❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲
❆◆❚❖◆■❖ ❊❉❙❖◆ P❊❘❊■❘❆ ❉❆ ❙■▲❱❆ ❋■▲❍❖
❆ ❚❘■●❖◆❖▼❊❚❘■❆ ❊❙❋➱❘■❈❆ ❊ ❖ ●▲❖❇❖ ❚❊❘❘❊❙❚❘❊
❆◆❚❖◆■❖ ❊❉❙❖◆ P❊❘❊■❘❆ ❉❆ ❙■▲❱❆ ❋■▲❍❖
❆ ❚❘■●❖◆❖▼❊❚❘■❆ ❊❙❋➱❘■❈❆ ❊ ❖ ●▲❖❇❖ ❚❊❘❘❊❙❚❘❊
❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛❞♦r✿
Pr♦❢✳ ▼s✳ ❋r❛♥❝✐s❝♦ ❱❛❧❞❡♠✐r♦ ❇r❛❣❛✳
Dados Internacionais de Catalogação na Publicação Universidade Federal do Ceará
Biblioteca do Curso de Matemática
S58t Silva Filho, Antonio Edson Pereira da
A trigonometria esférica e o globo terrestre / Antonio Edson Pereira da Silva Filho. – 2014. 50 f. : il., enc.; 31 cm
Dissertação (mestrado) – Universidade Federal do Ceará, Centro de Ciências, Departamento de Matemática, Programa de Pós-Graduação em Matemática em Rede Nacional, Juazeiro do Norte, 2014. Área de Concentração: Ensino de Matemática.
Orientação: Prof. Francisco Valdemiro Braga.
❆♦ ♠❡✉ q✉❡r✐❞♦ ♣❛✐
❆●❘❆❉❊❈■▼❊◆❚❖❙
❆ ❉❡✉s✱ ♣❡❧❛s ❜ê♥çã♦s ❡ s❛❜❡❞♦r✐❛ ❝♦♥❝❡❞✐❞❛s ❛ ♠❡✉ s❡r✳
❆ ♠❡✉ ♣❛✐✱ ❆♥t♦♥✐♦ ❊❞s♦♥ P❡r❡✐r❛ ❞❛ ❙✐❧✈❛✱ ♠❡s♠♦ ♥ã♦ ❡st❛♥❞♦ ♠❛✐s ❝♦♥♦s❝♦✱ ♣♦r s❡♠♣r❡ ♠❡ ♠♦str❛r ♦ ✈❡r❞❛❞❡✐r♦ ✈❛❧♦r ❞❛ ❡❞✉❝❛çã♦ ❡ ❛ ♠✐♥❤❛ ♠ã❡✱ ❆♥t♦♥✐❛ ❍♦❧❛♥❞❛ ▲❡✐t❡ ❞❛ ❙✐❧✈❛✱ sí♠❜♦❧♦ ❞♦ ✈❡r❞❛❞❡✐r♦ ❛♠♦r✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ❞❡❞✐❝❛çã♦ ❛♦ ❧♦♥❣♦ ❞❡ss❛ ✐♠♣♦♥❡♥t❡ r❡❛❧✐③❛çã♦✳
❆ ◆❡②❧②❛♥♥❡ ❆❧✈❡s ▼❡♥❞♦♥ç❛✱ ♠✐♥❤❛ ❡s♣♦s❛✱ ❡ ❛ ❆❧②❝❡ ❊♠✐❧② ❆❧✈❡s ❞❛ ❙✐❧✈❛✱ ♠✐♥❤❛ ✜❧❤❛✱ r❛③õ❡s ❞♦ ♠❡✉ ✈✐✈❡r✱ ♣♦r t♦❞♦ ❝❛r✐♥❤♦✱ ❛♠♦r✱ ❞❡❞✐❝❛çã♦ ❡✱ ❛❝✐♠❛ ❞❡ t✉❞♦✱ ♣❡❧❛s ❧✐çõ❡s ❞❡ ✈✐❞❛ ❢❛♠✐❧✐❛r✳
❆♦s ♠❡✉s ✐r♠ã♦s✱ s♦❜r✐♥❤♦s ❡ ❝✉♥❤❛❞♦s ♣❡❧❛ ♣❛r❝❡r✐❛ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦s ♠❛✐s ♦❜s❝✉r♦s✳
➚ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭❙❇▼✮ ♣❡❧❛ ✐❞❡❛❧✐③❛çã♦ ❡ ❝♦♦r❞❡♥❛çã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✭P❘❖❋▼❆❚✮✱ ♣r♦❣r❛♠❛ q✉❡✱ ❡♠ ♠é❞✐♦ ♣r❛③♦✱ ❝❛✉s❛rá ✉♠❛ ♣r♦❢✉♥❞❛ ♠❡❧❤♦r✐❛ ♥♦ ❡♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡♠ t♦❞♦s ♦s ♥í✈❡✐s✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚ ✲ ❏✉❛③❡✐r♦ ❞♦ ◆♦rt❡ ♣♦r s✉❛s ❧✐çõ❡s ❞❡ ♠❡str❡s✱ s❡♠♣r❡ ❞✐s♣♦st♦s ❛ s❡ ❞♦❛r ❡♠ ❢❛✈♦r ❞♦ ♠❡✉ ❝r❡s❝✐♠❡♥t♦ ✐♥t❡❧❡❝t✉❛❧ ❛♦ ❧♦♥❣♦ ❞❡ss❛ ❥♦r♥❛❞❛✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ▼ár✐♦ ❞❡ ❆ss✐s ❖❧✐✈❡✐r❛✱ ❡①❡♠♣❧♦ ❞❡ s❡r ❤✉♠❛♥♦✱ ♣♦r s❡✉ ❡s♣❧ê♥❞✐❞♦ ❡♠♣❡♥❤♦ ❛♦ ♣r♦❣r❛♠❛✳
❆ ❋r❛♥❝✐s❝♦ ❱❛❧❞❡♠✐r♦ ❇r❛❣❛✱ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣♦r t♦❞❛ ❝♦♥tr✐❜✉✐çã♦ ✐♥t❡❧❡❝t✉❛❧ ❡ ♣❡❧❛ ❞✐s♣♦s✐çã♦ ❛♦ ❧♦♥❣♦ ❞❛ r❡❛❧✐③❛çã♦ ❞❡ss❛ ♣❡sq✉✐s❛✳
❆♦ ❛♠✐❣♦ ❋r❛♥❝✐s❝♦ ❉✐❛s✱ ♣♦r s✉❛s ❝♦♥tr✐❜✉✐çõ❡s ♣r❡st❛❞❛s à r❡❛❧✐③❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳
❆ t♦❞♦s ♦s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ ♣♦r t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❡ ❛♣r❡♥❞✐③❛❞♦ ❛♦ ❧♦♥❣♦ ❞❡ss❛ ❝❛♠✐♥❤❛❞❛✳ Pr✐♥❝✐♣❛❧♠❡♥t❡✱ ❛♦s ❝♦❧❡❣❛s ❞❡ ✈✐❛❣❡♠ ❈❧❡✐t♦♥ ▼❡❧♦✱ ❘❛❢❛❡❧ ❇r❛③✱ ❲❡s❝❧❡② ❈✉♥❤❛ ❡ ❆♠✐❧t♦♥ ❳✐♠❡♥❡s✱ ✜❝❛ ❛ s❛✉❞❛❞❡ ❞❡ t♦❞♦s ♦s ♠♦♠❡♥t♦s q✉❡ ♣❛rt✐❧❤❛♠♦s✳
❆♦s ❆♠✐❣♦s ▼❛r❝❡❧♦ ❉❛♥t❛s✱ ❉❛②✈✐❞ ▼❛rq✉❡s ❡ ◆✐❧♦ ❈és❛r✱ ♣❡❧❛s ✈❡r❞❛❞❡✐r❛s ❛♠✐③❛❞❡s ❢r✉t♦ ❞❛ ♣r❡s❡♥ç❛ ❞❛ ▼❛t❡♠át✐❝❛ ❡♠ ♥♦ss❛s ✈✐❞❛s✳
❘❊❙❯▼❖
❆ tr✐❣♦♥♦♠❡tr✐❛ ❡s❢ér✐❝❛ s✉r❣✐✉ ❞❛s ♥❡❝❡ss✐❞❛❞❡s ❞❛ ❆str♦♥♦♠✐❛✱ ♥❛ ❜✉s❝❛ ❞❡ ❞❡s❝r❡✈❡r ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♦ s✐st❡♠❛ s♦❧❛r✳ ▼❡♥t❡s ❜r✐❧❤❛♥t❡s ❝♦♠♦ ❊✉❝❧✐❞❡s✱ ❆r✐st❛r❝♦ ❞❡ ❙❛♠♦s✱ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛✱ ❍✐♣❛r❝♦✱ ▼❡♥❡❧❛✉ ❞❡ ❆❧❡①❛♥❞r✐❛✱ Pt♦❧♦♠❡✉✱ ❡♥tr❡ ♦✉tr♦s✱ ❡st✉❞❛r❛♠ s♦❜r❡ ♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s ❛ tr✐❣♦♥♦♠❡tr✐❛ ❡s❢ér✐❝❛ ❜✉s❝❛♥❞♦ ✉♠❛ ❛ss♦❝✐❛çã♦ ❝♦♠ ♦ ❣❧♦❜♦ t❡rr❡str❡✳ ■♥✐❝✐❛r❡♠♦s ❝♦♠ ♦ ❡st✉❞♦ ❞♦s ❡❧❡♠❡♥t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✱ ❞♦♥❞❡ ❞❡✜♥✐r❡♠♦s ♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s ❡ ♣r♦✈❛r❡♠♦s s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s✱ ❝♦♠♦ s♦♠❛ ❞❛s ♠❡❞✐❞❛s ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❡ ❛ ❢ór♠✉❧❛ ❞❡ ●✐r❛r❞ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ s✉❛ ár❡❛✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡✲ s❡♥t❛♠♦s ❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s ❡ ❛s ♣r✐♥❝✐♣❛✐s r❡❧❛çõ❡s ❡♥tr❡ ♦s ❧❛❞♦s ❡ ♦s â♥❣✉❧♦s ❞❡ss❡s tr✐â♥❣✉❧♦s✱ ❝♦♠♦ ❛ ❧❡✐ ❞♦s s❡♥♦s ❡ ❧❡✐ ❞♦s ❝♦ss❡♥♦s✱ ❛❧é♠ ❞❡ ✉♠ ❜r❡✈❡ ❡st✉❞♦ ❞♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s r❡tâ♥❣✉❧♦s✳ ❋✐♥❛❧♠❡♥t❡✱ ❝♦♥s✐❞❡r❛♠♦s ❛ ❚❡rr❛ ❝♦♠♦ ✉♠❛ ❡s❢❡r❛✱ ❞❡♥♦♠✐♥❛❞❛ ❣❧♦❜♦ t❡rr❡str❡✱ s♦❜r❡ ❛ q✉❛❧ ❛❜♦r❞❛♠♦s ❞✐✈❡rs♦s ❝♦♥❝❡✐t♦s ❣❡♦❣rá✜✲ ❝♦s ❝♦♠♦ ♣❛r❛❧❡❧♦s✱ ♠❡r✐❞✐❛♥♦s✱ ❧❛t✐t✉❞❡s✱ ❧♦♥❣✐t✉❞❡s✱ ❛ ✜♠ ❞❡ ✉t✐❧✐③❛r ❞❛ tr✐❣♦♥♦♠❡tr✐❛ ❡s❢ér✐❝❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ❞✐stâ♥❝✐❛s ❡ â♥❣✉❧♦s s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ t❡rr❡str❡✱ ❝r✐❛♥❞♦ ♦ ❢♦rt❡ ❝❛rát❡r ✐♥t❡r❞✐s❝✐♣❧✐♥❛r ❡♥tr❡ ▼❛t❡♠át✐❝❛ ❡ ●❡♦❣r❛✜❛✳
❆❇❙❚❘❆❈❚
❚❤❡ s♣❤❡r✐❝❛❧ tr✐❣♦♥♦♠❡tr② ❝❛♠❡ ❢r♦♠ t❤❡ ♥❡❡❞s ♦❢ ❆str♦♥♦♠②✱ ✐♥ t❤❡ s❡❛r❝❤ ❢♦r ♠❛t❤❡✲ ♠❛t✐❝❛❧❧② ❞❡s❝r✐❜✐♥❣ t❤❡ s♦❧❛r s②st❡♠✳ ❇r✐❧❧✐❛♥t ♠✐♥❞s ❧✐❦❡ ❊✉❝❧✐❞❡s✱ ❆r✐st❛r❝♦ ♦❢ ❙❛♠♦s✱ ❆♣♦❧ô♥✐♦ ♦❢ P❡r❣❛✱ ❍✐♣❛r❝♦✱ ▼❡♥❡❧❛✉ ♦❢ ❆❧❡①❛♥❞r✐❛✱ Pt♦❧♦♠❡✉✱ ❛♥❞ ♦t❤❡rs✱ ❤❛✈❡ st✉❞✐❡❞ t❤❡ s♣❤❡r✐❝❛❧ tr✐❛♥❣❧❡s✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ st✉❞② t❤❡ ❢✉♥❞❛♠❡♥t❛❧ r❡s✉❧ts s♣❤❡r✐❝❛❧ tr✐❣♦♥♦✲ ♠❡tr② s❡❡❦✐♥❣ ❛♥ ❛ss♦❝✐❛t✐♦♥ ✇✐t❤ t❤❡ ❣❧♦❜❡✳ ❲❡ ❜❡❣✐♥ ✇✐t❤ t❤❡ st✉❞② ♦❢ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❡❧❡♠❡♥ts ♦❢ ❛ s♣❤❡r✐❝❛❧ s✉r❢❛❝❡✱ ✇❤❡r❡ ✇❡ ❞❡✜♥❡ t❤❡ s♣❤❡r✐❝❛❧ tr✐❛♥❣❧❡s ❛♥❞ ♣r♦✈❡ t❤❡✐r ✐♠♣♦rt❛♥t ♣r♦♣❡rt✐❡s✱ s✉❝❤ ❛s s✉♠ ♦❢ t❤❡ ♠❡❛s✉r❡s ♦❢ t❤❡ ✐♥t❡r♥❛❧ ❛♥❣❧❡s ❛♥❞ t❤❡ ●✐r❛r❞ ❢♦r♠✉❧❛ t♦ ❝❛❧❝✉❧❛t❡ ✐ts ❛r❡❛✳ ❚❤❡♥✱ ✇❡ ♣r❡s❡♥t t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ s♣❤❡r✐❝❛❧ tr✐❛♥❣❧❡s ❛♥❞ t❤❡ ♠❛✐♥ r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ t❤❡ s✐❞❡s ❛♥❞ ❛♥❣❧❡s ♦❢ t❤❡s❡ tr✐❛♥❣❧❡s✱ ❛s t❤❡ ❧❛✇ ♦❢ s✐♥❡s ❛♥❞ ❧❛✇ ♦❢ ❝♦s✐♥❡s✱ ❛♥❞ ❛ ❜r✐❡❢ st✉❞② ♦❢ s♣❤❡r✐❝❛❧ r❡❝t❛♥❣❧❡ tr✐❛♥❣❧❡s✳ ❋✐♥❛❧❧②✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❊❛rt❤ ❛s ❛ s♣❤❡r❡ ❝❛❧❧❡❞ ❡❛rt❤ ❣❧♦❜❡✱ ♦✈❡r ✇❤✐❝❤ ✇❡ ❛❞❞r❡ss ✈❛r✐♦✉s ❣❡✲ ♦❣r❛♣❤✐❝❛❧ ❝♦♥❝❡♣ts s✉❝❤ ❛s ♣❛r❛❧❧❡❧s✱ ♠❡r✐❞✐❛♥s✱ ❧❛t✐t✉❞❡s✱ ❧♦♥❣✐t✉❞❡s✱ ✐♥ ♦r❞❡r ♦❢ ✉s❡ ♦❢ s♣❤❡r✐❝❛❧ tr✐❣♦♥♦♠❡tr② t♦ ❝❛❧❝✉❧❛t❡ ❞✐st❛♥❝❡s ❛♥❞ ❛♥❣❧❡s ♦♥ t❤❡ ❊❛rt❤✬s s✉r❢❛❝❡✱ ❝r❡❛t✐♥❣ str♦♥❣ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r② ❝❤❛r❛❝t❡r ❜❡t✇❡❡♥ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ●❡♦❣r❛♣❤②✳
❙❯▼➪❘■❖
✶ ■◆❚❘❖❉❯➬➹❖
❖ ✐♥í❝✐♦ ❞♦s ❡st✉❞♦s s♦❜r❡ ❛ ❡s❢❡r❛ ❢♦✐ ❞❡s❡♥❝❛❞❡❛❞♦ ❞❡✈✐❞♦ às ♥❡❝❡ss✐❞❛❞❡s ❞❛ ❆str♦♥♦♠✐❛✱ q✉❡ ❜✉s❝❛✈❛ ❝♦♠♦ s❡r✐❛ ❛ ❢♦r♠❛ ❞❛ ❚❡rr❛ ❡ ♦ ♠♦✈✐♠❡♥t♦ ❞♦s ♣❧❛♥❡t❛s✳ ❖ ♥❛s❝✐♠❡♥t♦ ❞❛ tr✐❣♦♥♦♠❡tr✐❛✱ ✉♠❛ ❝r✐❛çã♦ ❞❛ ▼❛t❡♠át✐❝❛ ❣r❡❣❛✱ ❡st❡✈❡ ❛ss♦❝✐❛❞♦ ❛ ♣r❡✲ ✈✐sã♦ ❞❡ ❡❢❡♠ér✐❞❡s t❡rr❡str❡✱ ❛♦ ❝á❧❝✉❧♦ ❞♦ t❡♠♣♦ ❡ ♣❛r❛ s❡r ✉t✐❧✐③❛❞❛ ♥❛ ◆❛✈❡❣❛çã♦ ❡ ♥❛ ●❡♦❣r❛✜❛✳
❆ tr✐❣♦♥♦♠❡tr✐❛ ❡s❢ér✐❝❛✱ s♦❜r❡t✉❞♦ ♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s✱ ✈✐♥❤❛ s❡♥❞♦ ❡st✉❞❛❞❛ ♣♦r ❞✐✈❡rs♦s ♠❛t❡♠át✐❝♦s✱ q✉❡ ✈✐✈❡r❛♠ ❡♥tr❡ ✸✵✵ ❡ ✷✵✵ ❛✳❈✳✱ ❝♦♠♦ ❊✉❝❧✐❞❡s✱ ❆r✐st❛r❝♦ ❞❡ ❙❛♠♦s ❡ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛✱ ♥♦ ❡♥t❛♥t♦✱ ❍✐♣❛r❝♦✱ q✉❡ ✈✐✈❡✉ ❡♠ t♦r♥♦ ❞❡ ✶✷✵ ❛✳❈✱ é ❝♦♥s✐❞❡r❛❞♦ ♦ ♣r❡❝✉rs♦r✱ ♣♦r s❡r ♦ ♣r✐♠❡✐r♦ ❛ ❞❡t❡r♠✐♥❛r ❝♦♠ ♣r❡❝✐sã♦ ♦ ♥❛s❝❡r ❡ ♦ ♦❝❛s♦ ❞❡ ✈ár✐❛s ❡str❡❧❛s✳ ❯♠ ♣♦✉❝♦ ❞❡♣♦✐s✱ ❝❡r❝❛ ❞❡ ✶✵✵ ❛✳❈✱ ▼❡♥❡❧❛✉ ❞❡ ❆❧❡①❛♥❞r✐❛✱ ❛♣r❡s❡♥t❛✈❛ tr❛❜❛❧❤♦s ❞❡s❡♥✈♦❧✈✐❞♦s s♦❜r❡ ♦ ❛ss✉♥t♦✳ ❈♦♥t✉❞♦✱ ♦ á♣✐❝❡ ❞❛ ❚r✐❣♦♥♦♠❡tr✐❛ ❣r❡❣❛ ❢♦✐ ❛❧❝❛♥ç❛❞♦ ❝♦♠ ❈❧á✉❞✐♦ Pt♦❧♦♠❡✉✱ q✉❡ ✈✐✈❡✉ ❡♠ t♦r♥♦ ❞❡ ✶✺✵ ❞✳❈✳✱ ❝♦♠ ♦ s❡✉ tr❛❜❛❧❤♦ ❆❧♠❛❣❡st♦✱ q✉❡ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♠♦r❞✐❛❧ ❞❡s❝r❡✈❡r ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ s✐st❡♠❛ s♦❧❛r✳
◆❡st❡ tr❛❜❛❧❤♦✱ s❡rã♦ ♠♦str❛❞♦s r❡s✉❧t❛❞♦s ❝❧áss✐❝♦s ❞❛ tr✐❣♦♥♦♠❡tr✐❛ ❡s❢ér✐❝❛✱ ❝♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ ❛♣❧✐❝❛r ♥♦ ❡st✉❞♦ ❞♦ ❣❧♦❜♦ t❡rr❡str❡✳ P❛r❛ ✐st♦✱ ❞✐✈✐❞✐♠♦s ❛ ❞✐ss❡rt❛çã♦ ❡♠ três ❝❛♣ít✉❧♦s✳
◆♦ ❝❛♣ít✉❧♦ ✶✱ s❡rá tr❛t❛❞♦ s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❡ s❡✉s ❡❧❡♠❡♥t♦s✱ ❡♠ s❡✲ ❣✉✐❞❛ ❞❡✜♥✐r❡♠♦s tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ❡ ♣r♦✈❛r❡♠♦s s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s✱ ❛s q✉❛✐s r❡q✉❡r❡♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ré✈✐♦ ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❛ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s tr✐❡❞r♦s✳
◆♦ ❝❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s✳ ❉♦r❛✈❛♥t❡ é ❞❡s❡♥✈♦❧✈✐❞♦ ❛s ♣r✐♥❝✐♣❛✐s r❡❧❛çõ❡s ❡♥tr❡ ♦s ❧❛❞♦s ❡ ♦s â♥❣✉❧♦s ❞❡ss❡s tr✐â♥❣✉❧♦s✱ ❛ ❧❡✐ ❞♦s s❡♥♦s ❡ ❧❡✐ ❞♦s ❝♦ss❡♥♦s✱ ❞❡s❡♥✈♦❧✈✐❞❛s ♣❡❧♦ ❛❧❡♠ã♦ ❇❡ss❡❧ ✭✶✼✽✹ ✲ ✶✽✹✻✮✳ ❆✐♥❞❛ ♥❡ss❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❜r❡✈❡ ❡st✉❞♦ ❞♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s r❡tâ♥❣✉❧♦s✳
✷ ❆ ❙❯P❊❘❋❮❈■❊ ❊❙❋➱❘■❈❆ ❊ ❖❙
❚❘■➶◆●❯▲❖❙ ❊❙❋➱❘■❈❖❙
❊st✉❞❛r❡♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ ♦s ❡❧❡♠❡♥t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✱ ❞❡✜♥✐r❡♠♦s tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦✱ ♦♥❞❡ ❞❡st❛❝❛r❡♠♦s ❛s s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s✱ ❝♦♠♦ s♦♠❛ ❞❛s ♠❡❞✐❞❛s ❞♦s s❡✉s â♥❣✉❧♦s ✐♥t❡r♥♦s ❡ s✉❛ ár❡❛✳
✷✳✶ ❊❧❡♠❡♥t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛
❉❡✜♥✐çã♦ ✶ ❙❡❥❛ O ✉♠ ♣♦♥t♦ ❡ r ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✳ ❆ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❞❡❝❡♥tr♦ ❖ ❡ r❛✐♦ r é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞♦ ❡s♣❛ç♦ ❝✉❥❛ ❞✐stâ♥❝✐❛ ❛♦ ♣♦♥t♦ ❖ é ✐❣✉❛❧ ❛ r✳
❋✐❣✉r❛ ✷✳✶✿ ❙✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r
❖s ♣♦♥t♦s ❞♦ ❡s♣❛ç♦ ❝✉❥❛ ❞✐stâ♥❝✐❛ ❛♦ ♣♦♥t♦ O é ♠❡♥♦r ❞♦ q✉❡r sã♦ ✐♥t❡r✐♦r❡s à
s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✳ ❉❡ss❡ ♠♦❞♦✱ ❛ r❡✉♥✐ã♦ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❞❡ ❝❡♥tr♦O ❡ r❛✐♦r ❝♦♠
s❡✉s ♣♦♥t♦s ✐♥t❡r✐♦r❡s é ❝❤❛♠❛❞❛ ❞❡ ❡s❢❡r❛ ❞❡ ❝❡♥tr♦O ❡ r❛✐♦r✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ t❛❧
❡s❢❡r❛ é ❛ r❡✉♥✐ã♦ ❞❡ t♦❞♦s ♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛ ❞❡ ♦r✐❣❡♠O ❡ ❝♦♠♣r✐♠❡♥t♦r✳
■♥❞✐❝❛r❡♠♦s ❝♦♠ AB ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ AB✳
✷✳✶✳✶ ❆ ✐♥t❡rs❡çã♦ ❡♥tr❡ ✉♠ ♣❧❛♥♦ ❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛
❈♦♥s✐❞❡r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛S ❡ ✉♠ ♣❧❛♥♦ Πq✉❡ ❛ ❡♥❝♦♥tr❡✳ ❙❡ ❛ ✐♥t❡rs❡çã♦ Π∩S é ❡①❛t❛♠❡♥t❡ ✉♠ ♣♦♥t♦✱ ❞✐③❡♠♦s q✉❡ ♦ ♣❧❛♥♦ Π é t❛♥❣❡♥t❡ à s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ S✳
P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ Π∩S ♣♦ss✉✐r ♠❛✐s ❞♦ q✉❡ ✉♠ ♣♦♥t♦✱ ♦ ♣❧❛♥♦ Π é s❡❝❛♥t❡ à s✉♣❡r❢í❝✐❡
❡s❢ér✐❝❛ S✳ P♦❞❡♠♦s ❝❛r❛❝t❡r✐③❛r ♦ ♣❧❛♥♦ t❛♥❣❡♥t❡ ❝♦♥❢♦r♠❡ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✶✶
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ✉♠ ♣❧❛♥♦ Π ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ r❛✐♦ OT ❡♠ T✳ ❉❡✈❡♠♦s
♣r♦✈❛r q✉❡ ♥❡♥❤✉♠ ♦✉tr♦ ♣♦♥t♦ ❞❡ Π ❡stá ♥❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✳
❋✐❣✉r❛ ✷✳✷✿ P❧❛♥♦ t❛♥❣❡♥t❡ à s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛
❙❡❥❛ P ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❡ Π✱ ❝♦♠ P ❞✐st✐♥t♦ ❞❡T✳ ❈♦♠♦ Π é ♣❡r♣❡♥❞✐❝✉❧❛r
❛OT✱ t❡♠♦s q✉❡ ♦ tr✐â♥❣✉❧♦∆OP T é r❡tâ♥❣✉❧♦✱ ❝♦♠ ❤✐♣♦t❡♥✉s❛ OP ❡ ❝❛t❡t♦sOT ❡P T✳
▲♦❣♦ OP > OT =r ❡✱ ♣♦rt❛♥t♦✱P ♥ã♦ ❡stá ♥❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ Π ✉♠ ♣❧❛♥♦ t❛♥❣❡♥t❡ à s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ♥♦ ♣♦♥t♦ T✳ ❙✉✲
♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡Π ♥ã♦ s❡❥❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ r❛✐♦ OT✳
❋✐❣✉r❛ ✷✳✸✿ Π é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ r❛✐♦OT ❡♠ T
❙❡♥❞♦ F ♦ ♣é ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ Π✱ tr❛ç❛❞❛ ❛ ♣❛rt✐r ❞❡ O✱ t❡♠♦s q✉❡ F 6=T✱ ♣♦✐sΠ♥ã♦ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛OT✳ ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ♦ ♣♦♥t♦R ❞❛ r❡t❛F T t❛❧ q✉❡ F s❡❥❛ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡T R✳ ❉❡ss❛ ❢♦r♠❛✱ ♦s tr✐â♥❣✉❧♦s∆OF R❡ ∆OF T sã♦ ❝♦♥❣r✉❡♥✲
t❡s✱ ♣❡❧♦ ❝❛s♦ ❧❛❞♦ ✲ â♥❣✉❧♦ ✲ ❧❛❞♦✱ ❛❝❛rr❡t❛♥❞♦ q✉❡OR =OT =r✱ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱R
❡stá ♥❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✳ ▲♦❣♦✱ ♦ ♣❧❛♥♦Π✐♥t❡rs❡❝t❛ ❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❡♠ ❞♦✐s ♣♦♥t♦s
❞✐st✐♥t♦sR ❡T✱ ❝♦♥tr❛❞✐③❡♥❞♦ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ Π é ✉♠ ♣❧❛♥♦ t❛♥❣❡♥t❡✳
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✶✷
❚❡♦r❡♠❛ ✷✳✶✳✷ ❆ ✐♥t❡rs❡çã♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❝♦♠ ✉♠ ♣❧❛♥♦ ♣❛ss❛♥❞♦ ♣❡❧♦ s❡✉ ❝❡♥tr♦ é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ♠❡s♠♦ ❝❡♥tr♦ ❡ ♠❡s♠♦ r❛✐♦✳
❉❡♠♦♥str❛çã♦✿ ❉❛❞❛ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛S❝♦♠ ❝❡♥tr♦O ❡ r❛✐♦r ❡ ✉♠ ♣❧❛♥♦Πq✉❡
♣❛ss❛ ♣♦r O✱ ❛ ✐♥t❡rs❡çã♦ Π∩S é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡Π ❝✉❥❛ ❞✐stâ♥❝✐❛ ❛ O é ✐❣✉❛❧
❛r✱ ♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r✳
❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ♥♦s ❞❛rá ❛ s✐t✉❛çã♦ ♠❛✐s ❣❡r❛❧✳
❚❡♦r❡♠❛ ✷✳✶✳✸ ❙❡ ✉♠ ♣❧❛♥♦ ♣❛ss❛ ♣♦r ✉♠ ♣♦♥t♦ ❞♦ ✐♥t❡r✐♦r ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✱ ❡♥tã♦ ❛ ✐♥t❡rs❡çã♦ ❞♦ ♣❧❛♥♦ ❝♦♠ ❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❖ ❝❡♥tr♦ ❞❡ss❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ é ♦ ♣é ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ tr❛ç❛❞❛ ❛ ♣❛rt✐r ❞♦ ❝❡♥tr♦ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✳
❋✐❣✉r❛ ✷✳✹✿ P❧❛♥♦ s❡❝❛♥t❡ à s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ S ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r✳ ❙❡❥❛ Π
✉♠ ♣❧❛♥♦ q✉❡ ♥ã♦ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ S ❡ q✉❡ ♣❛ss❛ ♣♦r ✉♠ ♣♦♥t♦ Y ❞♦ s❡✉ ✐♥t❡r✐♦r✳
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✶✸
❖r❛✱ ❝♦♠♦ Y ❡stá ♥♦ ✐♥t❡r✐♦r ❞❡S✱ t❡♠♦s q✉❡ OY < r✳ ❆❧é♠ ❞✐ss♦✱ ♥♦ tr✐â♥❣✉❧♦
r❡tâ♥❣✉❧♦ ∆OF Y t❡♠♦s q✉❡ OY é ❤✐♣♦t❡♥✉s❛✱ ❡♥q✉❛♥t♦ q✉❡ OF é ❝❛t❡t♦✳ ▲♦❣♦ OF < OY < r✱ ♦✉ s❡❥❛✱F t❛♠❜é♠ ❡stá ♥♦ ✐♥t❡r✐♦r ❞❡ S✳
❆❣♦r❛✱ s❡❥❛ X ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ♥❛ ✐♥t❡rs❡çã♦ Π∩S✳ ❆ss✐♠✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦
tr✐â♥❣✉❧♦ ∆OF X✱ r❡tâ♥❣✉❧♦ ❡♠ F✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ t❡♠♦s q✉❡
OF2+F X2 =OX2 =r2
❡✱ ♣♦rt❛♥t♦✱F X =√r2−OF2✳
▲♦❣♦✱ X ❡stá ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ F ❡ r❛✐♦ √r2−OF2✳ Pr♦✈❛♠♦s ❛ss✐♠
q✉❡ ❛ ✐♥t❡rs❡çã♦Π∩S ❡stá ❝♦♥t✐❞❛ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ F ❡ r❛✐♦√r2−OF2✳
P❛r❛ ❝♦♠♣❧❡t❛r ❛ ❞❡♠♦♥str❛çã♦✱ ♣r❡❝✐s❛♠♦s ♠♦str❛r q✉❡ t♦❞♦ ♣♦♥t♦ ❞❡ss❛ ❝✐r✲ ❝✉♥❢❡rê♥❝✐❛ ♣❡rt❡♥❝❡ ❛ ✐♥t❡rs❡çã♦✳
❙❡❥❛X✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡♠Π✱ ❝♦♠ ❝❡♥tr♦F ❡ r❛✐♦√r2−OF2✳
◆♦✈❛♠❡♥t❡ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱
OX2 = OF2+F X2 = OF2+r2 −OF2 = r2.
P♦rt❛♥t♦ OX =r ❡ X ♣❡rt❡♥❝❡ à s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛S✳
❆ s❡❝çã♦ ♣❧❛♥❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❊st❛ ❝✐r❝✉♥❢❡rê♥✲ ❝✐❛ é ❝❤❛♠❛❞❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♠á①✐♠❛✶ q✉❛♥❞♦ ♦ ♣❧❛♥♦ q✉❡ ✐♥t❡rs❡❝t❛ ❛ s✉♣❡r❢í❝✐❡
❡s❢ér✐❝❛ ♣❛ss❛ ♣❡❧♦ s❡✉ ❝❡♥tr♦✳ ❊♠ ❝❛s♦ ❝♦♥trár✐♦✱ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♦❜t✐❞❛ é ❝✐r❝✉♥❢❡rê♥✲ ❝✐❛ ♠í♥✐♠❛✳
❆s ❡①tr❡♠✐❞❛❞❡s ❞♦ ❞✐â♠❡tr♦ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ q✉❡ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ q✉❡ ❛ ✐♥t❡rs❡❝t❛✱ sã♦ ♦s ♣♦❧♦s ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✭♠á①✐♠❛ ♦✉ ♠í♥✐♠❛✮✳
✶❍á ✉♠❛ ❢♦rt❡ r❛③ã♦ ♣❛r❛ ❡ss❡ ♥♦♠❡✿ ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s sã♦ ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❞❡ ♠❛✐♦r r❛✐♦
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✶✹
▼á①✐♠❛ ▼í♥✐♠❛
M
M′
P
P′
O
❋✐❣✉r❛ ✷✳✺✿ ❙❡❝çã♦ ♣❧❛♥❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛
◆❛ ❋✐❣✉r❛ ✷✳✺✱ P é ♦ ♣♦❧♦ ❞❡ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s✱ ♥♦ ❡♥t❛♥t♦ s♦✲
♠❡♥t❡ ✉♠❛ ❞❡❧❛s é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♠á①✐♠❛✳ ❉❡ss❡ ♠♦❞♦✱ ♦ ♣❧❛♥♦ q✉❡ ❞❡t❡r♠✐♥❛ ❡st❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♠á①✐♠❛ ❞✐✈✐❞❡ ❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❡♠ ❞✉❛s r❡❣✐õ❡s✱ ❛s q✉❛✐s ❝❤❛♠❛r❡♠♦s ❞❡ ❤❡♠✐s❢ér✐♦s✳
❋✐❣✉r❛ ✷✳✻✿ ❍❡♠✐❢ér✐♦s
✷✳✶✳✷ ❆ ❞✐stâ♥❝✐❛ ♥✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛
❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦sA❡B ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛Sé✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱
♦ ♠❡♥♦r ❞❡ t♦❞♦s ♦s ❝♦♠♣r✐♠❡♥t♦s ❞❛s tr❛❥❡tór✐❛s s♦❜r❡ S ❧✐❣❛♥❞♦ A ❛ B✳ ❖ s❡❣✉✐♥t❡
r❡s✉❧t❛❞♦ ❝❛r❛❝t❡r✐③❛ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✳
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✶✺
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ S ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❞❡ ❝❡♥tr♦ O✳ ❙❡♥❞♦ A ❡ B ❞♦✐s ♣♦♥t♦s
❞❡S q✉❡ ♥ã♦ sã♦ ♣♦❧♦s✱ ❝♦♥s✐❞❡r❡ ♦ ♣❧❛♥♦Π ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦s ♣♦♥t♦s ♥ã♦✲❝♦❧✐♥❡❛r❡s O✱ A ❡ B✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✶✳✷ ❛ ✐♥t❡rs❡çã♦ Π∩S é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♠á①✐♠❛✱ q✉❡ ♣❛ss❛
♣❡❧♦s ♣♦♥t♦sA❡B✱ ❛ q✉❛❧ é ú♥✐❝❛✱ ♣♦✐s ❞♦ ❝♦♥trár✐♦ t❡rí❛♠♦s ❞♦✐s ♣❧❛♥♦s ♣❛ss❛♥❞♦ ♣❡❧♦s
♣♦♥t♦s ♥ã♦✲❝♦❧✐♥❡❛r❡sO✱ A ❡ B✱ ♦ q✉❡ ♥ã♦ é ♣♦ssí✈❡❧✳
❉❡ss❡ ♠♦❞♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❞✐stâ♥❝✐❛ ♥✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❝♦♠♦ s❡❣✉❡✿ ❉❡✜♥✐çã♦ ✷ ❉❛❞♦s ❞♦✐s ♣♦♥t♦s A ❡ B s♦❜r❡ ✉♠ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✱ ❛ ❞✐stâ♥❝✐❛ d(A, B)
é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ♠❡♥♦r ❛r❝♦ AB ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♠á①✐♠❛ q✉❡ ♣❛ss❛ ♣♦r A ❡ B✳
B N
A
S
❋✐❣✉r❛ ✷✳✼✿ ❉✐stâ♥❝✐❛ ❡♥tr❡ A ❡ B
❖❜s❡r✈❛çã♦✿ ❆ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ t❡♠♦s q✉❡ t❛❧ ❛r❝♦ ❢❛③ ♦ ♣❛♣❡❧ ❞❡ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❞❛ ●❡♦♠❡tr✐❛ P❧❛♥❛✱ ❡♥q✉❛♥t♦ q✉❡ ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s ❢❛③❡♠ ❛♥❛❧♦❣✐❛ ❛s r❡t❛s✳
❖ ❝á❧❝✉❧♦ ❞❡ss❡ ❝♦♠♣r✐♠❡♥t♦ ♣♦❞❡ s❡r ❢❡✐t♦ ❛ ♣❛rt✐r ❞❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ AOB[✱
✉♠❛ ✈❡③ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❛r❝♦ é ♣r♦♣♦r❝✐♦♥❛❧ à ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ❝❡♥tr❛❧ ❝♦r✲ r❡s♣♦♥❞❡♥t❡✳ ❆ss✐♠✱ s❡♥❞♦ A ❡ B ❞♦✐s ♣♦♥t♦s ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❞❡ ❝❡♥tr♦ O ❡
r❛✐♦ r✱ ❛ ❞✐stâ♥❝✐❛ d(A, B) ❡♥tr❡A ❡ B é ❞❛❞❛ ♣♦r
d(A, B) = α
360◦ ·2πr
♦♥❞❡α é ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦AOB[ ❡♠ ❣r❛✉s✳
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✶✻
✷✳✶✳✸ ➶♥❣✉❧♦ ❡s❢ér✐❝♦
❉❡✜♥✐çã♦ ✸ ❖ â♥❣✉❧♦α❢♦r♠❛❞♦ ❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ♣❡❧❛ ✐♥t❡rs❡çã♦ ❞❡ ❞♦✐s ❛r❝♦s
❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s ❞❡♥♦♠✐♥❛✲s❡ â♥❣✉❧♦ ❡s❢ér✐❝♦✳ ❖s ❛r❝♦s ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s sã♦ ♦s ❧❛❞♦s ❡ ♦ s❡✉ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦✱ ♦ ✈ért✐❝❡ ❞♦ â♥❣✉❧♦ ❡s❢ér✐❝♦✳ ❆ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ❡s❢ér✐❝♦ é ✐❣✉❛❧ ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ❞✐❡❞r♦ ❢♦r♠❛❞♦ ♣❡❧♦s ♣❧❛♥♦s q✉❡ ❝♦♥tê♠ ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s✳
❋✐❣✉r❛ ✷✳✽✿ ➶♥❣✉❧♦ ❡s❢ér✐❝♦
◆❛ ❋✐❣✉r❛ ✷✳✾✱AP B[ é ✉♠ â♥❣✉❧♦ ❡s❢ér✐❝♦ ♥❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❞❡ ❝❡♥tr♦O ❡ r❛✐♦ OP✳ ❈♦♥s✐❞❡r❡ ♦ ♣❧❛♥♦ OAB q✉❡ ❞❡t❡r♠✐♥❛ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♠á①✐♠❛ ABC✱ ❝✉❥♦ ♣♦❧♦ é
♦ ♣♦♥t♦P✱ q✉❡ é ✈ért✐❝❡ ❞♦ r❡❢❡r✐❞♦ â♥❣✉❧♦ ❡s❢ér✐❝♦✳ ❈♦♠♦ ♦ â♥❣✉❧♦ ❞✐❡❞r♦ ❢♦r♠❛❞♦ ♣❡❧♦s
♣❧❛♥♦sAOP ❡BOP é ♠❡❞✐❞♦ ♣❡❧♦ â♥❣✉❧♦ ♣❧❛♥♦AOB[ ♦ q✉❛❧✱ ♣♦r s✉❛ ✈❡③✱ é ♠❡❞✐❞♦ ♣❡❧♦
❛r❝♦ ❆❇✱ s❡❣✉❡✲s❡ q✉❡ ✉♠ â♥❣✉❧♦ ❡s❢ér✐❝♦ é ♠❡❞✐❞♦ ♣❡❧♦ ♠❡♥♦r ❛r❝♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛⌢ ♠á①✐♠❛✱ ❝✉❥♦ ♣♦❧♦ é ♦ ✈ért✐❝❡ ❞♦ â♥❣✉❧♦✳ ❉❡ss❡ ♠♦❞♦✱ t❡r❡♠♦s q✉❡ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ â♥❣✉❧♦ ❡s❢ér✐❝♦ é ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛180◦✳
α
α
A B P
C O
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✶✼
✷✳✷ ❚r✐â♥❣✉❧♦s ❡s❢ér✐❝♦s
❖ ✐♥í❝✐♦ ❞♦ ❡st✉❞♦ ❞♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s ❢♦✐ ❛tr❡❧❛❞♦ à ❝r✐❛çã♦ ❞❛ ❚r✐❣♦♥♦♠❡✲ tr✐❛ ♣❡❧♦s ❣r❡❣♦s✱ ❛ q✉❛❧ s✉r❣✐✉ ❞❡✈✐❞♦ às ♥❡❝❡ss✐❞❛❞❡s ❞❛ ❆str♦♥♦♠✐❛✳ ❊ss❡ ❡st✉❞♦ ♥❛ ▼❛t❡♠át✐❝❛ ❣r❡❣❛ ✈✐♥❤❛ s❡♥❞♦ ❢❡✐t♦ ❞❡s❞❡ ♦s ú❧t✐♠♦s ♣✐t❛❣ór✐❝♦s✳ ❖ ♣ró♣r✐♦ ❊✉❝❧✐❞❡s✱ q✉❡ ✈✐✈❡✉ ❡♠ t♦r♥♦ ❞❡ ✸✵✵ ❛✳❈✳✱ ❡♠ ✉♠ ❞❡ s❡✉s tr❛❜❛❧❤♦s✱ ❖ ❋❡♥ô♠❡♥♦✱ ❡st✉❞♦✉ s✉❝✐♥✲ t❛♠❡♥t❡ ❛ ●❡♦♠❡tr✐❛ ❊s❢ér✐❝❛✳ P♦ré♠✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ❍✐♣❛r❝♦✷✱ ♦ ♠❛✐s ❡♠✐♥❡♥t❡ ❞♦s
❛strô♥♦♠♦s ❞❛ ❆♥t✐❣✉✐❞❛❞❡✱ q✉❡ ✈✐✈❡✉ ❡♠ t♦r♥♦ ❞❡ ✶✷✵ ❛✳❈✳✱ ❢♦✐ ♦ ♣r❡❝✉rs♦r ❞❡ss❡ ❡st✉❞♦✱ ✉♠❛ ✈❡③ q✉❡ ✉s❛♥❞♦ tá❜✉❛s tr✐❣♦♥♦♠étr✐❝❛s r❡s♦❧✈❡✉ tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s r❡tâ♥❣✉❧♦s ❝♦♠ ♣r♦❝❡ss♦s s❡♠❡❧❤❛♥t❡s às ✈ár✐❛s ❢ór♠✉❧❛s ❝♦♥❤❡❝✐❞❛s ❛t✉❛❧♠❡♥t❡✳
❯♠ ♣♦✉❝♦ ❞❡♣♦✐s ❞❡ ❍✐♣❛r❝♦✱ ▼❡♥❡❧❛✉ ❞❡ ❆❧❡①❛♥❞r✐❛✱ q✉❡ ✈✐✈❡✉ ❡♠ t♦r♥♦ ❞❡ ✶✵✵ ❛✳❈✳✱ ❥á ❛♣r❡s❡♥t❛✈❛ r❡s✉❧t❛❞♦s ❞❡s❡♥✈♦❧✈✐❞♦s s♦❜r❡ ❚r✐❣♦♥♦♠❡tr✐❛✱ ♠♦str❛❞♦ ❡♠ ♣❛rt❡s ❞♦ s❡✉ ❧✐✈r♦ ●❡♦♠❡tr✐❛ ❊s❢ér✐❝❛✳ ◆♦ tr❛❜❛❧❤♦ ❞❡ ▼❡♥❡❧❛✉✱ ❛♣❛r❡❝❡ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❛ ❞❡✜♥✐çã♦ ❞❡ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦✱ ❛❧é♠ ❞❡ ❞✐✈❡rs❛s ♣r♦♣♦s✐çõ❡s r❡❧❛t✐✈❛s ❛ ❡ss❡s tr✐â♥❣✉❧♦s ❡ s❡♠❡❧❤❛♥t❡s às ❡st❛❜❡❧❡❝✐❞❛s ♣♦r ❊✉❝❧✐❞❡s ♣❛r❛ ♦s tr✐â♥❣✉❧♦s ❞❛ ●❡♦♠❡tr✐❛ P❧❛♥❛✳ Pr♦✈♦✉ ❛✐♥❞❛ q✉❡✱ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ❝♦♥❝❡✐t♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ♥❛ ❣❡♦♠❡tr✐❛ ❡s❢ér✐❝❛✱ ♣♦✐s ❞♦✐s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s tê♠ â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❝♦♥❣r✉❡♥t❡s✱ ❡♥tã♦ ♦s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✱ r❡s✉❧t❛❞♦ ♣❛r❛ ♦ q✉❛❧ ♥ã♦ ❤á ♥❡♥❤✉♠ ❛♥á❧♦❣♦ ♥♦ ♣❧❛♥♦✳ ❊❧❡ ✉s♦✉✱ s❡♠ ♣r♦✈❛r✱ ♦ t❡♦r❡♠❛ ❞❛ ●❡♦♠❡tr✐❛ P❧❛♥❛✱ ❤♦❥❡ ❝♦♥❤❡❝✐❞♦ ♣♦r ❚❡♦r❡♠❛ ❞❡ ▼❡♥❡❧❛✉✱ ♣❛r❛ ❞❡❞✉③✐r ❞✐✈❡rs♦s r❡s✉❧t❛❞♦s s♦❜r❡ tr✐❣♦♥♦♠❡tr✐❛ ❡s❢ér✐❝❛✱ ♣r♦✈❛♥❞♦✱ t❛♠❜é♠✱ ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❞❡ss❡ t❡♦r❡♠❛ ♣❛r❛ tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s✳
✷✳✷✳✶ ❉❡✜♥✐çã♦
❉❡✜♥✐çã♦ ✹ ❯♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ é ❛ ♣♦rçã♦ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❧✐♠✐t❛❞❛ ♣❡❧♦s ❛r❝♦s ❞❡ três ❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s ♠❡♥♦r❡s ❞♦ q✉❡ ✉♠❛ s❡♠✐❝✐r❝✉♥❢❡rê♥❝✐❛✱ q✉❡ s❡ ✐♥t❡rs❡❝t❛♠ ❞♦✐s ❛ ❞♦✐s✳ ❖s ❛r❝♦s sã♦ ❝❤❛♠❛❞♦s ❧❛❞♦s ❡ ♦s ✈ért✐❝❡s ❞♦s três â♥❣✉❧♦s ❡s❢ér✐❝♦s sã♦ t❛♠❜é♠ ♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦✳
✷❈♦♥s✐❞❡r❛❞♦ ♦ ♣❛✐ ❞❛ ❚r✐❣♦♥♦♠❡tr✐❛✱ ♣♦r t❡r s✐❞♦ ♦ ♣✐♦♥❡✐r♦ ♥❛ ❡❧❛❜♦r❛çã♦ ❞❡ ✉♠❛ t❛❜❡❧❛ tr✐❣♦♥♦♠é✲
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✶✽
❋✐❣✉r❛ ✷✳✶✵✿ ❚r✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ABC
◆❛ ✜❣✉r❛ ✷✳✶✵✱ ♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ABC tê♠ A✱ B ❡ C ❝♦♠♦ ✈ért✐❝❡s ❡ ♦s ❛r❝♦s
♦♣♦st♦s ❝♦rr❡s♣♦♥❞❡♥t❡s a✱ b ❡ c❝♦♠♦ ❧❛❞♦s✳
❆❞❡♠❛✐s✱ ♦❜s❡r✈❡ q✉❡ ❧✐❣❛♥❞♦✲s❡ ♦s ✈ért✐❝❡s A✱B ❡C ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ❛♦
❝❡♥tr♦ O ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✱ ♦❜t❡♠♦s ♦ tr✐❡❞r♦ O✲ABC✱ ❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✷✳✶✶✳
C
B c
a
A b O
❋✐❣✉r❛ ✷✳✶✶✿ ❚r✐❡❞r♦ O✲ABC
❖ r❡s✉❧t❛❞♦ s❡❣✉✐♥t❡ ❝r✐❛ ✉♠❛ r❡❧❛çã♦ ✐♥trí♥s❡❝❛ ❡♥tr❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ❡ ♦ tr✐❡❞r♦ ❝♦♥str✉í❞♦ ❛ ♣❛rt✐r ❞❡❧❡✳
▲❡♠❛ ✷✳✷✳✶ ❆ t♦❞♦ tr✐❡❞r♦ ❝♦rr❡s♣♦♥❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦✱ q✉❡ ❡stá s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❝✉❥♦ ❝❡♥tr♦ é ♦ ✈ért✐❝❡ ❞❡ss❡ tr✐❡❞r♦✱ t❛❧ q✉❡ s❡✉s ❧❛❞♦s ♠❡❞❡♠ â♥❣✉❧♦s ❞❛s ❢❛❝❡s ❞♦ tr✐❡❞r♦ ❡ s❡✉s â♥❣✉❧♦s ♠❡❞❡♠ â♥❣✉❧♦s ❞✐❡❞r♦s ❢♦r♠❛❞♦ ♣❡❧❛s ❢❛❝❡s✳
❉❡♠♦♥str❛çã♦✿ ■♠❛❣✐♥❡♠♦s ✉♠ tr✐❡❞r♦ ❝✉❥♦ ✈ért✐❝❡ s❡❥❛ ♦ ❝❡♥tr♦ O ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡
❡s❢ér✐❝❛ S✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✶✳✷ ♦s ♣❧❛♥♦s q✉❡ ❢♦r♠❛♠ ♦ tr✐❡❞r♦✱ ♣❛ss❛♥❞♦ ♣♦r O✱ ❞❡t❡r✲
♠✐♥❛♠ s♦❜r❡ S ❛r❝♦s ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s✱ ♦s q✉❛✐s s❡ ❡♥❝♦♥tr❛♠ ❞♦✐s ❛ ❞♦✐s ❡
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✶✾
â♥❣✉❧♦s ❝❡♥tr❛✐s ❞♦s ❛r❝♦s ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♠á①✐♠❛ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ ♣❧❛♥♦ ❞❛ ❢❛❝❡ s♦❜r❡
S ❡✱ ♣♦rt❛♥t♦✱ t❡♥❞♦ ❝♦♠♦ ♠❡❞✐❞❛ ♦ ❛r❝♦ ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ s❡✉s ❧❛❞♦s✱ ♠❛s ❡ss❡s ❛r❝♦s
sã♦ ♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ❢♦r♠❛❞♦✱ ❝✉❥♦s ❧❛❞♦s ♠❡❞❡♠ ❛s ❢❛❝❡s ❞♦ tr✐❡❞r♦✳
P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ tr❛ç❛r♠♦s✱ ❛ ♣❛rt✐r ❞♦s ♣♦♥t♦s q✉❡ ❛s ❛r❡st❛s ❞♦ tr✐❡❞r♦ ❝♦rt❛♠
S ❡ q✉❡ sã♦ ♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦✱ r❡t❛s t❛♥❣❡♥t❡s ❛♦s ❛r❝♦s ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛
♠á①✐♠❛ q✉❡ ❢♦r♠❛♠ ♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦✱ ♦s â♥❣✉❧♦s ❢♦r♠❛❞♦s ♣♦r ❡ss❛s t❛♥❣❡♥✲ t❡s ♠❡❞❡♠ ♦s â♥❣✉❧♦s ❞❡ss❡ tr✐â♥❣✉❧♦❀ ♠❛s ❡ss❛s t❛♥❣❡♥t❡s sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛♦s r❛✐♦s ♥♦ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛✱ ♦✉ s❡❥❛✱ sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s às ❛r❡st❛s ❞♦ tr✐❡❞r♦ ❡♠ ✉♠ ♠❡s♠♦ ♣♦♥t♦ ❡✱ ♣♦rt❛♥t♦✱ sã♦ ❛s ♠❡❞✐❞❛s ❞♦s â♥❣✉❧♦s ❞♦s ❞✐❡❞r♦s q✉❡ ❢♦r♠❛♠ ♦ tr✐❡❞r♦✳ ❆ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦s â♥❣✉❧♦s ❞♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ♠❡❞❡♠ ♦s â♥❣✉❧♦s ❞✐❡❞r♦s ❞♦ tr✐❡❞r♦ ❝♦♥✲ s✐❞❡r❛❞♦✳
◆♦t❡♠♦s q✉❡ ♦ ▲❡♠❛ ❛♥t❡r✐♦r ♠♦str❛ q✉❡ ♣♦❞❡♠♦s ❞❡❞✉③✐r ❞✐✈❡rs❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s ❛tr❛✈és ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s tr✐❡❞r♦s✳
✷✳✷✳✷ ❉❡s✐❣✉❛❧❞❛❞❡s ❢✉♥❞❛♠❡♥t❛✐s ❞♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s
❚❡♦r❡♠❛ ✷✳✷✳✺ ✭❉❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ♣❛r❛ tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s✮ ◗✉❛❧q✉❡r ❧❛❞♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ é ♠❡♥♦r ❞♦ q✉❡ ❛ s♦♠❛ ❞♦s ♦✉tr♦s ❞♦✐s ❧❛❞♦s✳
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ABC s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠❛ ❡s❢❡r❛
❞❡ ❝❡♥tr♦ O ❡ r❛✐♦r✳ ▲✐❣❛♥❞♦✲s❡ ♦s ✈ért✐❝❡sA✱ B✱C ❞♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ❛♦ ❝❡♥tr♦ O ❞❛
❡s❢❡r❛✱ ♦❜t❡♠♦s ♦ tr✐❡❞r♦O✲ABC✳
❉❡ss❛ ❢♦r♠❛✱ ♣❡❧♦ ▲❡♠❛ ✷✳✷✳✶ ♣❛r❛ ❜❛st❛ ♣r♦✈❛r q✉❡ q✉❛❧q✉❡r â♥❣✉❧♦ ❞❛ ❢❛❝❡ ❞♦ tr✐❡❞r♦O✲ABC é ♠❡♥♦r q✉❡ ❛ s♦♠❛ ❞♦s ♦✉tr♦s ❞♦✐s✳
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✷✵
❞❡AOC[ ❡①✐st❡ ✉♠ ♣♦♥t♦ D✱ t❛❧ q✉❡ \AOD =AOB[✳ ❆ss✐♠✱ t♦♠❡♠♦s D ∈AC ❞❡ ♠♦❞♦
q✉❡OB =OD✳ ❉❡ss❡ ♠♦❞♦✱ t❡r❡♠♦s q✉❡ ♦s tr✐â♥❣✉❧♦s∆AOB ❡∆AODsã♦ ❝♦♥❣r✉❡♥t❡s✱
❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱AD=AB✳
B D
O
C A
P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ tr✐â♥❣✉❧♦ ∆ABC✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱
t❡♠♦s q✉❡✿
AC < AB+BC ✭✷✳✶✮
❡✱ ❝♦♠♦D∈AC✱ ❡♥tã♦✿
AC =AD+DC. ✭✷✳✷✮
❆ss✐♠✱ ✉s❛♥❞♦ ✭✷✳✷✮ ❡♠ ✭✷✳✶✮ t❡r❡♠♦s q✉❡✿
AD+DC < AB+BC.
▼❛s✱ ❝♦♠♦AD=AB t❡♠♦s q✉❡✿
DC < BC
❞♦♥❞❡ s❡❣✉❡ q✉❡✱
\
DOC <BOC.\ ✭✷✳✸✮
P♦rt❛♥t♦✱
[
AOC =\AOD+\DOC <AOB[ +\BOC.
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✷✶
❚❡♦r❡♠❛ ✷✳✷✳✻ ❆ s♦♠❛ ❞♦s três ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ❡stá ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡
0◦ ❡ 360◦✳
❉❡♠♦♥str❛çã♦✿ ◗✉❡ ❛ s♦♠❛ ❞♦s ❧❛❞♦s é ♠❛✐♦r ❞♦ q✉❡0◦ é ❡✈✐❞❡♥t❡✳ ❱❛♠♦s ♠♦str❛r q✉❡
❛ s♦♠❛ ❞♦s ❧❛❞♦s é ♠❡♥♦r ❞♦ q✉❡360◦✳ ❈♦♥❢♦r♠❡ ❛♥t❡r✐♦r♠❡♥t❡✱ ❝♦♥str✉✐r❡♠♦s ♦ tr✐❡❞r♦
O✲ABC✱ ❡ ✉s❛♥❞♦ ♦ ▲❡♠❛ ✷✳✷✳✶✱ ♠♦str❛r❡♠♦s q✉❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ❞❛s s✉❛s ❢❛❝❡s é
♠❡♥♦r ❞♦ q✉❡360◦✳
❈♦♥str✉✐♥❞♦ ❛ s❡♠✐rr❡t❛OC′ ♦♣♦st❛ ❞❡OC✱ ♦❜t❡r❡♠♦s ♦ tr✐❡❞r♦O✲ABC′ ❡✱ ♣❡❧♦
❚❡♦r❡♠❛ ✷✳✷✳✺✱ t❡♠♦s q✉❡✿
[
AOB <BOC\′+AOC\′. ✭✷✳✹✮
❋✐❣✉r❛ ✷✳✶✷✿ ❛ s❡♠✐rr❡t❛ OC′ é ♦♣♦st❛ ❞❡ OC
❖❜s❡r✈❡ q✉❡✱ ♦s â♥❣✉❧♦s \BOC ❡ BOC\′ sã♦ ❛❞❥❛❝❡♥t❡s s✉♣❧❡♠❡♥t❛r❡s✱ ❡ ♦ ♠❡s♠♦
♦❝♦rr❡ ❝♦♠ AOC[ ❡AOC\′✳ ▲♦❣♦✱ \
BOC+BOC\′+AOC[ +AOC\′ = 360◦. ✭✷✳✺✮
P♦rt❛♥t♦✱ ✉s❛♥❞♦ ✭✷✳✹✮ ❡♠ ✭✷✳✺✮✱ s❡❣✉❡ q✉❡✿
\
BOC+AOC[ +AOB <[ 360◦
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✷✷
✷✳✷✳✸ ❚r✐â♥❣✉❧♦s ❡s❢ér✐❝♦s ♣♦❧❛r❡s
❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦✳ ❈♦♥str✉❛♠♦s três ❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s t❛✐s
q✉❡ ♦s ✈ért✐❝❡s A✱ B ❡ C s❡❥❛ s❡✉s r❡s♣❡❝t✐✈♦s ♣♦❧♦s ❆❣♦r❛✱ s❡❥❛♠ A′ ❛ ✐♥t❡rs❡çã♦ ❞❛s
❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s q✉❡ ♣♦ss✉❡♠B ❡ C ❝♦♠♦ ♣♦❧♦s ❡✱ q✉❡ ❧♦❝❛❧✐③❛✲s❡ ♥♦ ♠❡s♠♦ ❤❡✲
♠✐s❢ér✐♦ ❞♦ ❧❛❞♦❇❈❀ ♣♦r⌢ B′ ❛ ✐♥t❡rs❡çã♦ ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s q✉❡ ♣♦ss✉❡♠A❡C
❝♦♠♦ ♣♦❧♦s ❡✱ q✉❡ ❧♦❝❛❧✐③❛✲s❡ ♥♦ ♠❡s♠♦ ❤❡♠✐s❢ér✐♦ ❞♦ ❧❛❞♦❆❈❀ ❡ ♣♦r⌢ C′ ❛ ✐♥t❡rs❡çã♦ ❞❛s
❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s q✉❡ ♣♦ss✉❡♠A ❡ B ❝♦♠♦ ♣♦❧♦s ❡✱ q✉❡ ❧♦❝❛❧✐③❛✲s❡ ♥♦ ♠❡s♠♦ ❤❡✲
♠✐s❢ér✐♦ ❞♦ ❧❛❞♦❆❇✳ ❉❡ss❡ ♠♦❞♦✱ ♦❜t❡♠♦s ✉♠ ♦✉tr♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦⌢ A′B′C′✱ ❝♦♥❢♦r♠❡
❛ s❡❣✉✐♥t❡✿
❉❡✜♥✐çã♦ ✺ ❖ tr✐â♥❣✉❧♦ ♣♦❧❛r ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ABC é ♦✉tr♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ A′B′C′ q✉❡ s❡ ♦❜tê♠ ❛ ♣❛rt✐r ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s ❝✉❥♦s ♣♦❧♦s sã♦ ♦s ✈ért✐❝❡s ❞♦
tr✐â♥❣✉❧♦ ABC✳
❋✐❣✉r❛ ✷✳✶✸✿ A′B′C′ é ♦ tr✐â♥❣✉❧♦ ♣♦❧❛r ❞❡ ABC
❆s ♣r♦♣r✐❡❞❛❞❡s ❢✉♥❞❛♠❡♥t❛✐s s♦❜r❡ ♦s tr✐â♥❣✉❧♦s ♣♦❧❛r❡s sã♦✿
❚❡♦r❡♠❛ ✷✳✷✳✼ ❙❡ A′B′C′ é ♦ tr✐â♥❣✉❧♦ ♣♦❧❛r ❞❡ ABC✱ ❡♥tã♦ ABC é ♦ tr✐â♥❣✉❧♦ ♣♦❧❛r
❞❡ A′B′C′✳
❉❡♠♦♥str❛çã♦✿ ❙❡♥❞♦ A ♣♦❧♦ ❞❡ B⌢′C′ ❡ C é ♣♦❧♦ ❞❡ A⌢′B′✱ ❡♥tã♦ ♦s ❛r❝♦s AB⌢′ ❡ CB⌢ ′
♠❡❞❡♠90◦✱ ❧♦❣♦✱B′ é ♣♦❧♦ ❞❡❆❈✳ ❊✈✐❞❡♥t❡♠❡♥t❡✱ ❞♦ ♠❡s♠♦ ♠♦❞♦⌢ A′ é ♣♦❧♦ ❞❡ ❇❈ ❡⌢ C′
é ♣♦❧♦ ❞❡ ❆❇✳ ❆ss✐♠✱ ♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦⌢ ABC é ✉♠ ❞♦s ♦✐t♦ tr✐â♥❣✉❧♦s ❢♦r♠❛❞♦s ♣❡❧❛s
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✷✸
❋✐❣✉r❛ ✷✳✶✹✿ ❚r✐â♥❣✉❧♦s ♣♦❧❛r❡s
P❛r❛ ❝♦♥❝❧✉✐r q✉❡ ♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ABC é ♦ tr✐â♥❣✉❧♦ ♣♦❧❛r ❞❡ A′B′C′✱ é
♥❡❝❡ssár✐♦ ✈❡r✐✜❝❛r q✉❡ A ❡ A′ ❡st❡❥❛♠ ❞♦ ♠❡s♠♦ ❧❛❞♦ q✉❡ B⌢′C′✱ q✉❡ B ❡ B′ ❡st❡❥❛♠
❞♦ ♠❡s♠♦ ❧❛❞♦ q✉❡A⌢′C′ ❡ q✉❡ C ❡ C′ ❡st❡❥❛♠ ❞♦ ♠❡s♠♦ ❧❛❞♦ q✉❡ A⌢′B′✳ P♦r ❞❡✜♥✐çã♦✱
B ❡ B′ ❡stã♦ ♥♦ ♠❡s♠♦ ❧❛❞♦ ❞❡ ❆❈ ❡⌢ D ❞✐st❛ ♠❡♥♦s ❞❡ 180◦ q✉❡ q✉❛❧q✉❡r ♣♦♥t♦ s♦❜r❡
⌢
❆❈✳ ❊♥tã♦✱ ✉♠❛ ✈❡③ q✉❡ B′ ♣♦r s❡r ♣♦❧♦ ❞❡ ❆❈ ❡stá ❛⌢ 90◦ ❞❡ q✉❛❧q✉❡r ♣♦♥t♦ ❞❡st❡ ❛r❝♦✱
♣♦rt❛♥t♦ B ❡B′ ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❡A⌢′C′✳
❉❡ ♠❛♥❡✐r❛ ✐♥t❡✐r❛♠❡♥t❡ ❛♥á❧♦❣❛✱ ♣♦❞❡ s❡r ♠♦str❛❞♦ q✉❡ A ❡ A′ ❡st❡❥❛♠ ❞♦
♠❡s♠♦ ❧❛❞♦ q✉❡B⌢′C′ ❡ C′ ❡st❡❥❛♠ ❞♦ ♠❡s♠♦ ❧❛❞♦ q✉❡A⌢′B′✳
❚❡♦r❡♠❛ ✷✳✷✳✽ ❊♠ ❞♦✐s tr✐â♥❣✉❧♦s ♣♦❧❛r❡s✱ ❝❛❞❛ â♥❣✉❧♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ✐❣✉❛❧ ❛♦ s✉✲ ♣❧❡♠❡♥t♦ ❞♦ ❧❛❞♦ ♦♣♦st♦ ❝♦rr❡s♣♦♥❞❡♥t❡✱ ❞♦ ♦✉tr♦ tr✐â♥❣✉❧♦✳
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ♦s tr✐â♥❣✉❧♦s ♣♦❧❛r❡s ABC ❡ A′B′C′ ❞❛ ✜❣✉r❛ ❛❜❛✐①♦✱
❜❛st❛ ♣r♦✈❛r q✉❡ Ab′ = 180◦−a✳
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✷✹
Pr♦❧♦♥❣❛♥❞♦✲s❡ ♦s ❛r❝♦s A⌢′B′ ❡ A⌢′C′ ❛té q✉❡ ❡♥❝♦♥tr❡♠ ♦ ❛r❝♦ ❇❈ ♥♦s ♣♦♥t♦s⌢
D ❡ E✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ss✐♠✱ ♦ â♥❣✉❧♦ Ab′ ♣♦ss✉✐ ♠❡❞✐❞❛ ✐❣✉❛❧ ❛ ❉❊✳ P♦r ♦✉tr♦ ❧❛❞♦✱⌢
s❡♥❞♦ B ♣♦❧♦ ❞❡ A⌢′E ❡ C ♣♦❧♦ ❞❡ A⌢′D✱ ❡♥tã♦✱ ♣♦r ❞❡✜♥✐çã♦ ❞❡ ❞✐stâ♥❝✐❛ ♣♦❧❛r✱ t❡♠✲s❡
⌢
❇❊=❉❈⌢ = 90◦✳ P♦rt❛♥t♦✱
a=❇❊⌢ +❉❈⌢ −❉❊⌢ = 180◦
−Ab′,
❡✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱
b
A′ = 180◦−a.
✷✳✷✳✹ ❙♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦
❖s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s ♣♦ss✉❡♠ ❞✐✈❡rs❛s ❡s♣❡❝✐✜❝✐❞❛❞❡s q✉❛♥❞♦ ❝♦♠♣❛r❛❞♦s ❝♦♠ ♦s tr✐â♥❣✉❧♦s ❞❛ ●❡♦♠❡tr✐❛ P❧❛♥❛✱ ✉♠❛ ❞❡ss❛s ❞✐❢❡r❡♥ç❛s é ♠♦str❛❞❛ ❛ s❡❣✉✐r✳
❚❡♦r❡♠❛ ✷✳✷✳✾ ❆ s♦♠❛ ❞❛s ♠❡❞✐❞❛s ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦ ❡stá ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ 180◦ ❡ 540◦✳
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ABC✱ ❞❡ â♥❣✉❧♦s ✐♥t❡r♥♦s Aˆ✱ Bˆ ❡ ˆ
C✱ ❡ ♦ s❡✉ tr✐â♥❣✉❧♦ ♣♦❧❛r A′B′C′✳ ❉❡ss❡ ♠♦❞♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✷✳✽ t❡♠♦s q✉❡✿ ˆ
A= 180◦−a′ ˆ
B = 180◦−b′ ˆ
C = 180◦−c′
♦ q✉❡ ❛❝❛rr❡t❛
ˆ
A+ ˆB+ ˆC = 540◦−(a′+b′+c′) ✭✷✳✻✮
P♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✷✳✷✳✻ t❡♠♦s q✉❡✿
0◦ < a′+b′ +c′ <360◦ ✭✷✳✼✮
P♦rt❛♥t♦✱ ❞❡ ✭✷✳✻✮ ❡ ✭✷✳✼✮ s❡❣✉❡ q✉❡
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✷✺
❖❜s❡r✈❛çã♦✿ ❖ t❡♦r❡♠❛ ❛❝✐♠❛✱ ❝❛r❛❝t❡r✐③❛ q✉❡ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠❛ ✐s♦♠❡tr✐❛✸ ❡♥tr❡ ❛
s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❡ ♦ ♣❧❛♥♦ ✭♦✉ ♠❡s♠♦ ❡♥tr❡ ✉♠❛ ♣❛rt❡ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❡ ✉♠❛ ♣❛rt❡ ❞♦ ♣❧❛♥♦✮✳
❈♦♥❢♦r♠❡ ♠♦str❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ é s❡♠♣r❡ s✉♣❡r✐♦r ❛ ❞♦✐s â♥❣✉❧♦s r❡t♦s✱ ♦ q✉❡ s✉❣❡r❡ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳
❉❡✜♥✐çã♦ ✻ ❙❡❥❛♠ Aˆ✱ Bˆ ❡ Cˆ ♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ABC✳ ❆
❞✐❢❡r❡♥ç❛ E = ˆA+ ˆB+ ˆC−180◦ é ❝❤❛♠❛❞❛ ❞❡ ❡①❝❡ss♦ ❡s❢ér✐❝♦ ❞♦ tr✐â♥❣✉❧♦ ABC✳
❆✐♥❞❛ ❞♦ ❚❡♦r❡♠❛ ✷✳✷✳✾✱ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡ s♦❜r❡ ♦s â♥❣✉❧♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦✿
❈♦r♦❧ár✐♦ ✷✳✷✳✶ ❊♠ t♦❞♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ❝❛❞❛ â♥❣✉❧♦ ✐♥t❡r♥♦ ❛✉♠❡♥t❛❞♦ ❞❡ 180◦ é
♠❛✐♦r ❞♦ q✉❡ ❛ s♦♠❛ ❞♦s ♦✉tr♦s ❞♦✐s â♥❣✉❧♦s✳
❉❡♠♦♥str❛çã♦✿ ❙❡♥❞♦ A′B′C′ ♦ tr✐â♥❣✉❧♦ ♣♦❧❛r ❞♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ABC✱ ♣❡❧♦ ❚❡♦✲
r❡♠❛ ✷✳✷✳✽ t❡♠♦s q✉❡✿
ˆ
A= 180◦−a′ ˆ
B = 180◦−b′ ˆ
C = 180◦−c′
❙♦♠❛♥❞♦ ❛s ❞✉❛s ú❧t✐♠❛s ✐❣✉❛❧❞❛❞❡s ♦❜t❡♠♦s q✉❡
ˆ
B+ ˆC <360◦−(b′+c′) ✭✷✳✽✮
❆❣♦r❛✱ ✉s❛♥❞♦ ❚❡♦r❡♠❛ ✷✳✷✳✺ ♥♦ tr✐â♥❣✉❧♦ A′B′C′ t❡r❡♠♦s q✉❡
a′ < b′+c′. ✭✷✳✾✮
❉❡ss❛ ♠❛♥❡✐r❛✱ ✉s❛♥❞♦ ✭✷✳✾✮ ♥❛ ✐♥❡q✉❛çã♦ ✭✷✳✽✮ ❝❤❡❣❛r❡♠♦s q✉❡
ˆ
B+ ˆC <360◦−a′
❡✱ ♣♦rt❛♥t♦✱
ˆ
B + ˆC < 180◦+ ˆA.
✸❯♠❛ ✐s♦♠❡tr✐❛ é ✉♠❛ ❛♣❧✐❝❛çã♦T :X −→Y ❡♥tr❡ ❞♦✐s ❡s♣❛ç♦s ♠étr✐❝♦s✱ q✉❡ ♣r❡s❡r✈❛ ❞✐stâ♥❝✐❛ ❡♥tr❡
s❡✉s ♣♦♥t♦s✱ ✐st♦ é✱dY(T(x1), T(x2)) =dX(x1, x2)♣❛r❛ q✉❛✐sq✉❡r x1, x2 ∈X✳ ❉❡ss❡ ♠♦❞♦✱ s❡ ❡①✐st✐ss❡
✉♠❛ ✐s♦♠❡tr✐❛ψ:S −→Π ❡♥tr❡ ❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛S ❡ ♦ ♣❧❛♥♦Π t❡r✐❛ q✉❡ ♣r❡s❡r✈❛r ❞✐stâ♥❝✐❛s✱ ✐st♦
é✱ t❡r✐❛ q✉❡ ❛♣❧✐❝❛r ❝✐r❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s ❞❡S ❡♠ r❡t❛s ❞❡Π✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✷✻
✷✳✷✳✺ ➪r❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦
❖ ❣❡ô♠❡tr❛ ❢r❛♥❝ês ❆❧❜❡rt ●✐r❛r❞✹ ✭✶✺✾✺ ✲ ✶✻✸✷✮ ✐♥t❡r❡ss♦✉✲s❡ ♣♦r ●❡♦♠❡tr✐❛
❊s❢ér✐❝❛✱ r❡❛❧✐③❛♥❞♦ ❞✐✈❡rs❛s ❝♦♥tr✐❜✉✐çõ❡s ♥❡st❡ r❛♠♦✳ ❊♥tr❡ ❡ss❛s✱ ●✐r❛r❞ ♠♦str♦✉ ❡♠ ✶✻✷✾✱ ❛ ❢ór♠✉❧❛ ❞❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ❡♠ t❡r♠♦s ❞♦ s❡✉ ❡①❝❡ss♦ ❡s❢ér✐❝♦✳
◆❡st❛ s❡çã♦✱ ♣r♦✈❛r❡♠♦s ❛ ❢ór♠✉❧❛ ♦❜t✐❞❛ ♣♦r ●✐r❛r❞✳ ❆♥t❡s ❞✐ss♦ ♣r❡❝✐s❛r❡♠♦s ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s✳
❈♦♥s✐❞❡r❡♠♦s ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ S ❞❡ ❝❡♥tr♦ ❡♠ O ❡ r❛✐♦ r✱ ❛ q✉❛❧ ♣❡r♠❛✲
♥❡❝❡rá ✜①❛ ♥♦ ❞❡❝♦rr❡r ❞❡st❛ s❡çã♦✳
❉❡✜♥✐çã♦ ✼ ❯♠ ❢✉s♦ é ✉♠❛ r❡❣✐ã♦ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❞✉❛s ❝✐r✲ ❝✉♥❢❡rê♥❝✐❛s ♠á①✐♠❛s✳ ❊ss❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ♣♦ss✉❡♠✱ ❡♠ ❝♦♠✉♠✱ ❞♦✐s ♣♦♥t♦s ❞✐❛♠❡tr❛❧✲ ♠❡♥t❡ ♦♣♦st♦s✱ q✉❡ sã♦ ❞✐t♦s ✈ért✐❝❡s ❞♦ ❢✉s♦✳ ❖ â♥❣✉❧♦α ❢♦r♠❛❞♦ ♣❡❧❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s
♠á①✐♠❛s é ♦ â♥❣✉❧♦ ❞♦ ❢✉s♦✱ ♦♥❞❡ ❡ss❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s sã♦ ♦s s❡✉s ❧❛❞♦s✳
O α
α
❋✐❣✉r❛ ✷✳✶✻✿ ❋✉s♦ ❞❡ â♥❣✉❧♦ α
❯♠ ❢✉s♦ ❞❡ â♥❣✉❧♦ π r❛❞✐❛♥♦s é ✉♠ ❤❡♠✐s❢ér✐♦ ❡✱ ♣♦rt❛♥t♦✱ s✉❛ ár❡❛ ❝♦♠♣r❡❡♥❞❡
❛ ♠❡t❛❞❡ ❞❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✱ ✐st♦ é✱ 2πr2✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ✉♠ ❢✉s♦ ❞❡ â♥❣✉❧♦ π
2 r❛❞✐❛♥♦s ♦❝✉♣❛ 1
4 ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✱ ❧♦❣♦ s✉❛ ár❡❛ éπr
2✳ ❉❡ ✉♠ ♠♦❞♦ ❣❡r❛❧✱ ❛ ár❡❛
❞❡ ✉♠ ❢✉s♦ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ s❡✉ â♥❣✉❧♦✳ ❆ss✐♠ s❡♥❞♦✱ s❡ ♦ â♥❣✉❧♦ ❞❡ ✉♠ ❢✉s♦ ♠❡❞❡ α
r❛❞✐❛♥♦s✱ ❛ ár❡❛ ❞❡ss❡ ❢✉s♦ é ✐❣✉❛❧ ❛2αr2✳
❉❡✜♥✐çã♦ ✽ ❉❛❞♦ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r X ♥❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✱ ❝❤❛♠❛✲s❡ ❛♥tí♣♦❞❛ ❞❡ X✱ ♦ ú♥✐❝♦ ♣♦♥t♦ X′ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ t❛❧ q✉❡ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ XX′ ❝♦♥té♠ ♦
❝❡♥tr♦ O✳
✹❊♠ ✶✻✷✻ ♣✉❜❧✐❝♦✉ ✉♠ tr❛t❛❞♦ s♦❜r❡ tr✐❣♦♥♦♠❡tr✐❛ ❝♦♥t❡♥❞♦ ❛s ♣r✐♠❡✐r❛s ❛❜r❡✈✐❛t✉r❛s sin✱tan✱sec✳
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✷✼
❉❛❞♦ ✉♠ ❢✉s♦ ϕ ♥❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✱ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ❛♥tí♣♦❞❛s ❞♦s
♣♦♥t♦s ❞❡ ϕ é ❛✐♥❞❛ ✉♠ ❢✉s♦ ϕ′✱ ❝❤❛♠❛❞♦ ❞❡ ❢✉s♦ ❛♥tí♣♦❞❛ ❞❡ ϕ✳ ❉❡ss❡ ♠♦❞♦✱ s✉r❣❡ ♦
❝♦♥❝❡✐t♦ ❞❡ ❢✉s♦ ❝♦♠♣❧❡t♦✳
❉❡✜♥✐çã♦ ✾ ❯♠ ❢✉s♦ ❝♦♠♣❧❡t♦ é ❛ r❡✉♥✐ã♦Φ = ϕ∪ϕ′✳
❋✐❣✉r❛ ✷✳✶✼✿ ❋✉s♦ ❝♦♠♣❧❡t♦
▲❡♠❛ ✷✳✷✳✷ ❙❡❥❛ Φ ✉♠ ❢✉s♦ ❝♦♠♣❧❡t♦✱ ❝✉❥♦ â♥❣✉❧♦ ♠❡❞❡ α r❛❞✐❛♥♦s✳ ◗✉❛❧q✉❡r ♣❧❛♥♦
q✉❡ ♣❛ss❡ ♣❡❧♦ ❝❡♥tr♦ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❛ ❞❡❝♦♠♣õ❡ ❡♠ ❞♦✐s ❤❡♠✐s❢ér✐♦s H ❡ H′✳ ❆s
♣❛rt❡sR ❡R′ ❞♦ ❢✉s♦ ❝♦♠♣❧❡t♦ Φ❝♦♥t✐❞♦s ❡♠ ❝❛❞❛ ✉♠ ❞♦s ❤❡♠✐s❢ér✐♦s ♣♦ss✉❡♠ ❛ ♠❡s♠❛
ár❡❛2αr2✳
❉❡♠♦♥str❛çã♦✿ ❇❛st❛ ♣r♦✈❛r q✉❡R ❡R′ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ár❡❛✱ ♣♦✐s ❛ss✐♠ s❡♥❞♦✱
ár❡❛ ❞❡Φ = ✭ár❡❛ ❞❡ R✮ ✰ ✭ár❡❛ ❞❡R′✮ = 2·✭ár❡❛ ❞❡ R✮
❡✱ ♣♦rt❛♥t♦✱
ár❡❛ ❞❡R = 1
2·✭ár❡❛ ❞❡ Φ✮= 2αr 2✳
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✷✽
❙❡♥❞♦ R ❡ R′ ✜❣✉r❛s ❛♥tí♣♦❞❛s✱ t♦❞♦ ♣♦♥t♦ ❞❡ R′ é ♦ ❛♥tí♣♦❞❛ ❞❡ ✉♠ ♣♦♥t♦ ❞❡
R❡ r❡❝✐♣r♦❝❛♠❡♥t❡✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱R =s∪t é ❛ r❡✉♥✐ã♦ ❞❡ ❞♦✐s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s
❝♦♠ ✉♠ ✈ért✐❝❡ ❝♦♠✉♠ ❡R′ =s′∪t′ é ❛ r❡✉♥✐ã♦ ❞♦s tr✐â♥❣✉❧♦s ❛♥tí♣♦❞❛s ❞❡s ❡t✳ ❉❡st❛
❢♦r♠❛✱ ❛♣❡♥❛s é ♣r❡❝✐s♦ ♣r♦✈❛r q✉❡ ♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦t ❡ ♦ s❡✉ ❛♥tí♣♦❞❛t′ tê♠ ❛ ♠❡s♠❛
ár❡❛✳ ❖❜s❡r✈❡ q✉❡ t ❡ t′ ♣♦ss✉❡♠ â♥❣✉❧♦s ✐❣✉❛✐s ❡ ❧❛❞♦s ❝♦♥❣r✉❡♥t❡s✱ ❞♦✐s ❛ ❞♦✐s✱ ♣♦ré♠
t ❡ t′ ♥ã♦ sã♦ ❝♦♥❣r✉❡♥t❡s✱ ✉♠❛ ✈❡③ q✉❡ ♥ã♦ é ♣♦ssí✈❡❧✱ ♣♦r ✉♠ ♠♦✈✐♠❡♥t♦ rí❣✐❞♦✱ ♠♦✈❡r
✉♠ ❞❡❧❡s ♥♦ ❡s♣❛ç♦ ❛té s♦❜r❡♣♦r✲s❡ ❡①❛t❛♠❡♥t❡ s♦❜r❡ ♦ ♦✉tr♦✱ ❡①❝❡t♦ ♥♦ ❝❛s♦ ❡①❝❡♣❝✐♦♥❛❧ ❞❡ q✉❡ ❡❧❡s s❡❥❛♠ ✐sós❝❡❧❡s✳ ❆s ✜❣✉r❛s s❡❣✉✐♥t❡s ♠♦str❛♠ ❞✉❛s t❡♥t❛t✐✈❛s ❞❡ s♦❜r❡♣♦rt ❡ t′✳ ◆✉♠❛ ❞❡❧❛s ❢❛③❡♠✲s❡ ❝♦✐♥❝✐❞✐r ♦s três ✈ért✐❝❡s✳ ◆❛ ♦✉tr❛✱ ❝♦✐♥❝✐❞❡✲s❡ ✉♠ â♥❣✉❧♦ ❞❡ t
❝♦♠ ♦ ♦✉tr♦ ✐❣✉❛❧ ❞❡t′✳
❋✐❣✉r❛ ✷✳✶✾✿ ❙♦❜r❡♣♦s✐çã♦ ❞♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s
◆❛ s❡❣✉♥❞❛ t❡♥t❛t✐✈❛✱ ♦❜s❡r✈❡ q✉❡ s❡ t ✭❡ ♣♦rt❛♥t♦ t′✮ ❢♦r ✐sós❝❡❧❡s ❡♥tã♦ t é
❝♦♥❣r✉❡♥t❡ ❛♦ s❡✉ ❛♥tí♣♦❞❛ t′✱ ❧♦❣♦ ❡st❡s ❞♦✐s tr✐â♥❣✉❧♦s tê♠ ❛ ♠❡s♠❛ ár❡❛✳ ◆♦ ❝❛s♦
❣❡r❛❧✱ ♣r♦❝❡❞❡✲s❡ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✳ ❖ ♣❧❛♥♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦s ♣♦♥t♦sA✱ B✱ C✱ ✈ért✐❝❡s
❞❡ t✱ ✐♥t❡rs❡❝t❛♠ ❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ s❡❣✉♥❞♦ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♠í♥✐♠❛✱ ❞❡t❡r♠✐♥❛❞♦
❛ss✐♠ ✉♠❛ ❝❛❧♦t❛ ❡s❢ér✐❝❛ q✉❡ ❝♦♥té♠ ♦ tr✐â♥❣✉❧♦ t✳ ❙❡❥❛ P ♣♦❧♦ ❞❛ ❝❛❧♦t❛✱ ♦ q✉❛❧ é ♦
♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛ ❝❛❧♦t❛ ❝♦♠ ❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ABC t✐r❛❞❛ ♣❡❧♦ ❝❡♥tr♦
❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❖s ❛r❝♦s ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♠á①✐♠❛ P❆✱⌢ P❇ ❡⌢ P❈ ♣♦ss✉❡♠ ♦ ♠❡s♠♦⌢ ❝♦♠♣r✐♠❡♥t♦✱ ❧♦❣♦ ♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦sP AB✱ P BC ❡ P AC sã♦ ✐sós❝❡❧❡s✳ ❆ss✐♠✱ s❡ ♦
♣♦❧♦P ❡st✐✈❡r ♥♦ ✐♥t❡r✐♦r ❞♦ tr✐â♥❣✉❧♦t=ABC✱ t❡♠✲s❡ q✉❡
ár❡❛ ❞❡ t= ✭ár❡❛ ❞❡ P AB✮ ✰ ✭ár❡❛ ❞❡P BC✮ ✰ ✭ár❡❛ ❞❡ P AC✮✳
❖r❛✱ ✉♠❛ ❝♦♥str✉çã♦ ✐♥t❡✐r❛♠❡♥t❡ ❛♥á❧♦❣❛ ♣♦❞❡ s❡r ❡❢❡t✉❛❞❛ ❝♦♠ ♦ tr✐â♥❣✉❧♦ ❛♥✲ tí♣♦❞❛t′ =A′B′C′✱ ❞❡❝♦♠♣♦♥❞♦✲♦ ❝♦♠♦ ✉♥✐ã♦ ❥✉st❛♣♦st❛ ❞♦s tr✐â♥❣✉❧♦s ❡s❢ér✐❝♦s ✐sós❝❡❧❡s
P′A′B′✱ P′B′C′ ❡ P′A′C′✱ ❝❛❞❛ ✉♠ ❞❡❧❡s ❛♥tí♣♦❞❛ ❞♦ s❡✉ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡♠ t✳ ❉♦♥❞❡✱
s❡❣✉❡✲s❡ q✉❡
❈❛♣ít✉❧♦ ✷✳ ❆ ❙✉♣❡r❢í❝✐❡ ❊s❢ér✐❝❛ ❡ ♦s ❚r✐â♥❣✉❧♦s ❊s❢ér✐❝♦s ✷✾
❋✐❣✉r❛ ✷✳✷✵✿ ❖ ♣♦❧♦ P ❞♦ tr✐â♥❣✉❧♦ ABC
P♦r ♦✉tr♦ ❧❛❞♦✱ ♣♦❞❡ ♦❝♦rr❡r ♦ ❝❛s♦ ❞❡ q✉❡ ♦ ♣♦❧♦ P ❡st❡❥❛ ❢♦r❛ ❞♦ tr✐â♥❣✉❧♦ t✳
◆❡st❡ ❝❛s♦✱
ár❡❛ ❞❡ t=✭ár❡❛ ❞❡ P AB✮ ✰ ✭ár❡❛ ❞❡P AC✮ − ✭ár❡❛ ❞❡ P BC✮✳
❆♥❛❧♦❣❛♠❡♥t❡ ♦❝♦rr❡ ❝♦♠ t′✱ ❡ ❝♦♥❝❧✉✐✲s❡ ❝♦♠♦ ❛❝✐♠❛ q✉❡
ár❡❛ ❞❡ t=ár❡❛ ❞❡ t′✳
❆❣♦r❛ ♣♦❞❡♠♦s ❞❡♠♦♥str❛r ❛ ❢ór♠✉❧❛ ❞❡ ●✐r❛r❞ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦✳
❚❡♦r❡♠❛ ✷✳✷✳✶✵ ✭❋ór♠✉❧❛ ❞❡ ●✐r❛r❞✮ ❙❡❥❛♠ α✱ β ❡ γ ♦s â♥❣✉❧♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦
❡s❢ér✐❝♦✱ ♠❡❞✐❞♦s ❡♠ r❛❞✐❛♥♦s✱ ❡♥tã♦ ❛ ár❡❛ A ❞❡ss❡ tr✐â♥❣✉❧♦ é ❞❛❞❛ ♣♦r
A =E·r2
♦♥❞❡ E =α+β+γ−π é ♦ ❡①❝❡ss♦ ❡s❢ér✐❝♦ ❞♦ tr✐â♥❣✉❧♦✳
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ✉♠ ❤❡♠✐s❢ér✐♦H q✉❡ ❝♦♥t❡♥❤❛ ♦ tr✐â♥❣✉❧♦ ❡s❢ér✐❝♦ ❞❛❞♦✳
Pr♦❧♦♥❣❛♥❞♦✱ ♥♦s ❞♦s s❡♥t✐❞♦s✱ ♦s ❧❛❞♦s q✉❡ ❢♦r♠❛♠ ♦ â♥❣✉❧♦α✱ ❛té ❡♥❝♦♥tr❛r❡♠ ♦ ❜♦r❞♦
❞❡ H✱ ♦❜t❡r❡♠♦s ✉♠❛ r❡❣✐ã♦ Rα ⊂ H✱ ❝✉❥❛ ár❡❛ é 2αr2✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ▲❡♠❛ ✷✳✷✳✷
