❈♦♥str✉çã♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❛
❈✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ◆♦✈❡ P♦♥t♦s
❆♣r❡s❡♥t❛r❡♠♦s✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ♦s ❡❧❡♠❡♥t♦s ❣❡♦♠étr✐❝♦s ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ❝♦♥str✉çã♦ ❞❡ ❞✉❛s ♠❛r❛✈✐❧❤♦s❛s ❞❡s❝♦❜❡rt❛s ❞❛ ❣❡♦♠❡tr✐❛✿ ❛ r❡t❛ ❞❡ ❊✉❧❡r ❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s✳ ■♥✐❝✐❛r❡♠♦s ❝♦♠ ✉♠ ❝♦♠❡♥tár✐♦ ❤✐stór✐❝♦ s♦❜r❡ ❡ss❛s ❞❡s❝♦❜❡rt❛s✱ s❡✉s ❞❡s❝♦❜r✐❞♦r❡s ❡ ♦ ❢ért✐❧ ♠♦♠❡♥t♦ q✉❡ ✈✐✈❡✉ ❛ ❝♦♠✉♥✐❞❛❞❡ ♠❛t❡♠át✐❝❛✱ s♦❜r❡t✉❞♦ ♦s ❣❡ô♠❡tr❛s✱ ♥♦ sé❝✉❧♦ ❳■❳✳
✶✳✶ ❘❡❝♦rt❡s ❍✐stór✐❝♦s ❞❛s ❉❡s❝♦❜❡rt❛s ●❡♦♠étr✐✲
❝❛s ❞♦ ❙é❝✉❧♦ ❳■❳
✶✳✶✳✶ ❆s ✧♣ré✲❞❡s❝♦❜❡rt❛s✧❞❡ ❊✉❧❡r ♥♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳❱■■■
❯♠ ❞♦s ♠❛✐s ❢❛s❝✐♥❛♥t❡s ❡❧❡♠❡♥t♦s ❞❛ ❣❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ é✱ s❡♠ ❞ú✈✐❞❛✱ ♦ tr✐✲ â♥❣✉❧♦✳ ❊ss❛ ✜❣✉r❛ ❣❡♦♠étr✐❝❛ q✉❡ é tã♦ út✐❧ ♣❛r❛ ❡♥r✐❥❡❝❡r ❛s ❝♦♥str✉çõ❡s✱ ❞❡✜♥✐r r❡❣✐õ❡s ♣❧❛♥❛s✱ ♦❢❡r❡❝❡r r❡❧❛çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✱ ❡♥tr❡ ✐♥ú♠❡r❛s ♦✉tr❛s ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s ♥♦ ❝♦t✐❞✐❛♥♦ ♣ós✲♠♦❞❡r♥♦ ❝♦♠♦ ♥♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ ●P❙✶✱ ♣♦r ❡①❡♠♣❧♦✱é ❝♦♥❤❡❝✐❞♦ ❜❡♠ ❛♥t❡s ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✷✳ ❙❡✉s ♣♦♥t♦s ♥♦tá✈❡✐s✸ t❛♠✲
❜é♠ sã♦ ❝♦♥❤❡❝✐❞♦s ❤á ❛❧❣✉♥s sé❝✉❧♦s ❛♥t❡s ❞❛ ♥♦ss❛ ❡r❛✱ ♣♦ré♠✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦
✶❖●❧♦❜❛❧ P♦s✐t✐♦♥✐♥❣ ❙②st❡♠✱ ❡♠ ♣♦rt✉❣✉ês✱ ❙✐st❡♠❛ ❞❡ ♣♦s✐❝✐♦♥❛♠❡♥t♦ ❣❧♦❜❛❧✳
✷❝♦❧❡tâ♥❡❛ ❞❡ ✶✸ ✈♦❧✉♠❡s ❡s❝r✐t❛ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛ ✭✸✻✵❛✳❝✳✲ ✷✾✺ ❛✳❝✳✮
q✉❡ s❡ t♦r♥♦✉ ❛ ❜❛s❡ ♣❛r❛ ❛ ❝❤❛♠❛❞❛ ❣❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛ ♣♦r ✈♦❧t❛ ❞❡ ✸✵✵ ❛♥♦s ❛♥t❡s ❞❛ ❡r❛ ✈✐❣❡♥t❡✳
✸P♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡ ❝❡✈✐❛♥❛s ✭s❡❣♠❡♥t♦s ❞❡ r❡t❛ ♥♦tá✈❡✐s ❞♦ tr✐â♥❣✉❧♦✮✳ ❖s ♠❛✐s ✉t✐❧✐③❛❞♦s
sã♦ ♦ ❜❛r✐❝❡♥tr♦✱ ♦ ✐♥❝❡♥tr♦✱ ♦ ❝✐r❝✉♥❝❡♥tr♦ ❡ ♦ ♦rt♦❝❡♥tr♦ ✭♠❛✐s ❞❡t❛❧❤❡s ♥❛ s✉❜s❡çã♦ ✶✳✷✳✶✱ ♣á❣✐♥❛ ✺✮✳
❛✉t♦r ❬✶✸❪ ❛té ♦ sé❝✉❧♦ ❳❱■■✱ ♥ã♦ ❡♥❝♦♥tr❛♠♦s ❛❧❣✉♠ r❡❣✐str♦ ❞❡ ♥♦✈❛s ❞❡s❝♦❜❡rt❛s r❡❧❡✈❛♥t❡s ❡ ❡st✉❞♦s s♦❜r❡ ❛ ❣❡♦♠❡tr✐❛ ❞♦ tr✐â♥❣✉❧♦ ❛té ♦ t❡♦r❡♠❛ ❞❡ ❈❡✈❛✹ ❡♠ ✶✻✼✽✳
❖ ❡st✉❞♦ ❞❛s ❝❡✈✐❛♥❛s ❡ ❞♦s ♣♦♥t♦s ♥♦tá✈❡✐s ♣❡❧❛ ✐♥t❡rs❡❝çã♦ ❞❡❧❛s ❞❡✜♥✐❞❛s✱ sã♦ ❛♣r❡s❡♥t❛❞♦s ❛♦s ❡st✉❞❛♥t❡s ❛✐♥❞❛ ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ❞❡ ❢♦r♠❛ ♠❡❝â♥✐❝❛ ❡ ❡♠ ♠✉✐t♦s ❝❛s♦s ❝♦♠♦ ♠❡r❛ ❝✉r✐♦s✐❞❛❞❡✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ♦ ❡❞✉❝❛♥❞♦ ♥ã♦ s✐♥t❛ ❛ ♠❡♥♦r ❛tr❛çã♦ ♥❡♠ s✐♠♣❛t✐❛ ♣♦r ❡ss❡ ❛ss✉♥t♦✳ ❊♠ s❡✉s ♠♦♠❡♥t♦s r❛ríss✐♠♦s ❞❡ ♣rát✐❝❛ ♣❡❞❛❣ó❣✐❝❛✱ ❛❧❣✉♥s ❡❞✉❝❛❞♦r❡s r❡❝♦rr❡♠ à ❝♦♥str✉çã♦ ❣❡♦♠étr✐❝❛ ❝♦♠ ré❣✉❛ ❡ ❝♦♠✲ ♣❛ss♦ ♣❛r❛ ✐❧✉str❛r ❡ss❡s ❜❡❧íss✐♠♦s ❡❧❡♠❡♥t♦s q✉❡ tr❛③❡♠ ♣❛r❛ ❛❧❣✉♥s ❞✐s❝í♣✉❧♦s ❛❧❣✉♠❛ ❛❧❡❣r✐❛ q✉❛♥❞♦ ✈ê❡♠ ❛ ❡①❛t✐❞ã♦ ❞❡ss❛ ❝♦♥str✉çã♦✳
❊ss❡ ♠ét♦❞♦ ❣❡♦♠étr✐❝♦ ❞❡ ❝♦♥str✉✐r ♦s ♣♦♥t♦s ♥♦tá✈❡✐s ❞♦ tr✐â♥❣✉❧♦ ❞❡s♣❡rt♦✉ ❡♠ ❊✉❧❡r✺ ❛❧❣✉♠❛s ❝✉r✐♦s✐❞❛❞❡s ❝♦♠♦ ❛ ❜r✐❧❤❛♥t❡ ♦❜s❡r✈❛çã♦ ❞❡ q✉❡ ♦s ♣♦♥t♦s ❜❛✲
r✐❝❡♥tr♦✱ ❝✐r❝✉♥❝❡♥tr♦ ❡ ♦rt♦❝❡♥tr♦ sã♦ ❝♦❧✐♥❡❛r❡s✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞♦ tr✐â♥❣✉❧♦✳ ❉❡♥♦♠✐♥❛♠♦s ❡st❛ r❡t❛ q✉❡ ❝♦♥té♠ ❡ss❡s ♣♦♥t♦s ♥♦tá✈❡✐s✱ ❡♠ s✉❛ ❤♦♠❡♥❛❣❡♠✱ ❞❡ ❘❡t❛ ❞❡ ❊✉❧❡r✳ ❉❡ ❛❝♦r❞♦✱ ♥♦✈❛♠❡♥t❡✱ ❝♦♠ ♦ ❛✉t♦r ❬✶✸❪✱ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ t❡♦r❡♠❛ ❢♦✐ ❛♥❛❧ít✐❝❛✱ ♦ q✉❡ ♥♦s ❧❡✈❛ ❛ ❝r❡r q✉❡ ❞❡✈❡ t❡r ❝♦♥str✉í❞♦ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ tr✐â♥❣✉❧♦s ❞✐❢❡r❡♥t❡s ❛té ❡♥❝♦♥tr❛r ❛ ❧❡✐✳
❊ss❛s ❞❡s❝♦❜❡rt❛s só r❛t✐✜❝❛♠ q✉ã♦ ❜r✐❧❤❛♥t❡ ❡r❛ ❛ ♠❡♥t❡ ❞❡ss❡ ✈❡r❞❛❞❡✐r♦ ❣ê♥✐♦ ❞❛ ▼❛t❡♠át✐❝❛✱ ♣♦✐s✱ ❝❤❡❣♦✉ à ❝♦♥❝❧✉sã♦ ❞❡ ✐♥ú♠❡r♦s tr❛❜❛❧❤♦s s❡♠ ✉♠❛ ❢❡rr❛♠❡♥t❛ t❡❝♥♦❧ó❣✐❝❛ ✐♠♣r❡s❝✐♥❞í✈❡❧ ♥♦s ❡st✉❞♦s ❛t✉❛✐s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ✉♠ ❝♦♠♣✉t❛❞♦r ♠✉♥✐❞♦ ❝♦♠ ✉♠ ♣r♦❣r❛♠❛ ❞❡ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛✳
✶✳✶✳✷ ❙é❝✉❧♦ ❳■❳✿ ❆ r❡✲❞❡s❝♦❜❡rt❛ ❞❛ ❣❡♦♠❡tr✐❛
◆ã♦ ❤á ❞ú✈✐❞❛s q✉❡ ♦s tr❛❜❛❧❤♦s ❣❡♦♠étr✐❝♦s ❞❡ ❊✉❧❡r ✐♠♣✉❧s✐♦♥❛r❛♠ s♦❜r❡♠❛✲ ♥❡✐r❛ ♦s ❡st✉❞♦s ♠❛✐s ❛♣r♦❢✉♥❞❛❞♦s ♥♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳■❳✳ ❆ ♣r✐♠❡✐r❛ ❞❡♠♦♥str❛✲ çã♦ s✐♥tét✐❝❛ ❞❛s ❞❡s❝♦❜❡rt❛s ❞❡ ❊✉❧❡r só ✈❡✐♦ ❡♠ ✶✽✵✸ ❝♦♠ ❈❛r♥♦t✻✱ q✉❡ ✐♥❛✉❣✉r♦✉
❛ss✐♠ ✉♠❛ ✈❡r❞❛❞❡✐r❛ ❝♦rr✐❞❛ ❛♦s ♥♦✈♦s ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞♦ tr✐â♥❣✉❧♦✱
✹Pr♦♣♦st♦ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ✐t❛❧✐❛♥♦ ●✐♦✈❛♥♥✐ ❈❡✈❛ ✭✶✻✹✼✲✶✼✸✹✮✱ ❝♦♥❢♦r♠❡ ♦ ❛✉t♦r ❬✶✶❪✱ ♥❛ ♣á✲
❣✐♥❛ ✸✷✱ ❡st❡ t❡♦r❡♠❛ ❛✜r♠❛ q✉❡ ❛s ❝❡✈✐❛♥❛s ✭s❡❣♠❡♥t♦s ❞❡ r❡t❛ q✉❡ ❧✐❣❛♠ ♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❛ ✉♠ ♣♦♥t♦ ❞♦ ❧❛❞♦ ♦♣♦st♦ ❛ ❡❧❡✱ q✉❡ t❡♠ ♦ s❡✉ ♥♦♠❡ ❡♠ ❤♦♠❡♥❛❣❡♠ ❛♦ ♣ró♣r✐♦ ●✐✲ ♦✈❛♥♥✐ ❈❡✈❛✮ ❣❡r❛❞❛s ♣❡❧♦s ✈ért✐❝❡s ❞❡ ✉♠ t✐â♥❣✉❧♦ A, B, C ❡ ♣❡❧♦s ♣♦♥t♦s ❞♦s ❧❛❞♦s ♦♣♦st♦s D, E, F✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ sã♦ ❝♦♥❝♦rr❡♥t❡s s❡BD·CE·AF =F B·DC·EA.
✺❖ ♠❛t❡♠át✐❝♦ ▲❡♦♥❤❛r❞ ❊✉❧❡r ✭✶✼✵✼✲✶✼✽✸✮ ♥❛s❝❡✉ ♥❛ s✉íç❛✱ ♥❛ ❇❛s✐❧é✐❛ ❡ s❡❣✉♥❞♦ ♦ ❛✉t♦r
❬✾❪✱ ❢♦✐ t❛❧✈❡③ ♦ ❤♦♠❡♠ q✉❡ ♠❛✐s ❝❛❧❝✉❧♦✉ ❡ ❡s❝r❡✈❡✉ ❡♠ t♦❞❛ ❛ ❤✐stór✐❛ ❞❛ ❤✉♠❛♥✐❞❛❞❡✳ ➱ ♦ r❡s♣♦♥sá✈❡❧ ♣♦r ✐♥ú♠❡r❛s ❝♦♥tr✐❜✉✐çõ❡s ♥❛ ➪❧❣❡❜r❛✱ ❆r✐t♠ét✐❝❛✱ ●❡♦♠❡tr✐❛ ❡ ❋ís✐❝❛ q✉❡ ❛té ❤♦❥❡ sã♦ ❢❡rr❛♠❡♥t❛s ♣♦❞❡r♦s❛s ♥❡ss❡s r❛♠♦s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳
✻▲❛③❛r❡ ▼✐❝♦❧❛s ▼❛r❣✉❡r✐t❡ ❈❛r♥♦t ✭✶✼✺✸✲✶✽✷✸✮✱ ♠❛t❡♠át✐❝♦ ❡ ❢ís✐❝♦✱ ❞❡♠♦♥str♦✉ ❛ ❡①✐stê♥❝✐❛
❞❛ r❡t❛ ❞❡ ❊✉❧❡r ♥♦ s❡✉ tr❛❜❛❧❤♦ ●❡♦♠❡tr✐❛ ❞❡ P♦s✐t✐♦♥✳
q✉❡ ❛té ❡♥tã♦✱ ♣❛r❡❝✐❛♠ ❝♦♥❝❧✉í❞♦s ♣❡❧♦s ❣r❡❣♦s✳
❙❡❣✉♥❞♦ ❬✻❪✱ ♠❛✐s ✉♠ ♣♦♥t♦ ♣❡rt❡♥❝❡♥t❡ ❛ r❡t❛ ❞❡ ❊✉❧❡r ❢♦✐ ❡♥❝♦♥tr❛❞♦ ♣♦r P♦♥✲ ❝❡❧❡t✼ ❡ ❇r✐❛♥❝❤♦♥✽ ❡♠ ✉♠ ❛rt✐❣♦ ♣✉❜❧✐❝❛❞♦ ❡♠ ❝♦♥❥✉♥t♦✱ ✐♥t✐t✉❧❛❞♦ ❘❡❝❤❡r❝❤❡s s✉r
❧❛ ❞ét❡r♠✐♥❛t✐♦♥ ❞✬✉♥❡ ❤②♣❡r❜♦❧❡ éq✉✐❧❛tér❡✱ ♥♦s ❆♥♥❛❧❡s ❞❡ ●❡r❣♦♥♥❡ ❞❡ ✶✽✷✵✲✶✽✷✶ ✭✈❡❥❛ ❬✼❪✮✳ ❚r❛t❛✲s❡ ❞♦ ❝❡♥tr♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❛ q✉❛❧ ♣❡rt❡♥❝❡♠ ♦s três ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s✱ ♦s três ♣♦♥t♦s ❝❤❛♠❛❞♦s ❞❡ ♣é ❞❡ ❛❧t✉r❛✾ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ s❡✉s
três ♣♦♥t♦s✶✵ ❞❡ ❊✉❧❡r✳ ❊ss❡ t❡♦r❡♠❛ é ♠❛✐s ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ t❡♦r❡♠❛ ❞❛ ❝✐r❝✉♥❢❡✲
rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s ♠❛s t❛♠❜é♠ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ❛❧❣✉♥s ❞♦s s❡✉s ✐❞❡❛❧✐③❛❞♦r❡s ❝♦♠♦ ❝ír❝✉❧♦ ❞❡ ❊✉❧❡r✶✶✱ ❝ír❝✉❧♦ ❞❡ P♦♥❝❡❧❡t✱ ❝ír❝✉❧♦ ❞❡ ❇r✐❛♥❝❤♦♥ ♦✉ ❝ír❝✉❧♦ ❞❡
❋❡✉❡r❜❛❝❤✶✷✱ ❡♥tr❡ ❡ss❡s✱ é ♠❛✐s ❝♦♠✉♠ ❡♥❝♦♥tr❛r♠♦s ♦s ❝ré❞✐t♦s ❞❛❞♦s✱ ❡q✉✐✈♦❝❛✲
❞❛♠❡♥t❡✱ ❛♦ ú❧t✐♠♦✳ ❙❡❣✉♥❞♦ ♦ ❛✉t♦r ❬✻❪✱ ♥♦✈❛♠❡♥t❡✱ ✐ss♦ s❡ ❞❡✈❡ ❛♦ ❢❛t♦ ❞❡ t❡r ♣r♦❞✉③✐❞♦ ✉♠❛ ♣❡q✉❡♥❛ ♠♦♥♦❣r❛✜❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦s ❡st✉❞♦s ❞❡ P♦♥❝❡❧❡t ❡ ❇r✐❛♥✲ ❝❤♦♥ ♥♦ ❛♥♦ ❞❡ ✶✽✷✷✱ ❝♦♠ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s✳
❋❡✉❡r❜❛❝❤✱ ♥❡ss❡ ♠❡s♠♦ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t♦✉ t❛♠❜é♠ ✐♠♣♦rt❛♥t❡s ❞❡s❝♦❜❡rt❛s s♦❜r❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s✱ ❡♥tr❡ ❛s q✉❛✐s✱ ❞❡st❛❝❛✲s❡ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ q✉❡ ❡st❛ é t❛♥❣❡♥t❡ às três ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡①✲✐♥s❝r✐t❛s✶✸ ❞♦ ♠❡s♠♦ tr✐â♥❣✉❧♦✳ ❊st❡
t❡♦r❡♠❛ é✱ ♣❛r❛ ♠✉✐t♦s✱ ✉♠ ❞♦s ♠❛✐s ❜❡❧♦s t❡♦r❡♠❛ ❞❛ ❣❡♦♠❡tr✐❛ ♣ós✲❊✉❝❧✐❞❡s✳
✶✳✶✳✸ ❯♠ ❜❡❧♦ ❞❡s❢❡❝❤❡ ♣❛r❛ ✉♠ sé❝✉❧♦ ❜r✐❧❤❛♥t❡
◆❛ s❡❣✉♥❞❛ ♠❡t❛❞❡ ❞♦ sé❝✉❧♦ ❳■❳✱ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ tr❛❜❛❧❤♦s ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s✱ ✐♠♣✉❧s✐♦♥❛❞♦s ♣♦r ❡ss❛s ❞❡s❝♦❜❡rt❛s q✉❡ ❛❜r✐r❛♠ ♥♦✈❛♠❡♥t❡ ♦ ❝❛♠♣♦ ❞❛ ♣❡sq✉✐s❛ ♥❛ ❣❡♦♠❡tr✐❛✳ ◆❛ s✉❛ ú❧t✐♠❛ ❞é❝❛❞❛✱ s❡❣✉♥❞♦ ♦ ❛✉t♦r ❬✶✶❪✱ ❛❧❣✉♥s t❡①t♦s✶✹ ✼❏❡❛♥ ❱✐❝t♦r P♦♥❝❡❧❡t ✭✶✼✽✽✲✶✽✻✼✮ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês q✉❡ s❡ ❞❡st❛❝♦✉ ♣r✐♥❝✐♣❛❧♠❡♥t❡
♥♦ ❡st✉❞♦ ❞❛ ●❡♦♠❡tr✐❛ Pr♦❥❡t✐✈❛✳
✽❈❤❛r❧❡s✲❏✉❧✐❡♥ ❇r✐❛♥❝❤♦♥ ✭✶✼✽✸✲✶✽✻✹✮ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ❝♦♥t❡♠♣♦râ♥❡♦ ❞❡ P♦♥❝❡❧❡t✳ ✾P♦♥t♦ ❞❡ ♠❡♥♦r ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦ ✈ért✐❝❡ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ ♦ s❡✉ ❧❛❞♦ ♦♣♦st♦✳ ❱❡r s❡çã♦ ✸✳✹✱
♣á❣✐♥❛ ✺✵✳
✶✵P♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ ❞❡✜♥✐❞♦ ♣♦r ✉♠ ✈ért✐❝❡ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ ♦ ♦rt♦❝❡♥tr♦ ❞❡ss❡ tr✐â♥✲
❣✉❧♦✳ ❱❡r ❛ s✉❜s❡çã♦ ✸✳✻✳✶✱ ♥❛ ♣á❣✐♥❛ ✺✸✳
✶✶❈♦♠♦ ❊✉❧❡r ❤❛✈✐❛ ❞❡s❝♦❜❡rt♦ q✉❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♣❛ss❛✈❛ ♣❡❧♦s ♣♦♥t♦s ♠é❞✐♦s ❡ ♦s ♣és ❞❛s
❛❧t✉r❛s✱ r❡st❛✈❛ ❛♣❡♥❛s ❞❡s❝♦❜r✐r q✉❡ ♣❛ss❛✈❛ t❛♠❜é♠ ♣❡❧♦s ♣♦♥t♦s ♠é❞✐♦s ❡♥tr❡ ♦ ♦rt♦❝❡♥tr♦ ❡ ♦ ✈ért✐❝❡ ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ♣♦r ✐ss♦✱ ❡♠ s✉❛ ❤♦♠❡♥❛❣❡♠✱ sã♦ ❞❛❞♦s ❛ ❡❧❡ ♦s ♥♦♠❡s ❞❡ss❡s ♣♦♥t♦s ❡ ❛té ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s✳
✶✷❑❛r❧ ❲✐❧❤❡❧♠ ❋❡✉❡r❜❛❝❤ ✭✶✽✵✵✲✶✽✸✹✮✱ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦ q✉❡ ❝♦♥tr✐❜✉✐✉ ❜❛st❛♥t❡ ❝♦♠ ❛ ❣❡♦✲
♠❡tr✐❛✱ ♠❡s♠♦ ❝♦♠ s❡✉ ❜r❡✈❡ t❡♠♣♦ ❞❡ ✈✐❞❛✳
✶✸❈✐r❝✉♥❢❡rê♥❝✐❛ ❝✉❥♦ ❝❡♥tr♦ é ❛ ✐♥t❡rs❡❝çã♦ ❞❛s ❜✐ss❡tr✐③❡s ❞♦s â♥❣✉❧♦s ❡①t❡r♥♦s ❞♦ tr✐â♥❣✉❧♦ ❡
é t❛♥❣❡♥t❡ ❛♦ ♠❡s♠♦✳
✶✹❙✉❛ ❛✜r♠❛çã♦ s❡ ❜❛s❡♦✉ ♥♦s ❡s❝r✐t♦s ❞❡ ❏♦♥❤ ❲❡❧❧❡s❧❡② ❘✉ss❡❧✱ ♥♦ s❡✉ ❧✐✈r♦ P✉r❡ ●❡♦♠❡tr②✱
❬✷✻❪✳
❞❡ ●❡♦♠❡tr✐❛ ♣r♦❥❡t✐✈❛ ❛♣r❡s❡♥t❛r❛♠ ✉♠❛ ❞❡s❝♦❜❡rt❛ ❢❛s❝✐♥❛♥t❡✿ ❛ ❝ô♥✐❝❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s✱ q✉❡✱ ❛ ❡①❡♠♣❧♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s✱ ❝♦♥té♠ ♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦ tr✐â♥❣✉❧♦✱ ♣♦ré♠✱ ❡♠ ❧✉❣❛r ❞♦s ♣és ❞❡ ❛❧t✉r❛ ❡ ❞♦s ♣♦♥t♦s ❞❡ ❊✉❧❡r✱ sã♦ ✉t✐❧✐③❛❞♦s ♦✉tr♦s ♣♦♥t♦s q✉❡ sã♦ ❣❡r❛❞♦s ♣♦r ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ♣❡rt❡♥❝❡♥t❡ ❛ r❡t❛ ❞❡ ❊✉❧❡r q✉❡✱ ❝♦♠ ✉♠ ❞♦s ✈ért✐❝❡s✱ ❞❡✜♥❡ ♦ ♣♦♥t♦ ♠é❞✐♦ ❡♥tr❡ ❡❧❡s ❡ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❡♥tr❡ ♦ ❧❛❞♦ ♦♣♦st♦ ❡ ❛ r❡t❛ q✉❡ ❝♦♥té♠ ❡ss❡ ♣♦♥t♦ ❡ ✉♠ ❞♦s ✈ért✐❝❡✱ ❝♦♠♣❧❡t❛♥❞♦✱ ❛ss✐♠✱ ♦s ♥♦✈❡ ♣♦♥t♦s✳
❆ ✈❛r✐❛çã♦ ❞❡ss❡ ♣♦♥t♦ ♥❛ r❡t❛ ❞❡ ❊✉❧❡r ❛♣r❡s❡♥t❛ ✈ár✐❛s ❢♦r♠❛s ❞❡ ❝ô♥✐❝❛s ❝♦♠♦ ❤✐♣ér❜♦❧❡s✱ ❡❧✐♣s❡s ❡ ❛té ♣❛rá❜♦❧❛s✱ q✉❡ ♥♦s ❢❛③ ❞❡❞✉③✐r q✉❡ s❡ ❡ss❡ ♣♦♥t♦ ✢✉t✉❛✱ ♣♦❞❡ ❝♦✐♥❝✐❞✐r ❝♦♠ ♦ ♦rt♦❝❡♥tr♦ ❡✱ ❝♦♠♦ s❛❜❡♠♦s✱ ❝♦♥str✉✐r❡♠♦s✱ ❛ss✐♠✱ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s✱ q✉❡ ♥❛❞❛ ♠❛✐s é q✉❡ ✉♠❛ ❢♦r♠❛ ❡s♣❡❝✐❛❧ ❞❡ ❈ô♥✐❝❛ ❞♦s ◆♦✈❡ P♦♥t♦s✳
❚♦❞❛✈✐❛✱ ❛ ❜❡❧❡③❛ ♣❧ást✐❝❛ ❞❡ss❛ ❞❡s❝♦❜❡rt❛✱ q✉❡✱ s❡❣✉♥❞♦ ❡ss❡ ❛✉t♦r✱ ❛té ♠❡❛❞♦s ❞♦ sé❝✉❧♦ ❳❳ ❢♦✐ ♠❛t❡r✐❛❧ ♦❜r✐❣❛tór✐♦ ♥❛s s❡❧❡çõ❡s ♣❛r❛ ♦ ❞♦✉t♦r❛♠❡♥t♦ ❡♠ ▼❛t❡♠á✲ t✐❝❛ ❞❡ ❯♥✐✈❡rs✐❞❛❞❡s ❝♦♠♦ ❖①❢♦r❞ ❡ ❈❛♠❜r✐❞❣❡✱ ❢♦✐ ❡sq✉❡❝✐❞❛ ❡ ♥❡❣❧✐❣❡♥❝✐❛❞❛ ♣❡❧❛s ■♥st✐t✉✐çõ❡s ❞❡ ❊♥s✐♥♦ ❙✉♣❡r✐♦r ❛ ♣♦♥t♦ ❞❡✱ ❡♠ ✷✵✵✷✱ ❡ss❡ ❛✉t♦r ❞❡s❝♦❜r✐r ❞❡ ❢♦r♠❛ ❛♥❛❧ít✐❝❛✱ ❛ ❈ô♥✐❝❛ ❞♦s ◆♦✈❡ P♦♥t♦s ❡ só ❞❡♣♦✐s ❞❡ ✉♠❛ ❜✉s❝❛ ✐♥t❡♥s❛✱ ♣❡r❝❡❜❡r q✉❡ ❡ss❛ ❞❡s❝♦❜❡rt❛ ❥á ❤❛✈✐❛ s✐❞♦ ♣✉❜❧✐❝❛❞❛ ❤á ♠❛✐s ❞❡ ✉♠ sé❝✉❧♦✳
✶✳✷ ❈♦♥str✉çã♦ ❞❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ◆♦✈❡ P♦♥t♦s
♥♦ ●❡♦❣❡❜r❛
◆♦ ❡♥s✐♥♦ ❞♦s ♣♦♥t♦s ♥♦tá✈❡✐s ❞♦ tr✐â♥❣✉❧♦✶✺✱ é ❝♦♠✉♠ ❡♥❝♦♥tr❛r♠♦s ♥♦s ❧✐✈r♦s
❞✐❞át✐❝♦s ❡ ♥❛ ♣rát✐❝❛ ❞♦❝❡♥t❡ ❞❡ ♣❛rt❡ ❞♦s ♣r♦❢❡ss♦r❡s✱ ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ♠❡❝â♥✐❝❛ ❞❡ss❡s ♣♦♥t♦s✳ ➱ ❝❡rt♦ q✉❡✱ ❡♥r✐q✉❡❝✐❞♦ ❝♦♠ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ✐❧✉str❛çõ❡s✱ ♣❛r❛ àq✉❡❧❡s ❡st✉❞❛♥t❡s q✉❡ ❥á ❞❡s♣❡rt❛r❛♠ ♦ ❞❡s❡❥♦ ♣❡❧❛ ❞❡s❝♦❜❡rt❛ ♥❛ ♠❛t❡♠át✐❝❛✱ ❛ t❡♦r✐❛ ❞❛ ❝♦♥str✉çã♦ ❞❡ss❡s ♣♦♥t♦s✱ ❛s ❝❡✈✐❛♥❛s✱ ❛s ❝✉r✐♦s✐❞❛❞❡s ❡ s✉❛s ✉t✐❧✐③❛çõ❡s sã♦ ❛❜s♦r✈✐❞❛s✱ ❝♦♥t✉❞♦✱ ♦ ❛❧✉♥♦ ♥ã♦ ❝♦♥stró✐ s❡✉s ♣ró♣r✐♦s ❝♦♥❤❡❝✐♠❡♥t♦s✳ ▼❡s♠♦ q✉❛♥❞♦ ♦s ❡❞✉❝❛❞♦r❡s r❡❝♦rr❡♠ à ❝♦♥str✉çã♦ ❣❡♦♠étr✐❝❛ ❝♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦✱ ♥ã♦ ❞á ♣❛r❛ tr❛❜❛❧❤❛r ✧❞✐♥❛♠✐❝❛♠❡♥t❡✧❝♦♠ ❡ss❛s ❝♦♥str✉çõ❡s ♣♦✐s✱ t♦r♥❛✲s❡ ✐♥✈✐á✈❡❧ ❛ r❡♣r♦❞✉çã♦ ❞❡ ✐♥ú♠❡r❛s s✐t✉❛çõ❡s ❞✐❢❡r❡♥t❡s ❝♦♠ ❡ss❛s ❢❡rr❛♠❡♥t❛s✳
◆♦ ♥♦ss♦ tr❛❜❛❧❤♦✱ ❡s❝♦❧❤❡♠♦s ✉♠ ❞♦s ♠✉✐t♦s ♣r♦❣r❛♠❛s ❣r❛t✉✐t♦s✱ ♦✉ s♦❢t✇❛r❡ ❧✐✈r❡s✱ ♦ ●❡♦❣❡❜r❛✱ ❝✉❥❛ ♣❧❛t❛❢♦r♠❛ ✐❧✉str❛ ❛s ✜❣✉r❛s ❞❡ss❡ tr❛❜❛❧❤♦✳ ❙❡❣✉♥❞♦ s✉❛ ♣á❣✐♥❛ ✈✐rt✉❛❧✶✻✱ é ✉♠ s♦❢t✇❛r❡ ❞❡ ♠❛t❡♠át✐❝❛ ❞✐♥â♠✐❝❛ ❣r❛t✉✐t♦ ❡ ♠✉❧t✐✲♣❧❛t❛❢♦r♠❛
♣❛r❛ t♦❞♦s ♦s ♥í✈❡✐s ❞❡ ❡♥s✐♥♦✱ q✉❡ ❝♦♠❜✐♥❛ ❣❡♦♠❡tr✐❛✱ á❧❣❡❜r❛✱ t❛❜❡❧❛s✱ ❣rá✜❝♦s✱
✶✺❈♦♥t❡ú❞♦ ❣❡r❛❧♠❡♥t❡ tr❛❜❛❧❤❛❞♦ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❡ ❛♣r♦❢✉♥❞❛❞♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ✶✻P♦❞❡ s❡r ❜❛✐①❛❞♦ ❣r❛t✉✐t❛♠❡♥t❡ ♥♦ ❡♥❞❡r❡ç♦ ❡❧❡trô♥✐❝♦ ✇✇✇✳❣❡♦❣❡❜r❛✳♦r❣✳
❡st❛tíst✐❝❛ ❡ ❝á❧❝✉❧♦ ❡♠ ✉♠ ú♥✐❝♦ s✐st❡♠❛✳
✶✳✷✳✶ ❖s ♣♦♥t♦s ♥♦tá✈❡✐s ❞♦ tr✐â♥❣✉❧♦
❆ ❝♦♥str✉çã♦ ❞♦ ❜❛r✐❝❡♥tr♦❚❛♠❜é♠ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❝❡♥tró✐❞❡✱ é t❛❧✈❡③✱ ♦ ♣♦♥t♦ ♠❛✐s ✉t✐❧✐③❛❞♦ ❞❡ ❢♦r♠❛ ♣rát✐❝❛ ♥♦ ♥♦ss♦ ❝♦t✐❞✐❛♥♦✱ s♦❜r❡ t✉❞♦ ♥❛ ❝♦♥str✉çã♦ ❝✐✈✐❧✱ ❥á q✉❡ ❡st❡ ♣♦♥t♦ é ♦ ❝❡♥✲ tr♦ ❞❡ ♠❛ss❛ ❞♦ tr✐â♥❣✉❧♦✳ ◆❡st❡ tr❛❜❛❧❤♦✱ r❡♣r❡s❡♥t❛r❡♠♦s ♦ ❜❛r✐❝❡♥tr♦ ♣♦r G✶✼✳
❊ss❡ ♣♦♥t♦ é ❡♥❝♦♥tr❛❞♦ tr❛ç❛♥❞♦✲s❡ ❛s ♠❡❞✐❛♥❛s✶✽ ❝♦rr❡s♣♦♥❞❡♥t❡s ❛ ❝❛❞❛ ✈ért✐❝❡
❡ ❡♥❝♦♥tr❛♥❞♦ ♦ s❡✉ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ q✉❡ é G✳
◆❛ ✉t✐❧✐③❛çã♦ ❞♦ ●❡♦❣❡❜r❛✱ ♦ ♣r♦❢❡ss♦r ♣♦❞❡✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ❧❡✈❛♥t❛r q✉❡st✐♦♥❛✲ ♠❡♥t♦s s♦❜r❡ ❝♦♠♦ ❡♥❝♦♥tr❛r ❡ss❡ ♣♦♥t♦ ♠é❞✐♦✳ ➱ ❝❧❛r♦ q✉❡ ♦ ♣r♦❣r❛♠❛ tr❛③ ❝♦♥s✐❣♦ ❡ss❛ ❢❡rr❛♠❡♥t❛ q✉❡✱ ❜❛st❛♥❞♦ ❡s❝♦❧❤❡r ♦s ❞♦✐s ♣♦♥t♦s✱ ❛♣r❡s❡♥t❛ ♦ s❡✉ ♣♦♥t♦ ♠é❞✐♦✳ P♦ré♠✱ ❛ s✉❣❡stã♦ ❞❡ ♣r♦❜❧❡♠❛s q✉❡ r❡q✉✐s✐t❡♠ ♦ r❛❝✐♦❝í♥✐♦ ❣❡♦♠étr✐❝♦ ❛♦ ❛❧✉♥♦✱ s♦❧✐❞✐✜❝❛rá ❡ss❡s ❝♦♥❤❡❝✐♠❡♥t♦s tr❛③❡♥❞♦ s❡❣✉r❛♥ç❛ ❡ s❛t✐s❢❛çã♦✳
❆ r❡♣r♦❞✉çã♦ ❞♦s ♣❛ss♦s ❞❛ ❝♦♥str✉çã♦ ❝♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦ é ✉♠❛ ❜♦❛ ❛t✐✈✐✲ ❞❛❞❡✱ ✐❧✉str❛❞❛ ♥❛ ✜❣✉r❛ ✶✳✶✱ q✉❡✱ ♥❡ss❡ ❝❛s♦ r❡♣r♦❞✉③ ❛ ❝♦♥str✉çã♦ ❞♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦✳
❋✐❣✉r❛ ✶✳✶✿ ■❧✉str❛çã♦ ❞❛ ❝♦♥str✉çã♦ ❞♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ✉♠ ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦
ABC✳
❯♠ ❢❛t♦ ❝✉r✐♦s♦ q✉❡ t❛♠❜é♠ ♣♦❞❡ s❡r ❧❡✈❛♥t❛❞♦ é q✉❡✱ ❛♦ ❝♦♥trár✐♦ ❞❡ ❛❧❣✉♥s
✶✼G ❞❡ ❝❡♥tr♦ ❣r❛✈✐t❛❝✐♦♥❛❧✱ ❥á q✉❡ ❡♠ ❝❛s♦ ❞❡ ❣r❛✈✐❞❛❞❡ ❝♦♥st❛♥t❡ ❡ ✉♠ ❝♦r♣♦ ❤♦♠♦❣ê♥❡♦✱ ♦
❝❡♥tr♦ ❞❡ ♠❛ss❛ s❡ ❝♦♥❢✉♥❞✐ ❝♦♠ ♦ ❝❡♥tr♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳
✶✽❘❡t❛s q✉❡ ♣❛ss❛♠ ♣♦r ✉♠ ✈ért✐❝❡ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ ♣❡❧♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡✉ ❧❛❞♦ ♦♣♦st♦✳
♣♦❧í❣♦♥♦s ♥ã♦✲❝♦♥✈❡①♦s✶✾✱ ♦ ❜❛r✐❝❡♥tr♦ ❞♦ tr✐â♥❣✉❧♦ é ✐♥t❡r♥♦✳ ❊ss❛ ✐♥❢♦r♠❛çã♦
♣♦❞❡ s❡r ✈❡r✐✜❝❛❞♦ ♥♦ ♣r♦❣r❛♠❛✳ ❖✉tr❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ ♣♦❞❡ s❡r ✈❡r✐✜❝❛❞❛✱ é q✉❡ G ❞✐✈✐❞❡ q✉❛❧q✉❡r ✉♠❛ ❞❛s ♠❡❞✐❛♥❛s ❡♠ ❞✉❛s ♣❛rt❡s ♣r♦♣♦r❝✐♦♥❛✐s t❛✐s q✉❡ ♦
s❡❣♠❡♥t♦ ❡♥tr❡ G ❡ ♦ ✈ért✐❝❡ é ♦ ❞♦❜r♦ ❞❛ ♦✉tr❛ ♣❛rt❡✱ ❝♦♠♦ ✐❧✉str❛ ❛ ✜❣✉r❛ ✶✳✷✳
❋✐❣✉r❛ ✶✳✷✿ ❙❡❣♠❡♥t♦ CG= 2GMC✳
❆ ❝♦♥str✉çã♦ ❞♦ ✐♥❝❡♥tr♦
❖ ✐♥❝❡♥tr♦ é ♦ ♣♦♥t♦ ❡q✉✐❞✐st❛♥t❡ ❛♦s s❡✉s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ♦✉ s❡❥❛✱ é ♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ✐♥s❝r❡✈❡ ❡ss❡ tr✐â♥❣✉❧♦✳ ❆ ❝♦♥str✉çã♦ q✉❡ ♥♦s ❢❛③ ❡♥✲ ❝♦♥tr❛r ❡ss❡ ♣♦♥t♦ é ❜❛st❛♥t❡ ✐♥t❡r❡ss❛♥t❡ ♣♦✐s ♣♦❞❡♠♦s tr❛❜❛❧❤❛r ✈ár✐♦s ❝♦♥❝❡✐t♦s ❝♦♠♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✱ ❜✐ss❡tr✐③✷✵ ❡ ❞✐stâ♥❝✐❛ ❡♥tr❡ ✉♠❛ r❡t❛ ❛ ✉♠ ♣♦♥t♦✳
❙❛❜❡✲s❡ q✉❡ ❛ ❜✐ss❡tr✐③✱ ❛❧é♠ ❞❡ s❡r ❛ q✉❡ ❞✐✈✐❞❡ ✉♠ â♥❣✉❧♦ ❡♠ ❞♦✐s â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s✱ t❛♠❜é♠ é ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❡q✉✐❞✐st❛♥t❡s ❛ ❞✉❛s r❡t❛s✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞✐stâ♥❝✐❛ ❡♥tr❡ ✉♠ ♣♦♥t♦ ❡ ✉♠❛ r❡t❛ ♣♦❞❡ s❡r ❛♣r❡s❡♥t❛❞♦ ❡ tr❛❜❛❧❤❛❞♦ ❝♦♠ ♦s ❡st✉❞❛♥t❡s✱ q✉❡ ❞❛rá s❡❣✉r❛♥ç❛ ♣❛r❛ ♦s ♠❡s♠♦s ✉t✐❧✐③❛r❡♠ ❛ ❢❡rr❛♠❡♥t❛ ✧r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r✧✳ P❛r❛ r❡♣r♦❞✉③✐r ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s s✉♣r❛❝✐t❛❞♦s✱ ❝♦♠♦ s✉❣❡stã♦✱ ❢❛r❡♠♦s ❡ss❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❝♦♥str✉çã♦ ❞❡ ❢♦r♠❛ ✐♥t✉✐t✐✈❛ ❡ ❞✐♥â♠✐❝❛✳ ❆♣r❡s❡♥t❛r❡♠♦s✱ ❡①❝❡♣❝✐♦♥❛❧♠❡♥t❡✱ ♥❡st❡ ❡①❡♠♣❧♦✱ ♦s ♣❛ss♦s ❛❜❛✐①♦ q✉❡ ❡stã♦ ✐❧✉str❛❞❛s ♥❛ ✜❣✉r❛ ✶✳✸✿
✶✳ ❆❝✐♦♥❡ ❛ ❢❡rr❛♠❡♥t❛ ✧❝♦♥tr♦❧❡ ❞❡s❧✐③❛♥t❡✧❡ ✜①❡ s❡✉s ❧✐♠✐t❡s ❡♥tr❡ ✵ ❡ ✶✵❀
✶✾❈❤❛♠❛♠♦s ❞❡ ♥ã♦✲❝♦♥✈❡①♦ ♦ ♣♦❧í❣♦♥♦ q✉❡ ❛♣r❡s❡♥t❛ ✈ért✐❝❡s ♥♦s ❞♦✐s s❡♠✐✲♣❧❛♥♦s ❞❡✜♥✐❞♦s
♣❡❧❛ r❡t❛ q✉❡ ❝♦♥té♠ ❛❧❣✉♠ ❞♦s ❧❛❞♦s✳
✷✵➱ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r ✉♠ ✈ért✐❝❡ ❞♦ tr✐â♥❣✉❧♦ ❞✐✈✐❞✐♥❞♦ ♦ â♥❣✉❧♦ ✐♥t❡r♥♦ ❣❡r❛❞♦ ♣❡❧♦s ❧❛❞♦s
❛❞❥❛❝❡♥t❡s ❛ ❡ss❡ ✈ért✐❝❡ ❡♠ ❞♦✐s â♥❣✉❧♦s ✐❣✉❛✐s✳ P♦❞❡♠♦s ❞✐③❡r t❛♠❜é♠ q✉❡✱ ❞❛❞❛s ❞✉❛s r❡t❛s ❝♦♥❝♦rr❡♥t❡s✱ ❛ ❜✐ss❡tr✐③ é ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❡q✉✐❞✐st❛♥t❡ ❛ ❡ss❛s r❡t❛s✳
❋✐❣✉r❛ ✶✳✸✿ ▲✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❡q✉✐❞✐st❛♥t❡s ❛♦s ❧❛❞♦s AB ❡ AC r❡♣r❡✲
s❡♥t❛❞♦s ♣❡❧❛ ❢❡rr❛♠❡♥t❛ ✧r❛str♦✧❛t✐✈❛❞❛ ♥♦ ●❡♦❣❡❜r❛✳
❋✐❣✉r❛ ✶✳✹✿ ❈♦♥str✉çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ♥♦ tr✐â♥❣✉❧♦ABC ♥♦ ●❡♦❣❡❜r❛✳
✷✳ ❈♦♠ ❛ ❢❡rr❛♠❡♥t❛ ✧❝✐r❝✉❧♦ ❞❛❞♦s ❝❡♥tr♦ ❡ r❛✐♦✧✱ tr❛❝❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝♦♠ ❝❡♥tr♦ ❡♠ A✭♦✉ B✱ ♦✉C✮ ❡ ✜①❡ ♦ r❛✐♦ ❂ ❛ ✭✈❛❧♦r ❞♦ ❝♦♥tr♦❧❡ ❞❡s❧✐③❛♥t❡✮✱ ❧♦❣♦
❛♣ós✱ ♠❛rq✉❡ ♦s ♣♦♥t♦s D ❡ E✱ ✐♥t❡rs❡❝çõ❡s ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝♦♠ ♦s ❧❛❞♦s AC ❡ AB ❞♦ tr✐â♥❣✉❧♦❀
✸✳ ❈♦♠ ❛ ❢❡rr❛♠❡♥t❛ ✧r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r✧❛t✐✈❛❞❛✱ tr❛❝❡ ❛s r❡t❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛♦s ❧❛❞♦s q✉❡ ♣❛ss❡♠ ♣❡❧♦s ♣♦♥t♦D❡E❡ ♠❛rq✉❡ ♦ ♣♦♥t♦F✱ ✐♥t❡rs❡❝çã♦ ❡♥tr❡
❛s r❡t❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s❀
✹✳ P♦s✐❝✐♦♥❡ ♦ ❝♦♥tr♦❧❡ ❞❡s❧✐③❛♥t❡ ♥❛ ♣♦s✐çã♦ 0✱ ❛t✐✈❡ ❛ ❢❡rr❛♠❡♥t❛ ✧r❛str♦✧✱
❛❝✐♦♥❡✲♦ ♥♦ ♣♦♥t♦ F ❡ ❧✐❣✉❡ ❛ ✧❆♥✐♠❛çã♦✧❞♦ ❝♦♥tr♦❧❡ ❞❡s❧✐③❛♥t❡✳
❆ ✜❣✉r❛ ✶✳✹✱ ✐❧✉str❛ ❛ ❝♦♥str✉çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ♥♦ tr✐â♥❣✉❧♦✳ ❆s ♠❡s♠❛s ❛t✐✈✐❞❛❞❡s r❡❛❧✐③❛❞❛s ♥❛ s❡çã♦ ❛♥t❡r✐♦r ❝♦♠ ♦ ❜❛r✐❝❡♥tr♦ ♣♦❞❡rã♦ s❡r r❡♣r♦✲
❞✉③✐❞❛s ❝♦♠ ♦ ✐♥❝❡♥tr♦✳ ❆❧é♠ ❞❡ ♦r✐❡♥t❛r ♦s ❛❧✉♥♦s ♣❛r❛ ❛✈❡r✐❣✉❛r s❡I é r❡❛❧♠❡♥t❡
✐♥t❡r♥♦ ❡ s❡ é ♦ ú♥✐❝♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛s três ❜✐ss❡tr✐③❡s ❝♦♠ ❛ ♠❛♥✐♣✉❧❛çã♦ ❞♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ é ✉♠❛ ❜♦❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❝♦♥str✉✐r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡s✲ s❛s ♣r♦♣r✐❡❞❛❞❡s✳
❆ ❞❡♠♦♥str❛çã♦ ❛❧❣é❜r✐❝❛ ❞❡ss❛s ♣r♦♣r✐❡❞❛❞❡s s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ♥♦ ❝❛♣ít✉❧♦ ✸✱ s❡çã♦ ✸✳✷✱ ♣á❣✐♥❛ ✹✼✳
❆ ❝♦♥str✉çã♦ ❞♦ ❝✐r❝✉♥❝❡♥tr♦
❖ ❝✐r❝✉♥❝❡♥tr♦ é ♦ ♣♦♥t♦ O ❡q✉✐❞✐st❛♥t❡ ❞♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ❧♦❣♦ é ♦
❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ♦ ❝✐r❝✉♥s❝r❡✈❡✳ P❛r❛ ❡♥❝♦♥tr❛r ❡st❡ ♣♦♥t♦✱ é ♥❡❝❡ssár✐♦ tr❛ç❛r ❛s ♠❡❞✐❛tr✐③❡s✷✶ ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ❡ ♦❜t❡r ♦ ú♥✐❝♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦
❡♥tr❡ ❡❧❛s✳
❆ ✜❣✉r❛ ✶✳✺✱ ✐❧✉str❛ ❛ ❝♦♥str✉çã♦ ❞❡ss❡ ♣♦♥t♦ q✉❡✱ ♣❡❧❛ s✉❛ s✐♠♣❧✐❝✐❞❛❞❡✱ ❞❡✐①❛✲ ♠♦s ❝♦♠♦ ❡①❡r❝í❝✐♦ ❛♦s ❧❡✐t♦r❡s q✉❡ ♣♦❞❡♠ ❛✐♥❞❛ ❝♦♥str✉✐r ♦✉tr❛s ❛t✐✈✐❞❛❞❡s ✐♥t❡r❛✲ t✐✈❛s ❝♦♠ ♦ s♦❢t✇❛r❡✳ ❆s ❞❡♠♦♥str❛çõ❡s ❛❧❣é❜r✐❝❛s ❡stã♦ ❛♣r❡s❡♥t❛❞❛s ♥❛ s❡çã♦ ✸✳✸✱ ♣á❣✐♥❛ ✹✾✳
❋✐❣✉r❛ ✶✳✺✿ ❚r✐â♥❣✉❧♦ ❆❇❈ ❝♦♠ ♦ ❝✐r❝✉♥❝❡♥tr♦ O✳
✷✶❘❡t❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛♦s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❡ ❝r✉③❛♠ ❡ss❡s ❧❛❞♦s ♥♦ s❡✉ ♣♦♥t♦ ♠é❞✐♦✳
P♦❞❡♠♦s t❛♠❜é♠ ❞❡✜♥✐r ♠❡❞✐❛tr✐③ ❝♦♠♦ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❡q✉✐❞✐st❛♥t❡ ❞❛s ❡①tr❡♠✐❞❛❞❡s ❞❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❬✶✷❪✳
❆ ❝♦♥str✉ç❛♦ ❞♦ ♦rt♦❝❡♥tr♦
❖ ♦rt♦❝❡♥tr♦ é ♦ ú♥✐❝♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛s ❛❧t✉r❛s✷✷ ❞❡ ✉♠ tr✐â♥❣✉❧♦✳ P❡❧❛
❡t✐♠♦❧♦❣✐❛ ❞❛ ♣❛❧❛✈r❛✱ q✉❡ é ✉♠❛ ❝♦♠♣♦s✐çã♦ ❞❡ ♦rt♦ ✭❞♦ ❣r❡❣♦ ♦rt❤♦s✱ q✉❡✱ ♥❛ ❣❡♦♠❡tr✐❛✱ s✐❣♥✐✜❝❛ r❡t♦✮ ❡ ❝❡♥tr♦✱ ♣♦r r❛③õ❡s ó❜✈✐❛s✱ ♣❡r❝❡❜❡✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡ s❡ tr❛t❛ ❞❡ ✉♠ ♣♦♥t♦ q✉❡ é ❡♥❝♦♥tr❛❞♦ ♣❡❧♦ ❝r✉③❛♠❡♥t♦ ❞❡ s❡❣♠❡♥t♦s ♦rt♦❣♦♥❛✐s✱ q✉❡✱ ♥♦ s❡✉ ❝❛s♦✱ ♣❛ss❛ ♣❡❧♦s ✈ért✐❝❡s ♦♣♦st♦s✳
P❛r❛ tr❛ç❛r♠♦s ❡ss❛s ❛❧t✉r❛s ♥♦ ●❡♦❣❡❜r❛✱ ❞❡ ❢♦r♠❛ s✐♠♣❧❡s✱ ❝♦♥str✉í♠♦s ❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ✉♠ ❞♦s ❧❛❞♦s q✉❡ ♣❛ss❛ ♣❡❧♦ ✈ért✐❝❡ ♦♣♦st♦ ❛ ❡ss❡ ❧❛❞♦✱ ✐♥❝❧✉s✐✈❡✱ ♥♦ ❝❛s♦ ❞♦ tr✐â♥❣✉❧♦ ♦❜t✉sâ♥❣✉❧♦✷✸✱ q✉❡✱ ♥♦s ❧❛❞♦s q✉❡ ❝♦♠♣õ❡ t❛❧ â♥❣✉❧♦✱ ♥ã♦ ❛♣r❡s❡♥t❛
♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦✱ ♣♦✐s✱ ♥❡ss❡ ❝❛s♦✱ ❛ ♣❡r♣❡♥❞✐❝✉❧❛r ♥ã♦ ❝♦♥❝♦rr❡ ❝♦♠ ❡ss❡ ❧❛❞♦ ❞♦ tr✐â♥❣✉❧♦✳ P❛r❛ ♦❜s❡r✈❛r♠♦s ♦ ♣é ❞❛ ❛❧t✉r❛ ❡♠ q✉❛❧q✉❡r t✐♣♦ ❞❡ tr✐â♥❣✉❧♦✱ é ♥❡❝❡ssár✐♦ q✉❡ ♦ ❡st✉❞❛♥t❡ tr❛❝❡ ❛ r❡t❛ q✉❡ ❝♦♥té♠ ♦ ❧❛❞♦ q✉❡ ❝♦♠♣õ❡ ♦ â♥❣✉❧♦ ♦❜t✉s♦ ❡ ♠❛rq✉❡ ❛ ✐♥t❡rs❡❝çã♦ ❞❡ss❛ r❡t❛ ❝♦♠ ❛ s✉❛ ♣❡r♣❡♥❞✐❝✉❧❛r q✉❡ ♣❛ss❛ ♣❡❧♦ ✈ért✐❝❡ ♦♣♦st♦✳
❋✐❣✉r❛ ✶✳✻✿ ❈♦♥str✉çã♦ ❞♦ ♦rt♦❝❡♥tr♦H ❡①t❡r✐♦r ❛♦ tr✐â♥❣✉❧♦ABC ♥♦ ●❡♦❣❡❜r❛✳
◆❛ ✜❣✉r❛ ✶✳✻✱ ✈❡♠♦s ✉♠ ❝❛s♦ ❞♦ ♦rt♦❝❡♥tr♦ H ❞♦ tr✐â♥❣✉❧♦ ♦❜t✉sâ♥❣✉❧♦ ABC✳
❊ss❛ ❝♦♥str✉çã♦ ♣❡r♠✐t❡ ❛♦s ❡❞✉❝❛♥❞♦s✱ ♠❛♥✐♣✉❧❛r ♦s ✈ért✐❝❡s ❛té ❝♦✐♥❝✐❞✐✲❧♦s ❝♦♠
H✳ ❆ ❢❡rr❛♠❡♥t❛ ✧➶♥❣✉❧♦✧❞❛rá ❛ ✐♥❢♦r♠❛çã♦ ♦❜✈✐❛ q✉❡ ❡ss❡ â♥❣✉❧♦ é ❞❡ ✾✵♦✳ ❆s
❞❡♠♦♥str❛çõ❡s ❛❧❣é❜r✐❝❛s ❢♦r❛♠ ❛♣r❡s❡♥t❛❞❛s ♥❛ s❡çã♦ ✸✳✹✱ ♣á❣✐♥❛ ✺✵✳
✷✷❆ ❛❧t✉r❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ✈ért✐❝❡ é ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❞❡✜♥✐❞❛ ♣♦r ❡ss❡ ✈ért✐❝❡
❡ ✉♠ ♣♦♥t♦ ♣❡rt❡♥❝❡♥t❡ à r❡t❛ ❛ q✉❛❧ ❝♦♥té♠ s❡✉ ❧❛❞♦ ♦♣♦st♦✱ ❞❡ ♠❡♥♦r ❝♦♠♣r✐♠❡♥t♦✱ ♦✉ s❡❥❛✱ é ♦ s❡❣♠❡♥t♦ ♣❡r♣❡♥❞✐❝✉❧❛r à r❡t❛ q✉❡ ❝♦♥té♠ ✉♠ ❞♦s ❧❛❞♦s q✉❡ ♣❛ss❛ ♣❡❧♦ ✈ért✐❝❡ ♦♣♦st♦ ❛ ❡ss❡ ❧❛❞♦✳
✷✸❚r✐â♥❣✉❧♦ q✉❡ ❛♣r❡s❡♥t❛ ✉♠ ❞❡ s❡✉s â♥❣✉❧♦s ❝♦♠ ✈❛❧♦r ♠❛✐♦r q✉❡ ✾✵♦✱ ♥❡ss❡ ❝❛s♦✱ ❡♠ ✉♠ ❞♦s
❧❛❞♦s q✉❡ ❢♦r♠❛♠ ❡ss❡ â♥❣✉❧♦ ♦❜t✉s♦✱ ♦ ♣é ❞❛ ❛❧t✉r❛ ✭❡①tr❡♠✐❞❛❞❡ ❞❛ ❛❧t✉r❛ ♦♣♦st❛ ❛♦ ✈ért✐❝❡✮ ❡stá ❧♦❝❛❧✐③❛❞♦ ♥♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❡ss❡ ❧❛❞♦✳
◆♦s ❧✐✈r♦s ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛✱ ♥ã♦ ❡♥❝♦♥tr❛♠♦s ❛❧❣✉♠❛ ✧✉t✐✲ ❧✐❞❛❞❡✧❡s♣❡❝✐❛❧ ♣❛r❛ ❡ss❡ ♣♦♥t♦✳ P❛r❛ ♦ ❡❞✉❝❛❞♦r q✉❡ t❡♠ ❝♦♠♦ ❜❛s❡ ❛ ❡❞✉❝❛çã♦ ❝♦♥str✉t✐✈❛ ✭♦✉ ❝♦♥str✉t✐✈✐st❛✮✱ é ❜❛st❛♥t❡ ❝♦♠♣❧✐❝❛❞♦ ❛♣r❡s❡♥t❛r ✉♠ ❝♦♥❝❡✐t♦✱ ♣r♦✲ ♣r✐❡❞❛❞❡ ♦✉ ❡♥t❡ ❣❡♦♠étr✐❝♦ s❡♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ♣rát✐❝❛✳ ❊ss❛ ✧❝❛rê♥❝✐❛✧♣♦❞❡r✐❛ s❡r ❛❜r❛♥❞❛❞❛ s❡ ❢♦ss❡♠ ✐♥❝❧✉í❞❛s ♥♦ ❝✉rrí❝✉❧♦ ❞❡st❡ ♥í✈❡❧ ❡❞✉❝❛❝✐♦♥❛❧✱ ❝♦♥t❡ú❞♦s ♠❛✐s ❛♣r♦❢✉♥❞❛❞♦s ❞❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ q✉❡ tr❛r✐❛♠ ♠❛✐s ❜❡❧❡③❛ ❡ ❝✉r✐♦s✐❞❛❞❡✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ r❡t❛ ❞❡ ❊✉❧❡r✱ q✉❡ ✈❡r❡♠♦s ❛ s❡❣✉✐r✳
✶✳✷✳✷ ❆ ❝♦♥str✉çã♦ ❞❛ r❡t❛ ❞❡ ❊✉❧❡r
❋✐❣✉r❛ ✶✳✼✿ ❈♦♥str✉çã♦ ❞♦s q✉❛tr♦ ♣♦♥t♦s ♥♦tá✈❡✐s O, G, H ❡ I ❞♦ tr✐â♥❣✉❧♦ ABC
♥♦ ●❡♦❣❡❜r❛✳
P❛r❛ ❝♦♥str✉✐r ❛ r❡t❛ ❞❡ ❊✉❧❡r ♥♦ ●❡♦❣❡❜r❛✱ é ✐♠♣♦rt❛♥t❡✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ❡♥✲ ❝♦♥tr❛r ♦s ✹ ♣♦♥t♦s ♥♦tá✈❡✐s ❞♦ tr✐â♥❣✉❧♦✳ ❊ss❛ ❝♦♥str✉çã♦✱ ❛♣❛r❡♥t❡♠❡♥t❡ ❝❛✉s❛rá ✉♠❛ ❝❡rt❛ ♣♦❧✉✐çã♦ ✈✐s✉❛❧ q✉❡ ♣♦❞❡rá s❡r ❛♠❡♥✐③❛❞❛ ❝♦♠ ❛ ♣❛❞r♦♥✐③❛çã♦ ❞❡ ❝♦r❡s ❞✐❢❡r❡♥t❡s ♣❛r❛ ❝❛❞❛ ❣r✉♣♦ ❞❡ r❡t❛s✱ ❝♦♠♦ ✐❧✉str❛♠♦s ♥❛ ✜❣✉r❛ ✶✳✼✳ ❖ ❡st✉❞❛♥t❡ ♣❡r❝❡❜❡rá q✉❡✱ ❛♦ ♠❛♥✐♣✉❧❛r ♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦✱ ♦s ♣♦♥t♦s s❡ ❛♣r♦①✐♠❛♠ ♦✉ s❡ ❛❢❛st❛♠✱ ✉♠ ❞♦s ♦✉tr♦s✳ ❈❛❜❡ ❛♦ ❡❞✉❝❛❞♦r✱ ❧❡✈❛♥t❛r ✉♠ q✉❡st✐♦♥❛♠❡♥t♦✿ s❡rá q✉❡ ❡①✐st❡ ✉♠ t✐♣♦ ❞❡ tr✐â♥❣✉❧♦ ❝✉❥♦s ♣♦♥t♦s ♥♦tá✈❡✐s ❝♦✐♥❝✐❞❡♠❄ ❡ss❛ ❛t✐✈✐❞❛❞❡ ✐♥st✐❣❛rá ♦ ❛❧✉♥♦ ❛ ❜✉s❝❛r✱ ❝♦♠ ♦ ♠á①✐♠♦ ❞❡ t❡♥t❛t✐✈❛s ♣♦ssí✈❡❧✱ ✉♥✐r ❡ss❡s ♣♦♥t♦s✳
❆t✐✈✐❞❛❞❡s ❝♦♠♦ ❡ss❛✱ ❧❡✈❛♥t❛rã♦ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ tr❛❜❛❧❤❛r ❛❧❣✉♥s ♦✉tr♦s ❝♦♥✲ ❝❡✐t♦s ❝♦♠♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s ❡ ❡q✉✐❧át❡r♦s✳ ❯♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ❡♥❝♦♥tr❛r✱ r❛♣✐❞❛♠❡♥t❡✱ ♣♦s✐çõ❡s ❝♦♥✈❡♥✐❡♥t❡s ♣❛r❛ ♦s ✈ért✐❝❡s t❛✐s q✉❡ r❡s♦❧✈❛ ❡ss❡ ♣r♦❜❧❡♠❛ é ❛❝✐♦♥❛r ♦ ❜♦tã♦ ✧â♥❣✉❧♦✧♥♦s três ✈ért✐❝❡s ❡✱ ❛ ♠❡❞✐❞❛ q✉❡ ❛ ❞✐❢❡✲ r❡♥ç❛ ❡♥tr❡ ♦s â♥❣✉❧♦s ❢♦r❡♠ ❞✐♠✐♥✉✐♥❞♦✱ ❡ss❡s ♣♦♥t♦s s❡ ❛♣r♦①✐♠❛♠✱ ❛té q✉❛♥❞♦ s❡ t♦r♥❛♠ ♦ ♠❡s♠♦ ♣♦♥t♦✱ ♦ q✉❡ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ♦s â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s ❡♥tr❡ s✐✱ ♦✉ s❡❥❛✱ ❞❡ ✻✵♦✳ P❛r❛ t❛♥t♦✱ ♦ ❛❧✉♥♦ t❡rá q✉❡ ❧❡✈❛♥t❛r✱ ♣r❡❧✐♠✐♥❛r♠❡♥t❡✱ ❛ ❝♦♥❥❡❝t✉r❛
❞❡ q✉❡ ❡ss❡ ❝❛s♦ ♦❝♦rr❡rá ❝♦♠ ♦ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦✳
❋✐❣✉r❛ ✶✳✽✿ ❚r✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❝♦♠ ♦s q✉❛tr♦ ♣♦♥t♦s ♥♦tá✈❡✐s O, G, H ❡ I ❞♦
tr✐â♥❣✉❧♦ ABC ❝♦✐♥❝✐❞❡♥t❡s r❡♣r❡s❡♥t❛❞♦s ♥♦ ●❡♦❣❡❜r❛✳
P♦ré♠✱ ❝♦♥✈é♠ s❛❧✐❡♥t❛r q✉❡ ❡♥❝♦♥tr❛r ❡ss❡ ♣♦s✐❝✐♦♥❛♠❡♥t♦ ❞♦s ✈ért✐❝❡s✱ r❡q✉✐✲ s✐t❛ ✉♠❛ ♣❡rí❝✐❛ ♠❛s ❛✜♥❛❞❛ ♣♦r ♣❛rt❡ ❞♦s ❛♣r❡♥❞✐③❡s✱ ❡ ♣♦❞❡ ❞✉r❛r ✉♠ t❡♠♣♦ ♠❛✐♦r✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ s❡ ♦ ❛rr❡❞♦♥❞❛♠❡♥t♦ ❞♦ â♥❣✉❧♦ ❡st✐✈❡r ❡♠ ✷ ❝❛s❛s ❞❡❝✐♠❛✐s✱ ♥❡ss❡ ❝❛s♦✱ ♣♦❞❡♠♦s ♣❡r♠✐t✐r q✉❡ ❤❛❥❛ ✉♠ ❛rr❡❞♦♥❞❛♠❡♥t♦ ♣❛r❛ ✶ ♦✉ ❛té ♥❡♥❤✉♠❛ ❝❛s❛ ❞❡❝✐♠❛❧✳
❙❡ ♦ ♣r♦❢❡ss♦r ♥ã♦ q✉✐s❡r ❧❡✈❛♥t❛r ❞❡ ✐♠❡❞✐❛t♦ ❡ss❛ ❝♦♥❥❡❝t✉r❛✱ ♣♦❞❡ t❛♠❜é♠ s✉✲ ❣❡r✐r q✉❡ s❡ tr❛❝❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ❡♠ q✉❛❧q✉❡r ✉♠ ❞♦s ♣♦♥t♦s ♥♦tá✈❡✐s✱ ❡①❝❡t♦ ♦ ❝✐r❝✉♥❝❡♥tr♦✱ ❝♦♠♦ ✐❧✉str❛♠♦s ♥❛ ✜❣✉r❛ ✶✳✽✱ q✉❡ ♣❛ss❡ ♣♦r q✉❛❧q✉❡r ✉♠ ❞♦s ✈ért✐❝❡s✳ ◆❡ss❡ ❝❛s♦✱ ♦ ❡st✉❞❛♥t❡ ❞❡✈❡ ♠♦✈❡r ♦s ✈ért✐❝❡s ❛té q✉❡ t♦❞♦s ❡st❡❥❛♠ ✐♥❝❧✉✲ s♦s ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ q✉❛♥❞♦ ❡❧❡ ❝♦♥❝❧✉✐r✱ ♦❜s❡r✈❛rá q✉❡ ♦s ♣♦♥t♦s ♥♦tá✈❡✐s ❡st❛rã♦ ❝♦✐♥❝✐❞✐♥❞♦✳ ❊ss❛ ❛t✐✈✐❞❛❞❡ ♣♦❞❡ s❡r r❡♣❡t✐❞❛ ✈❛r✐❛♥❞♦ ♦ ♣♦♥t♦ ♥♦tá✈❡❧ ❡ ♦ ✈ért✐❝❡ q✉❡ ❞❡t❡r♠✐♥❛rã♦ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❆♣ós ❡ss❛ ❡t❛♣❛✱ ♦ ♣r♦❢❡ss♦r ♣♦❞❡✱ ❡♥tã♦✱ s✉❣❡r✐r ❛ ♠❡❞✐❞❛ ❞♦s â♥❣✉❧♦s ♦✉ ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ♣❛r❛ q✉❡ ♦s ❡st✉❞❛♥t❡s t✐r❡♠ s✉❛s ❝♦♥❝❧✉sõ❡s✳
❆♣ós ♠❛♥✐♣✉❧❛r ♠✉✐t♦ ❡ss❛ ú❧t✐♠❛ ❝♦♥str✉çã♦✱ ♦ ♣r♦❢❡ss♦r ♣♦❞❡ s✉❣❡r✐r ❛♦s ❡❞✉✲ ❝❛♥❞♦s q✉❡ ♦❜s❡r✈❡♠ ♦ ♣♦s✐❝✐♦♥❛♠❡♥t♦ ❞❡H, I, G❡O ❡ ♣r♦❝✉r❡ ❛❧❣✉♠ ❛❧✐♥❤❛♠❡♥t♦
❡♥tr❡ ❡❧❡s✳ ❖r❛✱ s❛❜❡♠♦s q✉❡ s❡♠♣r❡ ❤❛✈❡rá ❛❧✐♥❤❛♠❡♥t♦ ❡♥tr❡ ❞♦✐s ❞❡❧❡s✱ ♣♦r ✐ss♦✱ é ✐♠♣♦rt❛♥t❡ ♣r♦❝✉r❛r ❡ss❡ ❛❧✐♥❤❛♠❡♥t♦ ❡♠✱ ♥♦ ♠í♥✐♠♦ três ❞❡❧❡s✳ ❯♠❛ ❜♦❛ s✉❣❡s✲ tã♦ é tr❛ç❛r ✉♠❛ r❡t❛ ❞❡t❡r♠✐♥❛❞❛ ♣♦r ❞♦✐s ❞♦s q✉❛tr♦ ♣♦♥t♦s ❡ ♦❜s❡r✈❛r s❡ ❡①✐st❡ t❛❧ ❛❧✐♥❤❛♠❡♥t♦✳ ❆♣ós ❛❧❣✉♠❛s t❡♥t❛t✐✈❛s✱ ❡❧❡s ♣❡r❝❡❜❡rã♦ q✉❡ t❡rã♦ ❞✐✜❝✉❧❞❛❞❡s ♣❛r❛ tr❛ç❛r ✉♠❛ r❡t❛ q✉❡ ❝♦♥t❡♥❤❛ I ❡ ♦✉tr♦s ❞♦✐s ♣♦♥t♦s✳ P♦❞❡♠♦s ❡♥tã♦ ✐♥s❡r✐r ♦
❝♦♥❝❡✐t♦ ❞❛ r❡t❛ ❞❡ ❊✉❧❡r✳
❱✐♠♦s ♥❛ s❡çã♦ ✶✳✶✳✶✱ ♣á❣✐♥❛ ✶✱ q✉❡ ❊✉❧❡r ❞❡s❝♦❜r✐✉ ❝♦♠ ✉♠ ♠ét♦❞♦ ♠✉✐t♦ s❡♠❡❧❤❛♥t❡✱ t♦❞❛✈✐❛✱ s❡♠ ♦ ❝♦♠♣✉t❛❞♦r✱ ♥❛❞❛ ♣rát✐❝♦✱ q✉❡ ♦s ♣♦♥t♦s O, G ❡ H
❋✐❣✉r❛ ✶✳✾✿ ❚r✐â♥❣✉❧♦ ABC ❝♦♠ ♦s q✉❛tr♦ ♣♦♥t♦s ♥♦tá✈❡✐sO, G, H ❡ I ❡ ❛ r❡t❛ ❞❡
❊✉❧❡r ♥♦ ●❡♦❣❡❜r❛✳
♣❡rt❡♥❝❡♠ ❛ ✉♠❛ r❡t❛ q✉❡ r❡❝❡❜❡✉ ♦ ♥♦♠❡ ❞❡ ❘❡t❛ ❞❡ ❊✉❧❡r✳ ❚♦❞❛✈✐❛✱ é ♥❡❝❡ssár✐♦ ❛ ❞❡♠♦♥str❛çã♦ ❛❧❣é❜r✐❝❛ ❞❡ss❛ ❞❡s❝♦❜❡rt❛✱ q✉❡ ❛♣r❡s❡♥t❛r❡♠♦s ♥❛ s❡çã♦ ✸✳✺✱ ♣á❣✐♥❛ ✺✶✳
❖ ✐♥❝❡♥tr♦ ❡ ❛ r❡t❛ ❞❡ ❊✉❧❡r
❆ ✜❣✉r❛ ✶✳✾✱ ✐❧✉str❛ ❛ r❡t❛ ❞❡ ❊✉❧❡r ❡ ♦s ♣♦♥t♦s ♥♦tá✈❡✐s ❞♦ tr✐â♥❣✉❧♦ ❡ ❡ss❛ ❝♦♥s✲ tr✉çã♦ ✐♥st✐❣❛ ✉♠ q✉❡st✐♦♥❛♠❡♥t♦ q✉❡ ♣♦❞❡ s❡r ❧❡✈❛♥t❛❞♦ ♣❛r❛ ♦ ❡st✉❞❛♥t❡✿ s❡rá q✉❡ I ♣♦❞❡ ♣❡rt❡♥❝❡r à r❡t❛ ❞❡ ❊✉❧❡r❄
❉✐❛♥t❡ ❞❡ss❡ ♣r♦❜❧❡♠❛✱ ❡s♣❡r❛✲s❡ q✉❡ ♦ ❡st✉❞❛♥t❡ ❞❡s♣❡rt❡ ❛ ❝✉r✐♦s✐❞❛❞❡ ❞❡ ❡♥✲ ❝♦♥tr❛r ❡ss❛ s✐t✉❛çã♦ ❡s♣❡❝✐❛❧ ❡✱ ❛♣ós s❡❣✉✐❞❛s t❡♥t❛t✐✈❛s ♣♦❞❡rá ❡♥❝♦♥tr❛r ♦ ú♥✐❝♦ ❝❛s♦ ❡♠ q✉❡ ♦❜s❡r✈❛♠♦s ♦s q✉❛tr♦ ♣♦♥t♦s ❛❧✐♥❤❛❞♦s✳ ❈♦♠ ❛✉①í❧✐♦ ❞❛ ✐❧✉str❛çã♦ ♥❛ ✜❣✉r❛ ✶✳✶✵✱ ♠❡❞✐♥❞♦ ♦s â♥❣✉❧♦s ❞❡ss❡ tr✐â♥❣✉❧♦✱ ❝♦♥❝❧✉✐r❡♠♦s q✉❡ ❡ss❡ é ✐sós❝❡❧❡s✳
➱ ✐♠♣♦rt❛♥t❡ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ss❛ ❝♦♥❥❡❝t✉r❛ ❝r✐❛❞❛ ♣❡❧♦s ❡❞✉❝❛♥❞♦s ♥❡ss❛ ❡t❛♣❛✱ ♠✉✐t♦ ❡❧❡♠❡♥t❛r ♣♦r s✐♥❛❧✱ ❥á q✉❡✱ ♥♦ ❝❛s♦ ❞♦ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s✱ s❛❜❡♠♦s ❞❛s s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ q✉❡ ❛ ❛❧t✉r❛✱ ❛ ♠❡❞✐❛♥❛✱ ❛ ♠❡❞✐❛tr✐③ ❡ ❛ ❜✐ss❡tr✐③ ❡♠ r❡❧❛çã♦ ❛♦ ✈ért✐❝❡ ❝✉❥♦ â♥❣✉❧♦ é ♦♣♦st♦ à ❜❛s❡ sã♦ t♦❞♦s ❝♦✐♥❝✐❞❡♥t❡s✳ ❈♦♠♦ I ♣❡rt❡♥❝❡
à ❜✐ss❡tr✐③✱ ❧♦❣♦ ♣❡rt❡♥❝❡ ❛ t♦❞❛s ❛s ♦✉tr❛s r❡t❛s✱ ❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ t❛♠❜é♠ ♣❡rt❡♥❝❡ à r❡t❛ ❞❡ ❊✉❧❡r ❛ q✉❛❧ ❝♦✐♥❝✐❞❡ ❝♦♠ ❡ss❛s r❡t❛s✱ ❝♦♥❢♦r♠❡ ✈✐♠♦s ♥❛ ú❧t✐♠❛ ✜❣✉r❛ ♠❡♥❝✐♦♥❛❞❛✳
➱ ✐♠♣♦rt❛♥t❡ s❛❧✐❡♥t❛r♠♦s q✉❡✱ s❡ ❡ss❡ tr✐â♥❣✉❧♦ t❛♠❜é♠ ❢♦r ❡q✉✐❧át❡r♦✱ ❡♥tã♦ ♥ã♦ t❡r❡♠♦s ❛ r❡t❛ ❞❡ ❡✉❧❡r✱ ♣♦✐s✱ ❝♦♠♦ ✈✐♠♦s ♥♦ ✐♥í❝✐♦ ❞❡st❛ s✉❜s❡çã♦✱ ♦s q✉❛tr♦ ♣♦♥t♦s ❝♦✐♥❝✐❞❡♠✱ ♥ã♦ ❤❛✈❡♥❞♦✱ ❛ss✐♠✱ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ r❡t❛✳
❊ss❛s ✐♥❢♦r♠❛çõ❡s✱ ❞❡♠♦♥str❛çõ❡s ❡ ❝♦♠❡♥tár✐♦s s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ♥❛ s❡çã♦ ✸✳✺✱
❋✐❣✉r❛ ✶✳✶✵✿ ❚r✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❝♦♠ ♦s q✉❛tr♦ ♣♦♥t♦s ♥♦tá✈❡✐s O, G, H ❡ I ♥❛ r❡t❛
❞❡ ❊✉❧❡r r❡♣r❡s❡♥t❛❞♦s ♥♦ ●❡♦❣❡❜r❛✳
✺✶✳
✶✳✷✳✸ ❆ ❝♦♥str✉çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s
❆ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦s ♥♦✈❡ ♣♦♥t♦s✱ ❝♦♥❢♦r♠❡ ❛♣r❡s❡♥t❛❞♦ ♥❛ s❡çã♦ ✶✳✶✱ ♣á❣✐♥❛ ✶✱ ❝♦♥té♠ ♥♦✈❡ ♣♦♥t♦s ❡s♣❡❝✐❛✐s✱ s❡✐s ❞♦s q✉❛✐s ❥á ❛♣r❡s❡♥t❛❞♦s✿ ❖s três ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s ❡ ♦s três ♣és ❞❡ ❛❧t✉r❛✳ ❖s ♦✉tr♦s três ♣♦♥t♦s sã♦ ♦s P♦♥t♦s ❞❡ ❊✉❧❡r✳
❆ ❝♦♥str✉çã♦ ❞♦s ♣♦♥t♦s ❞❡ ❊✉❧❡r
❊ss❡s ♣♦♥t♦s ❞✐✈✐❞❡♠ ♦ s❡❣♠❡♥t♦ ❝✉❥❛s ❡①tr❡♠✐❞❛❞❡s é ♦ ♦rt♦❝❡♥tr♦ ❡ ✉♠ ❞♦s ✈ért✐❝❡s ❡♠ ❞♦✐s s❡❣♠❡♥t♦s ❝♦♥❣r✉❡♥t❡s✳ ◆♦ ●❡♦❣❡❜r❛✱ ❝♦♠ ❛ ❢❡rr❛♠❡♥t❛ ✧P♦♥t♦ ▼é❞✐♦✧❛t✐✈❛❞❛✱ é s✉✜❝✐❡♥t❡ ♠❛r❝❛r ❡ss❛s ❡①tr❡♠✐❞❛❞❡s✱ ❝♦♠♦ ✐❧✉str❛♠♦s ♥❛ ✜❣✉r❛ ✶✳✶✶✱ ♣❛r❛ q✉❡ ♦s P♦♥t♦s ❞❡ ❊✉❧❡r EA✱ EB ❡ EC✱ ❡♠ r❡❧❛çã♦ ❛♦s ✈ért✐❝❡s A, B ❡ C✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s❡❥❛♠ ❞❡✜♥✐❞♦s✳
❋✐❣✉r❛ ✶✳✶✶✿ P♦♥t♦s ❞❡ ❊✉❧❡r ❞♦ tr✐â♥❣✉❧♦ ABC r❡♣r❡s❡♥t❛❞♦s ♥♦ ●❡♦❣❡❜r❛✳
