Open A construção do pentágono regular segundo Euclides

❘❡❣✉❧❛r s❡❣✉♥❞♦ ❊✉❝❧✐❞❡s

♣♦r

❆❧❡① ❈r✐st♦♣❤❡ ❈r✉③ ❞❛ ❙✐❧✈❛

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉♦✉t♦r P❡❞r♦ ❆♥t♦♥✐♦ ❍✐♥♦❥♦s❛ ❱❡r❛

❡ ❝♦✲♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉♦✉t♦r ❋❡r♥❛♥❞♦ ❆♥t♦♥✐♦ ❳❛✈✐❡r ❞❡ ❙♦✉③❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tí✲ t✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❏✉❧❤♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇

†_{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

◗✉❡r♦ ❛❣r❛❞❡❝❡r ❛ ❈❆P❊❙✱ ❙❇▼✱ P❘❖❋▼❆❚✱ ❯❋P❇✱ ❈❈❊◆✲❯❋P❇✱ ❉▼❀

t❛♠❜é♠ ❛♦ Pr♦❢❡ss♦r ❉♦✉t♦r ❏♦ã♦ ▼❛r❝♦s ❞♦ Ó✱ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢❡ss♦r ❉♦✉✲ t♦r P❡❞r♦ ❆♥t♦♥✐♦ ❍✐♥♦❥♦s❛ ❱❡r❛✱ ❛♦ ♠❡✉ ❝♦✲♦r✐❡♥t❛❞♦r Pr♦❢❡ss♦r ❉♦✉t♦r ❋❡r♥❛♥❞♦ ❆♥t♦♥✐♦ ❳❛✈✐❡r ❞❡ ❙♦✉③❛✱ ❛♦ ♣r♦❢❡ss♦r ❡①t❡r♥♦ à ❜❛♥❝❛ ♦ Pr♦❢❡ss♦r ❉♦✉t♦r ❏♦r❣❡ ❆♥t♦♥✐♦ ❍✐♥♦❥♦s❛ ❱❡r❛ ✭❯❋❘P❊✮❀

à ❙❡❝r❡tár✐❛ ❞❡ ❊❞✉❝❛çã♦ ❆r❧❡✐❞❡ ❆❧❜✉q✉❡rq✉❡ ●✉❡rr❛✱ ❛ Pr❡❢❡✐t✉r❛ ▼✉♥✐❝✐♣❛❧ ❞❡ ❚✐♠❜❛ú❜❛✱ ❛♦ ●❡st♦r ❞❛ ❊s❝♦❧❛ ❞❡ ❘❡❢❡rê♥❝✐❛ ❡♠ ❊♥s✐♥♦ ▼é❞✐♦ ❞❡ ❚✐♠❜❛ú❜❛ ♦ Pr♦❢❡ss♦r ▼❡str❡ ❆♥t♦♥✐♦ ❇❛r❜♦③❛❀

❛♦s ❝♦❧❡❣❛s ❞❡ ❝❛♠✐♥❤❛❞❛ ▲❡♦♥❛r❞♦ ❘♦❞r✐❣✉❡s ❞❡ ❆r❛ú❥♦✱ ❙❛❧❛t✐❡❧ ❉✐❛s ❞❛ ❙✐❧✈❛✱ ▼ár❝✐♦ ❆❧✈❡s ▼❛r✐♥❤♦ ❡ ●✐❧❞❡❝✐ ❏♦sé ❏✉st✐♥♦❀

❛♦ ❣r❛♥❞❡ ❛♠✐❣♦ ●✉t❡♠❜❡r❣ ♣❡❧♦ ✐♥❝❡♥t✐✈♦✱ ❡ ❛♦ P❡✳▼❛r✐s❛❧❞♦ ❇❛r❜♦s❛ ❞❡ ▲✐♠❛ ♣❡❧♦ ❛♣♦✐♦ ❡s♣✐r✐t✉❛❧❀

❞❡♠❛✐s ♣r♦❢❡ss♦r❡s ❞♦ ♣♦❧♦ ❯❋P❇❀

❉❡❞✐❝❛tór✐❛

◆❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❝♦♥str✉çõ❡s ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r✱ s❡♥❞♦ ❛ ♣r✐♥❝✐♣❛❧ ❞❡❧❛s ✉♠❛ ❝♦♥str✉çã♦ ❞❡ ❊✉❝❧✐❞❡s ❡♥❝♦♥tr❛❞❛ ♥♦ s❡✉ ❧✐✈r♦ ❖s ❊❧❡♠❡♥t♦s✳ ❆♣r❡s❡♥t❛♠♦s✱ t❛♠❜é♠✱ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡st❛ ❝♦♥tr✉çã♦✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ P♦❧í❣♦♥♦s r❡❣✉❧❛r❡s✱ P❡♥tá❣♦♥♦✱ ❊✉❝❧✐❞❡s✱ ❙❡❣♠❡♥t♦ ➪✉r❡♦✳

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ ♣r❡s❡♥t s♦♠❡ ❝♦♥str✉❝t✐♦♥s ♦❢ t❤❡ r❡❣✉❧❛r ♣❡♥t❛❣♦♥✱ t❤❡ ♠❛✐♥ ♦♥❡ ✐s ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❊✉❝❧✐❞ ❢♦✉♥❞ ✐♥ ❤✐s ❜♦♦❦ ❚❤❡ ❊❧❡♠❡♥ts✳ ❲❡ ❛❧s♦ ♣r❡s❡♥t s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤✐s ❝♦♥str✉❝t✐♦♥✳

❑❡②✇♦r❞s✿ ❘❡❣✉❧❛r P♦❧②❣♦♥s✱ P❡♥t❛❣♦♥✱ ❊✉❝❧✐❞✱ ●♦❧❞❡♥ ▼❡❛♥✳

✶ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛ ✲ ❯♠ ♣♦✉❝♦ ❞❡ ❍✐stór✐❛ ✸ ✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❆❧❡①❛♥❞r✐❛ ❡ ❊✉❝❧✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ ❖❜r❛s P❡r❞✐❞❛s ❡ ❖s ❊❧❡♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹

✷ ❆ ❝♦♥str✉çã♦ ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r ✼ ✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ ❖s P♦❧í❣♦♥♦s r❡❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ ❈♦♥str✉çõ❡s ❞♦ P❡♥tá❣♦♥♦ ❘❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✹ ❖✉tr❛s ❈♦♥str✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✸ ❆♣❧✐❝❛çõ❡s ✹✸

✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✷ ❊❧❡♠❡♥t♦s ◆♦tá✈❡✐s ❞♦ P♦❧í❣♦♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✸ ❈á❧❝✉❧♦ ❞♦ ▲❛❞♦ ❡ ❆♣ót❡♠❛ ❞♦ P♦❧í❣♦♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

❆ ❖ ❙❡❣♠❡♥t♦ ➪✉r❡♦ ✺✹

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✼

■♥tr♦❞✉çã♦

❆ ♠❛t❡♠át✐❝❛ é ✉♠❛ ❝✐ê♥❝✐❛ ❞❡ ❛s♣❡❝t♦ ú♥✐❝♦ ❞♦ ♣❡♥s❛♠❡♥t♦ ❤✉♠❛♥♦ ❡ s❡ ❞✐✲ ❢❡r❡ ❞❡ t♦❞❛s ❛s ♦✉tr❛s ❝✐ê♥❝✐❛s✱ ❡❧❛ ♣♦ss✉✐ ✐♥ú♠❡r♦s r❛♠♦s ♣❛r❛ t♦❞♦s ♦s ♣♦ssí✈❡✐s ❡st✉❞❛♥t❡s ❞❡ss❛ ♠❛❣♥í✜❝❛ ❝✐ê♥❝✐❛ ❡①❛t❛✳ ❉❡ s❡✉s r❛♠♦s✱ ❛ ❣❡♦♠❡tr✐❛ é ✉♠ ❞❡ss❡s q✉❡ t❡♠ ❛s ♠❛✐s ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s✱ q✉❡ sã♦ ✉t✐❧✐③❛❞❛s ❞♦s ♣r✐♠ór❞✐♦s ❞❛ ❝✐✈✐❧✐③❛çã♦ ❤✉♠❛♥❛ ❛té ♦ ♠♦♠❡♥t♦ ❛t✉❛❧✳

❆ ❣❡♦♠❡tr✐❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ t❡♠ ❣r❛♥❞❡ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡ ❛ ❢❛❧t❛ ❞❡ ✉♠ ♠❛♥✉❛❧ ❞❡ss❛ ár❡❛ ❞❛ ♠❛t❡♠át✐❝❛ ❢❡③ ❝♦♠ q✉❡ ❛❧❣✉♥s ♠❛t❡♠át✐❝♦s ♣r♦❞✉③✐ss❡♠ ♦s s❡✉s ♣r✐♠❡✐r♦s ♠❛♥✉❛✐s ❞❡ ❡♥s✐♥♦ ✭♣r✐♠ór❞✐♦s ❞♦s ❧✐✈r♦s✮✳ ❉❡♥tr❡ ❡ss❡s ❛✉t♦r❡s ❞❡st❛❝❛✲s❡ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛ ♦ q✉❛❧ ❡s❝r❡✈❡✉ ✈ár✐♦s ❧✐✈r♦s✱ ❡♥tr❡ ❡st❡s ❖s ❊❧❡♠❡♥t♦s✳

❊st❛ ❞✐ss❡rt❛çã♦ ❢❛③ ✉♠ ❛♣❛♥❤❛♥❞♦ ❤✐stór✐❝♦ ❞♦ ♣♦✉❝♦ ❞♦ q✉❡ s❡ s❛❜❡ s♦❜r❡ ❊✉❝❧✐❞❡s✳ ◆♦ ❝❛♣ít✉❧♦ ■✱ s❡rá ❞❛❞❛ ✐♥❢♦r♠❛çõ❡s s♦❜r❡✿ ❆❧❡①❛♥❞r✐❛✱ ♦s ❧✐✈r♦s ♣❡r❞✐❞♦s ❞❡ ❊✉❝❧✐❞❡s✱ ♦ ❧✐✈r♦ ❖s ❊❧❡♠❡♥t♦s ❞♦ q✉❛❧ ❢❛r❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ s✐♥tét✐❝❛ ❞❡ ❝❛❞❛ ❝❛♣ít✉❧♦✱ ✐♥❢♦r♠❛♥❞♦ ♦ s❡✉ ❝♦♥t❡ú❞♦ ♣❛r❛ q✉❡ ♦ ❧❡✐t♦r✱ q✉❡ q✉❡✐r❛ ♠❛✐s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦ ♠❡s♠♦✱ t❡♥❤❛ ✉♠❛ r❡❢❡rê♥❝✐❛✳

◆♦ ❝❛♣ít✉❧♦ ■■✱ s❡rã♦ ❞❡s❡♥✈♦❧✈✐❞♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s✱ ❝♦♠ ❞❡♠♦♥str❛çõ❡s ❞❡ ✐♥s❝r✐çã♦ ❡ ❝✐r❝✉♥s❝r✐çã♦ ❞❡st❡s ♣♦❧í❣♦♥♦s ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛❀ ❧♦❣♦ ❛♣ós é ❢❡✐t❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❧ó❣✐❝❛ ❞❡ t❡♦r❡♠❛s ❡ ❝♦♥str✉çõ❡s ✉s❛❞❛s ♣♦r ❊✉❝❧✐❞❡s✱ ❞♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ ✐♥s❝r✐çã♦ ❞❡ q✉❛❞r✐❧át❡r♦s ❡♠ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ✐♥❝❧✉✐♥❞♦ ❛ ❝♦♥str✉çã♦ ❞♦ s❡❣♠❡♥t♦ á✉r❡♦✱ ❞❛❞♦ ✉♠ s❡❣♠❡♥t♦ q✉❛❧q✉❡r✱ ❝♦♥str✉çã♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❡♠ q✉❡ ♦s â♥❣✉❧♦s ❞❛ ❜❛s❡ é ♦ ❞♦❜r♦ ❞♦ â♥❣✉❧♦ ❞♦ ✈ért✐❝❡✱ ❞❛❞♦ ✉♠ s❡❣♠❡♥t♦ q✉❛❧q✉❡r✱ ❛s ❞❡♠♦♥str❛çõ❡s ❢❡✐t❛s t♦❞❛s sã♦ ❝♦♠ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ár❡❛s✳ ❊❧❛s ❥✉st✐✜❝❛♠✱ ♠❛t❡♠❛t✐❝❛♠❡♥t❡✱ ❛ ❝♦♥str✉çã♦ ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r ❢❡✐t❛ ❝♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦✱ ❞❡♣♦✐s ❛♣r❡s❡♥t❛♠♦s ♠❛✐s três ❝♦♥str✉çõ❡s ❞✐❢❡r❡♥t❡s✳ ❖ ❝❛♣ít✉❧♦ é ✜♥❛❧✐③❛❞♦ ❝♦♠ ❛ ❞❡♠♦♥str❛çã♦ q✉❡ ♦ s❡❣♠❡♥t♦ á✉r❡♦ ❞❛ ❞✐❛❣♦♥❛❧ ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r é ❝♦♥❣r✉❡♥t❡ ❝♦♠ ♦ ❧❛❞♦ ❞❡ss❡ ♣♦❧í❣♦♥♦✳

❝♦♠ r❡s♦❧✉çã♦ ❞❡ q✉❛tr♦ ♣r♦❜❧❡♠❛s q✉❡ sã♦✿ ❝❛❧❝✉❧❛r ❛ ❛♣ót❡♠❛ ❡ ♦ ❧❛❞♦ ❞♦s ♣♦❧í✲ ❣♦♥♦s ❤❡①á❣♦♥♦✱ ❞❡❝á❣♦♥♦✱ ♣❡♥tá❣♦♥♦ ❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❞❡ ♥ ❧❛❞♦s✳

❖ s❡❣♠❡♥t♦ á✉r❡♦ é ❝♦♠❡♥t❛❞♦ ✈ár✐❛s ✈❡③❡s ♥❡st❡ tr❛❜❛❧❤♦✱ ♣♦r ✐ss♦ ✐♥❝❧✉í♠♦s ✉♠ ❛♣ê♥❞✐❝❡ ❢❛❧❛♥❞♦ s♦❜r❡ ♦ ♠❡s♠♦✱ ❞❡♠♦♥str❛♥❞♦ ♦ ✈❛❧♦r ❞❡❧❡✱ ❛ s✉❛ ✐♥❝♦♠❡♥s✉✲ r❛❜✐❧✐❞❛❞❡ ❡ ♦♥❞❡ é ❡♥❝♦♥tr❛❞♦ ❡♠ ✈ár✐❛s s✐t✉❛çõ❡s ♥❛ ♥❛t✉r❡③❛✳

❈❛♣ít✉❧♦ ✶

❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛ ✲ ❯♠ ♣♦✉❝♦

❞❡ ❍✐stór✐❛

✶✳✶ ■♥tr♦❞✉çã♦

◆❡st❡ ❝❛♣ít✉❧♦ ❢❛❧❛r❡♠♦s s✐♥t❡t✐❝❛♠❡♥t❡ ♦ q✉❡ ♦❝♦rr✐❛ ♥❛ é♣♦❝❛ ❞❡ ❊✉❝❧✐❞❡s✳ ❉✐ss❡rt❛r❡♠♦s ✉♠ ♣♦✉❝♦ ❞❡st❡ ♠❛t❡♠át✐❝♦ ❢❛s❝✐♥❛♥t❡ ❡ s♦❜r❡ ♦ s❡✉ ❧✐✈r♦ q✉❡ s❡r✈❡ ❞❡ ❜❛s❡ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ✉♠❛ ár❡❛ ❞❛ ♠❛t❡♠át✐❝❛ q✉❡ ♣r♦✈♦❝❛ ❢❛s❝í♥✐♦ ❡♠ ✈ár✐♦s ❡st✉❞✐♦s♦s ✲ ❛ ❣❡♦♠❡tr✐❛✳

✶✳✷ ❆❧❡①❛♥❞r✐❛ ❡ ❊✉❝❧✐❞❡s

❆ ♠♦rt❡ ❞❡ ❆❧❡①❛♥❞r❡✱ ♦ ●r❛♥❞❡✱ ❧❡✈♦✉ ❛ ❝♦♥✢✐t♦s ♠♦rt❛✐s ❡♥tr❡ ♦s ❣❡♥❡r❛✐s ❞♦ ❡①ér❝✐t♦ ❣r❡❣♦✳ ❖ ❝♦♥tr♦❧❡ ❞❛ ♣❛rt❡ ❡❣í♣❝✐❛ ❡st❛✈❛ ♥❛s ♠ã♦s ❞♦s Pt♦❧♦♠❡✉s✱ ♦s ❣♦✈❡r♥❛♥t❡s ♠❛❝❡❞ô♥✐❝♦s ❞♦ ❊❣✐t♦✳ Pt♦❧♦♠❡✉ ■ ❛ss❡♥t♦✉ ♦ ❛❧✐❝❡r❝❡ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❆❧❡①❛♥❞r✐❛ ✲ ▼✉s❡✉ ❡ ❇✐❜❧✐♦t❡❝❛ ✲ ❡❧❛ ❢♦✐ ✜♥❛♥❝✐❛❞❛ t❛♥t♦ ♣♦r Pt♦❧♦♠❡✉ ■ ❝♦♠♦ ♣❡❧♦ s❡✉ ✜❧❤♦ Pt♦❧♦♠❡✉ ■■✳ ❉❡st❡ ❝❡♥tr♦ ❛❝❛❞ê♠✐❝♦ ❡❧❡ r❡❝r✉t♦✉ ✉♠ ❣r✉♣♦ ❞❡ sá❜✐♦s✱ ♦s ❞❡ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ❡ ❡♠ ❞✐✈❡rs❛s ár❡❛s✳ ❊♥tr❡ ❡ss❡s sá❜✐♦s s❡ ❡♥❝♦♥tr❛ ❊✉❝❧✐❞❡s ✲ ♦ ❛✉t♦r ❞❡ ✉♠ ❞♦s ❧✐✈r♦s ♠❛✐s ❡st✉❞❛❞♦ ❞❡♣♦✐s ❞❛ ❇í❜❧✐❛✱ ❖s ❊❧❡♠❡♥t♦s ✭❙t♦✐❝❤✐❛✮✳ ➱ ✐♥t❡r❡ss❛♥t❡ ❢r✐s❛r q✉❡ ❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❆❧❡①❛♥❞r✐❛✱ ❛♣ós ✉♥s ✹✵ ❛♥♦s✱ ♦st❡♥t❛✈❛ ♠❛✐s ❞❡ ✻✵✵✳✵✵✵ r♦❧♦s ❞❡ ♣❛♣✐r♦ ❡ ❡♠ ✸✵✵ ❛✳❈✳ ❛ ✉♥✐✈❡rs✐❞❛❞❡ ❛❜r✐✉ ❛s ♣♦rt❛s ❡✱ ❝♦♠ ✐ss♦✱ ❛ ❝✐❞❛❞❡ ❞❡ ❆❧❡①❛♥❞r✐❛ s❡ t♦r♥♦✉ ❛ ♠❡tró♣♦❧❡ ❞❡ ❛♠♣❧♦ ❝♦♥❤❡❝✐♠❡♥t♦

✐♥t❡❧❡❝t✉❛❧ ❣r❡❣♦ ✭✈❡❥❛ ❬✶❪✮✳

P♦✉❝♦ s❡ s❛❜❡ s♦❜r❡ ❛ ✈✐❞❛ ❞❡ ❊✉❝❧✐❞❡s✱ ❛♣❡♥❛s q✉❡✱ s❡♠ ❞ú✈✐❞❛✱ ❡❧❡ ❢♦✐ ♣r♦❢❡ss♦r ❞❛ ♣r✐♠❡✐r❛ ✉♥✐✈❡rs✐❞❛❞❡ q✉❡ ❥á s❡ ♦✉✈✐✉ ❢❛❧❛r✱ ❛ ❢❛♠♦s❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❆❧❡①❛♥❞r✐❛✳ ❚r❛t❛✲s❡ ❞❛ ♣r✐♠❡✐r❛ ✐♥st✐t✉✐çã♦ ❡♠ q✉❡ s✉❛ ♦r❣❛♥✐③❛çã♦ t❡♠ s❡♠❡❧❤❛♥ç❛ ❝♦♠ ❛s q✉❡ ❡①✐st❡♠ ❛t✉❛❧♠❡♥t❡✳ ➱ ✉♠❛ ✐♥❢❡❧✐❝✐❞❛❞❡ q✉❡ ♥ã♦ ❡①✐st❡♠ r❡❣✐str♦s ❤✐stór✐❝♦s ❞❛ ❞❛t❛ ❡ ❞♦ ❧♦❝❛❧ ❞♦ ♥❛s❝✐♠❡♥t♦ ❞❡ ❊✉❝❧✐❞❡s✱ ♠❛s s❛❜❡✲s❡ q✉❡✱ ♣r♦✈❛✈❡❧♠❡♥t❡✱ s✉❛ ❢♦r♠❛çã♦ ❢♦✐ ♥❛ ❡s❝♦❧❛ ♣❧❛tô♥✐❝❛ ❞❡ ❆t❡♥❛s✳ ❆s ♦❜r❛s q✉❡ s♦❜r❡✈✐✈❡r❛♠ ❞♦ ❞❡st✐♥♦ trá❣✐❝♦ ❞♦ ❡sq✉❡❝✐♠❡♥t♦ sã♦✿ ❖s ❡❧❡♠❡♥t♦s✱ ❖s ❞❛❞♦s✱ ❉✐✈✐sã♦ ❞❡ ✜❣✉r❛s✱ ❖s ❋❡♥ô♠❡♥♦s ❡ Ó♣t✐❝❛✳ s❡❣✉♥❞♦ ❬✶❪ ❛♣❛r❡❝❡ ✉♠ r❡tr❛t♦ ❞❡ ❊✉❝❧✐❞❡s ❡♠ ✈ár✐♦s ❧✐✈r♦s ❞❡ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛✱ ❡ss❛ ✐♠❛❣❡♠ é ❞❡ ❊✉❝❧✐❞❡s ❞❡ ❙❛♠❛r❛ ✭✈❡❥❛ ❬✶❪✱ ❬✷❪✱ ❬✸❪✮✳

✶✳✸ ❖❜r❛s P❡r❞✐❞❛s ❡ ❖s ❊❧❡♠❡♥t♦s

❉♦ q✉❡ ❊✉❝❧✐❞❡s ♣r♦❞✉③✐✉✱ ♠❛✐s ❞❛ ♠❡t❛❞❡ ❢♦✐ ♣❡r❞✐❞♦✱ ❜❡♠ ❝♦♠♦ ✉♠❛ ♦❜r❛ s♦❜r❡ ❝ô♥✐❝❛s ❞❡ q✉❛tr♦ ✈♦❧✉♠❡s ❬✶❪✱ t❛♠❜é♠ ✉♠ tr❛t❛❞♦ s♦❜r❡ ▲✉❣❛r❡s ●❡♦♠étr✐❝♦s ❙ó❧✐✲ ❞♦s✭♥♦♠❡ ❣r❡❣♦ ♣❛r❛ ❛s s❡❝çõ❡s ❝ô♥✐❝❛s✮✱ ❞❡ ❆r✐st❡✉✱ ❛ ♦❜r❛ ❞❡❧❡ ❢♦✐ s✉♣❡r❛❞❛ ♣❡❧❛ ❛ ❞❡ ❆♣♦❧ô♥✐♦✳ ❊♥tr❡ ❛s ♦❜r❛s q✉❡ t❛♠❜é♠ ❢♦r❛♠ ♣❡r❞✐❞❛s t❡♠♦s ✉♠❛ s♦❜r❡ ▲✉❣❛r❡s ●❡♦♠étr✐❝♦s ❞❡ ❙✉♣❡r❢í❝✐❡✱ ♦✉tr❛ s♦❜r❡ Ps❡✉❞❛r✐❛ ✭♦✉ ❢❛❧á❝✐❛s✮ ❡ ✉♠❛ t❡r❝❡✐r❛ s♦❜r❡ P♦r✐s♠❛✱ s❡❣✉♥❞♦ P❛♣✉s ✉♠ ♣♦r✐s♠❛ é ❛❧❣♦ ✐♥t❡r♠❡❞✐ár✐♦ ❡♥tr❡ ✉♠ t❡♦r❡♠❛✱ ✉♠❛ ♣r♦♣♦st❛ ❞❡ ❞❡♠♦♥str❛çã♦✱ ♦✉ ✉♠❛ ❝♦♥str✉çã♦ ♣❛r❛ r❡s♦❧✈❡r ✉♠ ♣r♦❜❧❡♠❛❀ ♦✉tr♦s ❞❡s❝r❡✈❡♠ ❝♦♠♦ ✉♠❛ ♣r♦♣♦s✐çã♦ ❡♠ q✉❡ s❡ ❞❡t❡r♠✐♥❛ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❛s q✉❛♥✲ t✐❞❛❞❡s ❝♦♥❤❡❝✐❞❛s ❡ ✈❛r✐á✈❡✐s ♦✉ ✐♥❞❡t❡r♠✐♥❛❞❛s✱ t❛❧✈❡③ ❛ ♠❡❧❤♦r ♣r♦①✐♠✐❞❛❞❡ ❞❡ ✉♠❛ ✐❞❡✐❛ ❞❡ ❢✉♥çã♦ ✭✈❡❥❛ ❬✶❪✱ ❬✷❪✮✳

❖s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s é✱ s❡♠ ❞ú✈✐❞❛✱ ✉♠ ❞♦s ❧✐✈r♦s ♠❛✐s ❝✐t❛❞♦ ❡ ❡st✉❞❛❞♦✳ ◆❡♥❤✉♠ ❧✐✈r♦✱ ❝♦♠ ❡①❝❡çã♦ ❞❛ ❜í❜❧✐❛ ❡①❡r❝❡✉ ✉♠❛ ❣r❛♥❞❡ ✐♥✢✉ê♥❝✐❛ ♥♦ ♣❡♥s❛♠❡♥t♦ ❝✐❡♥tí✜❝♦❀ s❡❣✉♥❞♦ ❛s ✐♥❢♦r♠❛çõ❡s ❞❡s❝r✐t❛s ❡♠ ❬✶❪ ❡ ❬✷❪✱ ❛♠❜♦s ❛✜r♠❛♠ q✉❡ ❛♥t❡s ❞❡st❡ ❊❧❡♠❡♥t♦s✱ ❡①✐st✐r❛♠ ♦✉tr♦s ❝♦♠♦ ❛ ✈❡rsã♦ ❞❡ ❍✐♣ó❝❛tr❡s ❞❡ ❈❤✐♦s✱ ♠❛s ♥ã♦ é ❡♥❝♦♥tr❛❞❛ ♥❡♥❤✉♠❛ ✈❡rsã♦ ❛♥t❡r✐♦r ❛ ❞❡ ❊✉❝❧✐❞❡s✱ ♣♦✐s ❞❡ tã♦ ❜❡♠ ❡❧❛❜♦r❛❞❛ ❢♦✐ ❛ q✉❡ s♦❜r❡✈✐✈❡✉ ❞❡rr✉❜❛♥❞♦ ♦s s❡✉s ❝♦♥❝♦rr❡♥t❡s ❞❛ é♣♦❝❛✳ ❯♠❛ ✈✐sã♦ ❣❡r❛❧ ❞❡ ❛❧❣✉♥s ♣r♦❢❡ss♦r❡s q✉❡ ❛❝r❡❞✐t❛♠ q✉❡ ❡ss❡ ❧✐✈r♦ é ✉♠ ❝♦♠♣ê♥❞✐♦ ❞❡ ❣❡♦♠❡tr✐❛✱ ❡♥tr❡t❛♥t♦ ❡ss❡ ❛❝❤❛❞♦ ❢❛❧❛ ❞❡ ✉♠ t❡①t♦ ✐♥tr♦❞✉tór✐♦ ❝♦❜r✐♥❞♦ t♦❞❛ ❛ ♠❛t❡♠át✐❝❛ ❡❧❡♠❡♥t❛r✳ ❊❧❡ ❛❜♦r❞❛ ♦s s❡❣✉✐♥t❡s ❝♦♥t❡ú❞♦s✿ ❛r✐t♠ét✐❝❛ ✭t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✮✱ ❣❡♦♠❡tr✐❛ s✐♥tét✐❝❛ ✭♣♦♥t♦s✱ r❡t❛s✱ ♣❧❛♥♦s✱ ❝ír❝✉❧♦s ❡ ❡s❢❡r❛s✮ ❡ á❧❣❡❜r❛✳ ❚♦❞♦ ❡ss❡

❝♦♥t❡ú❞♦ ❢♦✐ ❞✐str✐❜✉í❞♦ ❛♦ ❧♦♥❣♦ ❞❡ ✶✸ ❧✐✈r♦s ♦✉ ❝❛♣ít✉❧♦s❀ ♦♥❞❡ s❡ ❞❡s❡♥✈♦❧✈❡♠ ✹✻✺ ♣r♦♣♦s✐çõ❡s✳ ❙❡rã♦ ❢❡✐t♦s ❛❧❣✉♥s ❝♦♠❡♥tár✐♦s ❛ r❡s♣❡✐t♦ ❞♦s tr❡③❡ ❧✐✈r♦s ✐♥❞✐❝❛♥❞♦✲♦s ♣♦r ❛❧❣❛r✐s♠♦s r♦♠❛♥♦s✳

❖ ▲✐✈r♦ ■ ♣♦ss✉✐ ♦ s❡❣✉✐♥t❡ ❝♦♥t❡ú❞♦✿ ✷✸ ❞❡✜♥✐çõ❡s✱ ✺ ♣♦st✉❧❛❞♦s✱ ✾ ♥♦çõ❡s ❝♦✲ ♠✉♥s ♥❡❝❡ssár✐❛s ♣❛r❛ ❛s ❞❡♠♦♥str❛çõ❡s ❡ ✹✽ ♣r♦♣♦s✐çõ❡s✳ ❆s ♣r✐♠❡✐r❛s ✷✸ ♣r♦♣♦s✐✲ çõ❡s tr❛t❛♠ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ tr✐â♥❣✉❧♦ ❡ ✐♥❝❧✉❡♠ ♦s três t❡♦r❡♠❛s ❞❡ ❝♦♥❣r✉ê♥❝✐❛✳ ❆s ♣r♦♣♦s✐çõ❡s ■ ✷✼ ❛ ■ ✸✷ ❡st❛❜❡❧❡❝❡♠ ❛ t❡♦r✐❛ ❞❛s ♣❛r❛❧❡❧❛s ❡ ♣r♦✲ ✈❛♠ q✉❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ✐❣✉❛❧ ❛ s♦♠❛ ❞❡ ❞♦✐s â♥❣✉❧♦s r❡t♦s✳ ❆s ♦✉tr❛s ♣r♦♣♦s✐çõ❡s ❢❛❧❛♠ s♦❜r❡ ♣❛r❛❧❡❧♦❣r❛♠♦s✱ tr✐â♥❣✉❧♦s✱ q✉❛❞r❛❞♦s✱ r❡✲ ❧❛çã♦ ❡♥tr❡ ár❡❛s ❡ ❛ ❞❡♠♦♥str❛çã♦ ❡✉❝❧✐❞✐❛♥❛ ❞♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❝♦♠ t❛♠❜é♠ ❛ s✉❛ r❡❝í♣r♦❝❛✳ ❖ ♠❛t❡r✐❛❧ ❞❡ss❡ ❧✐✈r♦ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♣❡❧♦s ♣✐t❛❣ór✐❝♦s ❛♥t✐❣♦s ❬✷❪✳ ❖ ▲✐✈r♦ ■■ é ♣❡q✉❡♥♦ ❝♦♠♣❛r❛❞♦ ❝♦♠ ♦ ❧✐✈r♦ ■✳ ❊st❡ ♣♦ss✉✐ ✶✹ ♣r♦♣♦s✐çõ❡s q✉❡ ❧✐❞❛♠ ❝♦♠ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ár❡❛s ✭❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ár❡❛s✮ ❡ ❝♦♠ ❛ á❧❣❡❜r❛ ❣❡♦♠é✲ tr✐❝❛✳

❖ ▲✐✈r♦ ■■■ ♣♦ss✉✐ tr✐♥t❛ ❡ ♥♦✈❡ ♣r♦♣♦s✐çõ❡s✳ ❈♦♥té♠ ♠✉✐t♦ ❞♦s t❡♦r❡♠❛s s♦❜r❡ ❝ír❝✉❧♦s✱ ❝♦r❞❛s✱ s❡❝❛♥t❡s✱ t❛♥❣❡♥t❡s ❡ ♠❡❞✐❞❛s ❞❡ â♥❣✉❧♦s ❛ss♦❝✐❛❞♦s q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞♦s t❡①t♦s ❞❡ ❣❡♦♠❡tr✐❛ ❡❧❡♠❡♥t❛r✳

❖ ▲✐✈r♦ ■❱ t❡♠ ❡①❛t❛♠❡♥t❡ ❞❡③❡ss❡✐s ♣r♦♣♦s✐çõ❡s✱ q✉❡ ♠♦str❛♠ ❛ ❝♦♥str✉çã♦ ❝♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦✱ ❞❡ ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s ❞❡ três✱ q✉❛tr♦✱ ❝✐♥❝♦✱ s❡✐s ❡ q✉✐♥③❡ ❧❛❞♦s✱ ❜❡♠ ❝♦♠♦ ❛ ✐♥s❝r✐çã♦ ❡ ❛ ❝✐r❝✉♥s❝r✐çã♦ ❞❡ss❡s ♣♦❧í❣♦♥♦s ♥✉♠ ❝ír❝✉❧♦ ❞❛❞♦✳

❖ ▲✐✈r♦ ❱ é ✉♠❛ ❡①♣♦s✐çã♦ ❞❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ❞❡ ❊✉❞♦①♦✱ ❛♣❧✐❝á✈❡❧ ❛ ❣r❛♥✲ ❞❡③❛s ❝♦♠❡♥s✉rá✈❡✐s ❡ ✐♥❝♦♠❡♥s✉rá✈❡✐s❀ ❛❧é♠ ❞✐ss♦✱ ❢❛③ ✉♠❛ ❞❡✜♥✐çã♦ ❞❡ ♣r♦♣♦rçã♦✳

❖ ▲✐✈r♦ ❱■ ❛♣❧✐❝❛ ❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ❡✉❞♦①✐❛♥❛ à ❣❡♦♠❡tr✐❛ ♣❧❛♥❛✳ ❊♥❝♦♥tr❛♠✲ s❡ ♥❡ss❡ ❧✐✈r♦ ♦s t❡♦r❡♠❛s ❢✉♥❞❛♠❡♥t❛✐s ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s❀ ❝♦♥str✉çõ❡s ❞❡ t❡r❝❡✐r❛✱ q✉❛rt❛ ❡ ♠é❞✐❛s ♣r♦♣♦r❝✐♦♥❛✐s❀ ❛s r❡s♦❧✉çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ❡q✉❛çõ❡s q✉❛✲ ❞rát✐❝❛s❀ ♦ t❡♦r❡♠❛ ❞❛ ❜✐ss❡tr✐③ ✐♥t❡r♥❛❀ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❚❡♦r❡♠❛ P✐tá❣♦r❛s✱ ❡♥tr❡ ♦✉tr♦s✳

❖s ▲✐✈r♦s ❱■■✱ ❱■■■ ❡ ■❳ tr❛t❛♠ ❞❛ t❡♦r✐❛ ❡❧❡♠❡♥t❛r ❞♦s ♥ú♠❡r♦s✱ ♥❡ss❡s ❝❛♣í✲ t✉❧♦s ❞❡s❡♥✈♦❧✈❡♠ ♦ ❛❧❣♦r✐t♠♦ ❡✉❝❧✐❞✐❛♥♦ ♣❛r❛ ❛❝❤❛r ♦ ♠❞❝ ❡ ♦ ♠♠❝❀ ❛s ♣r♦♣♦rçõ❡s ❝♦♥tí♥✉❛s ❡ ♣r♦❣r❡ssõ❡s ❣❡♦♠étr✐❝❛s r❡❧❛❝✐♦♥❛❞❛s❀ ♦ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❛r✐t✲ ♠ét✐❝❛❀ ❛ ♣r♦✈❛ ❞❡ q✉❡ ♦s ♥ú♠❡r♦s ♣r✐♠♦s sã♦ ✐♥✜♥✐t♦s ❝♦♠ ✉s♦ ❞❛ r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦✳

❖ ▲✐✈r♦ ❳ ❡st✉❞❛ ♦s ✐rr❛❝✐♦♥❛✐s✱ s❡❣♠❡♥t♦s ❞❡ r❡t❛ ✐♥❝♦♠❡♥s✉rá✈❡✐s ❝♦♠ ✉♠

s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❞❛❞♦✱ ♥❡❧❡ ❡①✐st❡ ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❝❛❧❝✉❧❛r ❝❡♠ t❡r♥♦s ♣✐t❛❣ór✐❝♦s✳ ❖s ❧✐✈r♦s ❳■✱ ❳■■ ❡ ❳■■ ❢❛❧❛♠ s♦❜r❡ ❛ ❣❡♦♠❡tr✐❛ só❧✐❞❛ ✭❡s♣❛❝✐❛❧✮✱ ❝♦♠ ❡①❝❡çã♦ ❞❛ ❡s❢❡r❛ ❡ ♦ ú❧t✐♠♦ ❛ ✐♥s❝r✐çã♦ ❞♦s ❝✐♥❝♦ ♣♦❧✐❡❞r♦s ❞❡ P❧❛tã♦ ❡♠ ✉♠❛ ❡s❢❡r❛✳

❈♦♠ ❡ss❡ r✐q✉íss✐♠♦ ♠❛t❡r✐❛❧✱ ❖s ❊❧❡♠❡♥t♦s t♦r♥❛✲s❡ ♦ ❧✐✈r♦ ♠❛✐s ❝♦♠♣❧❡t♦ ❞❛ é♣♦❝❛ ❡♠ ♣r❡♣❛r❛r ♦ ❡st✉❞❛♥t❡ à ✐♥✐❝✐❛çã♦ ❞♦s ❡st✉❞♦s ♠❛t❡♠át✐❝♦s ♦♥❞❡ é ❝♦♠✲ ♣❛r❛❞♦ ❝♦♠♦ ♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ❛♣r❡♥❞❡r ♠❛t❡♠át✐❝❛✱ ❛ss✐♠ ❝♦♠♦ sã♦ ❛s ❧❡tr❛s ❞♦ ❛❧❢❛❜❡t♦ sã♦ ♣❛r❛ ❛♣r❡♥❞❡r ❛ ❧❡r ✭♣❛r❛ ♠❛✐♦r❡s ❞❡s❝r✐çõ❡s ❞❡st❡s ❧✐✈r♦s ✈❡❥❛ ❬✶❪✱ ❬✷❪✮✳

❈❛♣ít✉❧♦ ✷

❆ ❝♦♥str✉çã♦ ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r

✷✳✶ ■♥tr♦❞✉çã♦

◆❡st❡ ❝❛♣ít✉❧♦✱ ♠♦str❛r❡♠♦s ❛ ❝♦♥str✉çã♦ ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r ✐♥s❝r✐t♦ ❡♠ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ q✉❡ é ♣♦♥t♦ ❛❧t♦ ❞♦ ▲✐✈r♦ ■❱ ❞❡ ❖s ❊❧❡♠❡♥t♦s✳ ❊ss❛ ❝♦♥str✉çã♦ ♥ã♦ ✉s❛ s❡♠❡❧❤❛♥ç❛✱ s❡ ❜❛s❡✐❛ ❛♣❡♥❛s ♥❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ár❡❛s ❬✽❪✳ P❛r❛ s✉❛ ❝♦♥str✉çã♦ ❡❧❡ ✉t✐❧✐③❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❧ó❣✐❝❛ ❞❡ ♣r♦♣♦s✐çõ❡s q✉❡ sã♦ ❡st✉❞❛❞❛s ❛♦ ❧♦♥❣♦ ❞♦s ▲✐✈r♦s ■✱ ■■✱ ■■■ ❡ ■❱✱ sã♦ ❛s ♣r♦♣♦s✐çõ❡s ■❱✳✶✵ ❡ ■❱✳✶✶ ♦♥❞❡ é ❢❡✐t❛ ❡ss❛ ❝♦♥str✉çã♦✳ ❆ s❡q✉ê♥❝✐❛ ❧ó❣✐❝❛✱ ♥❡st❡ tr❛❜❛❧❤♦✱ s❡rá ❛ ♠❡s♠❛ ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ❊✉❝❧✐❞❡s ❝♦♠♦ ♦ ❢❡③ ❡♠ s❡✉ ❧✐✈r♦✳ ❆♦ ❧❡✐t♦r✱ ♠✉✐t❛s ✈❡③❡s✱ ♣❡r❣✉♥t❛✲s❡ ❞❡ ♦♥❞❡ ✈❡♠ ❝❡rt❛s ❛✜r♠❛çõ❡s✳ ❊✉❝❧✐❞❡s✱ ❡♠ ❝❛❞❛ ❧✐✈r♦✱ ❡s❝r❡✈❡ ❞❡✜♥✐çõ❡s✱ ♣♦st✉❧❛❞♦s ❡ ♥♦çõ❡s ❝♦♠✉♥s q✉❡ sã♦ ✉t✐❧✐③❛❞♦s ❡♠ s✉❛s ❞❡♠♦♥str❛çõ❡s ❡ ❝♦♥str✉çõ❡s s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✐♥❢♦r♠á✲❧❛ ♥❛ ♠❡s♠❛✱ ❥á q✉❡ ❡stá s❡♥❞♦ ❛♣❧✐❝❛❞❛ s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❞❡♠♦♥strá✲❧❛✱ ❛♣❡♥❛s ❝♦♠♦ ❛❝❡✐tá✈❡❧✳ ❚♦❞❛s ❡ss❛s ❞❡✜♥✐çõ❡s✱ ♣♦st✉❧❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✹❪✳

✷✳✷ ❖s P♦❧í❣♦♥♦s r❡❣✉❧❛r❡s

❉❡✜♥✐çã♦ ✷✳✶ ❯♠ ♣♦❧í❣♦♥♦ é r❡❣✉❧❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t❡♠ t♦❞♦s ♦s s❡✉s ❧❛❞♦s ❝♦♥❣r✉❡♥t❡s ❡ t♦❞♦s ♦s s❡✉s â♥❣✉❧♦s ✐♥t❡r♥♦s ❝♦♥❣r✉❡♥t❡s✳

❖✉ s❡❥❛✱ ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r é ❡q✉✐❧át❡r♦ ❡ ❡q✉✐â♥❣✉❧♦✳

❉❡✜♥✐çã♦ ✷✳✷ ❉❡♥♦♠✐♥❛✲s❡ ❛r❝♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❛ ❝❛❞❛ ✉♠❛ ❞❛s ♣❛rt❡s ❡♠ q✉❡ ❡❧❛ é ❞✐✈✐❞✐❞❛ ♣♦r ✉♠ ♣❛r ❞❡ s❡✉s ♣♦♥t♦s ✭✈❡❥❛ ✜❣✉r❛ ✷✳✶✮✳

❋✐❣✉r❛ ✷✳✶✿

❉❡✜♥✐çã♦ ✷✳✸ ❉❡♥♦♠✐♥❛✲s❡ ❝♦r❞❛ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❛ q✉❛❧q✉❡r s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ t❡♥❤❛ s✉❛s ❡①tr❡♠✐❞❛❞❡s s♦❜r❡ ❡❧❛✳

◗✉❛♥❞♦ ✉♠❛ r❡t❛ ❝♦rt❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❢♦r♠❛♥❞♦ ✉♠❛ ❝♦r❞❛ ❡❧❛ é ❝❤❛♠❛❞❛ ❞❡ s❡❝❛♥t❡ ✭✈❡❥❛ ✜❣✉r❛ ✷✳✷✮✳

❋✐❣✉r❛ ✷✳✷✿

❉❡✜♥✐çã♦ ✷✳✹ ❉❡♥♦♠✐♥❛✲s❡ s❡❣♠❡♥t♦ ❞❡ ✉♠ ❝ír❝✉❧♦ ❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ✉♠ ❛r❝♦ ❡ ✉♠❛ ❝♦r❞❛ ❞❡ s✉❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✭✈❡❥❛ ✜❣✉r❛ ✷✳✸✮✳

❋✐❣✉r❛ ✷✳✸✿

❚❡♦r❡♠❛ ✷✳✺ ❉✐✈✐❞✐♥❞♦✲s❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡♠ ♥ ✭♥ > ✸✮ ❛r❝♦s ❝♦♥❣r✉❡♥t❡s✱

t❡♠♦s✿

✶✲ t♦❞❛s ❛s ❝♦r❞❛s ❞❡t❡r♠✐♥❛❞❛s ♣♦r ❞♦✐s ♣♦♥t♦s ❞❡ ❞✐✈✐sã♦ ❝♦♥s❡❝✉t✐✈♦s✱ r❡✉♥✐❞❛s✱ ❢♦r♠❛♠ ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❞❡ ♥ ❧❛❞♦s ✐♥s❝r✐t♦ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛❀

✷✲ ❛s t❛♥❣❡♥t❡s tr❛ç❛❞❛s ♣❡❧♦s ♣♦♥t♦s ❞❡ ❞✐✈✐sã♦ ❞❡t❡r♠✐♥❛♠ ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❞❡ ♥ ❧❛❞♦s ❝✐r❝✉♥s❝r✐t♦s à ❝✐r❝✉♥❢❡rê♥❝✐❛✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ A1, A2,· · · , An₋1, An ♦s n ♣♦♥t♦s ❞❡ ❞✐✈✐sã♦ ❞❛ ❝✐r❝✉♥❢❡✲ rê♥❝✐❛ C ✭✈❡❥❛ ✜❣✉r❛ ✷✳✹✮✳ ❖ ♣♦❧í❣♦♥♦ A1A2· · ·An₋1An é ❞❡n ❧❛❞♦s é ✐♥s❝r✐t♦✱ ♣♦✐s

❋✐❣✉r❛ ✷✳✹✿

t♦❞♦s ♦s ✈ért✐❝❡s ♣❡rt❡♥❝❡♠ à ❝✐r❝✉♥❢❡rê♥❝✐❛ C✳ ❙❡♥❞♦

❛r❝♦A1A2 ≡❛r❝♦A2A3 ≡ · · · ≡❛r❝♦An₋1An≡❛r❝♦AnA1 ✭✷✳✶✮

❡♥tã♦

A1A2 ≡A2A3 ≡ · · · ≡An₋1An ≡AnA1 ✭✷✳✷✮

P♦✐s✱ ♥✉♠❛ ♠❡s♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❛r❝♦s ❝♦♥❣r✉❡♥t❡s s✉❜t❡♥❞❡♠ ❝♦r❞❛s ❝♦♥❣r✉❡♥t❡s✳ ❖s â♥❣✉❧♦sA1, A2, A3, A4,· · · , An₋1, Ansã♦ ❝♦♥❣r✉❡♥t❡s✱ ♣♦✐s ❝❛❞❛ ✉♠ ❞❡❧❡s é â♥❣✉❧♦ ✐♥s❝r✐t♦ ❡♠ C ❡ t❡♠ ♣♦r ♠❡❞✐❞❛ ♠❡t❛❞❡ ❞❛ s♦♠❛ ❞❡ (n₋2) ❞♦s ❛r❝♦s ❝♦♥❣r✉❡♥t❡s ❡♠ q✉❡ C ✜❝♦✉ ❞✐✈✐❞✐❞❛✳ ❉❡ ✭✷✳✶✮ ❡ ✭✷✳✷✮ ❝♦♥❝❧✉í♠♦s q✉❡ A1A2· · ·An₋1An é ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❞❡n ❧❛❞♦s ✐♥s❝r✐t♦ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ C✱ ✐st♦ ♣r♦✈❛ ❛ ♣r✐♠❡✐r❛ ♣❛rt❡

❞♦ t❡♦r❡♠❛✳

❋✐❣✉r❛ ✷✳✺✿

P❡❧♦s ♣♦♥t♦s ❞❛ ❞✐✈✐sã♦✱ q✉❡ t❛♠❜é♠ sã♦ ✈ért✐❝❡s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r q✉❛❧✲ q✉❡r ✐♥s❝r✐t♦ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛C✱A1, A2, A3, A4,· · · , An₋1, An❝♦♥❞✉③✐♠♦s t❛♥❣❡♥t❡s ❛ C ♥❡ss❡s ✈ért✐❝❡s ❡ ♦❜t❡♠♦s ♦ ♣♦❧í❣♦♥♦ A′

1A′2A′3A′4· · ·A′n₋1A′n ❞❡ n ❧❛❞♦s ❝✐r❝✉♥s✲ ❝r✐t♦ ❛ C ✭✈❡❥❛ ✜❣✉r❛ ✷✳✺✮✳

❖s tr✐â♥❣✉❧♦s∆A′

1A1A2, ∆A′2A2A3, ∆A′3A3A4, ∆A′4A4A5, · · · ,∆A′n₋1An₋1An❡ ∆A′

nAnA1sã♦ tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s✱ ♣♦✐s ❝❛❞❛ ✉♠ ❞♦s â♥❣✉❧♦sA1, A2, A3, A4,· · · , An₋1

❡An❞❡st❡s tr✐â♥❣✉❧♦s✱ t❡♠ ♠❡❞✐❞❛ ✐❣✉❛❧ ❛ ♠❡t❛❞❡ ❞❛ ♠❡❞✐❞❛ ❞❡ ✉♠❛ ❞❛s ♣❛rt❡s ❝♦♥✲

❣r✉❡♥t❡s ❛r❝♦A1A2✱ ❛r❝♦A2A3✱ ❛r❝♦A3A4✱ ❛r❝♦A4A5✱· · ·✱ ❛r❝♦An₋1An✱ ❛r❝♦AnA1

❡♠ q✉❡ ❢♦✐ ❞✐✈✐❞✐❞❛ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✭sã♦ â♥❣✉❧♦s ❞♦ s❡❣♠❡♥t♦ ♦✉ s❡♠✐✲✐♥s❝r✐t♦s✮ ❡ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦ ❆▲❆✱ ✈✐st♦ q✉❡ s❡♥❞♦ A1A2A3A4 · · · An₋1An ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ✭❥á ❞❡♠♦♥str❛❞♦✮✱ ❡ ♦s ❧❛❞♦s A1A2, A2A3, A3A4, A4A5,· · · , An₋1An, AnA1

❞❡st❡s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❉❛ ❝♦♥❣r✉ê♥❝✐❛ ❞♦s tr✐â♥❣✉❧♦s ❞❡❝♦rr❡ q✉❡ _Aˆ′

1 ≡

ˆ

A′

2 ≡ Aˆ′3 ≡ Aˆ′3 ≡ · · · ≡ An′ˆ₋1 ≡ Aˆn′ ✭✶✮ ❀ ❡✱ ♣♦r s♦♠❛ ❝♦♥✈❡♥✐❡♥t❡✱ t❡♠♦s✿ A′1A′2 ≡

A′

2A′3 ≡ A′3A′4 ≡ A′4A5′ ≡ · · · ≡ A′n₋1An′ ≡ A′nA′1 ✭✷✮✳ ❉❡ ✭✶✮ ❡ ✭✷✮ ❝♦♥❝❧✉í♠♦s q✉❡

A1A2A3A4· · ·An₋1An é ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❞❡n ❧❛❞♦s ❝✐r❝✉♥s❝r✐t♦ à ❝✐r❝✉♥❢❡rê♥❝✐❛

C ✭✈❡❥❛ ❬✻❪✮✳

❚❡♦r❡♠❛ ✷✳✻ ❚♦❞♦ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r é ✐♥s❝r✐tí✈❡❧ ♥✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳

❋✐❣✉r❛ ✷✳✻✿

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ A1A2A3A4· · ·An₋1An ♦ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ✭✈❡❥❛ ✜❣✉r❛ ✷✳✻✮✱ ♣❡❧♦s ♣♦♥t♦s A1A2A3 tr❛❝❡♠♦s ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ C ❡ s❡❥❛ O ♦ s❡✉ ❝❡♥tr♦❀ ♣r♦✈❡♠♦s

q✉❡ C ♣❛ss❛ ♣❡❧♦s ❞❡♠❛✐s ✈ért✐❝❡s A4, A5,· · · , An₋1, An ❞♦ ♣♦❧í❣♦♥♦✳ ❈♦♠❡❝❡♠♦s ♣r♦✈❛♥❞♦ q✉❡ A4 ∈ C✳ ❈♦♥s✐❞❡r❡♠♦s ♦s tr✐â♥❣✉❧♦s ∆ OA2A1 ❡ ∆ OA3A4✳ ❊st❡s

tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦ ▲❆▲✱ ♣♦✐sA1A2 ≡A3A4 ✭❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦ r❡✲

❣✉❧❛r✮✱ OA2 ≡OA3 ✭r❛✐♦s ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✮ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ♦ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s∆

A2OA3 ✭â♥❣✉❧♦s ❞❛ ❜❛s❡ ❝♦♥❣r✉❡♥t❡s✮✱ ❡ ❛✐♥❞❛✱ q✉❡ ♦s â♥❣✉❧♦s Aˆ2 ❡Aˆ3 ❞♦ ♣♦❧í❣♦♥♦

sã♦ ❝♦♥❣r✉❡♥t❡s✱ ♣♦r ❞✐❢❡r❡♥ç❛ ❞❡❝♦rr❡ q✉❡ OAˆ2A1 ≡ OAˆ3A4✳

∆OA2A1 ≡ ∆OA3A4 =⇒OA1 ≡ OA4 =⇒ A4 ∈ C✳

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ t❡♠♦s q✉❡ A5 ∈ C ✭❜❛st❛ ❝♦♥s✐❞❡r❛r ∆OA3A2 ❡ ∆OA4A5✮✱ · · ·✱

An₋1∈C❡An∈C✱ ❡ ♦ ♣♦❧í❣♦♥♦A1A2A3A4· · ·An₋1✭❜❛st❛ ❝♦♥s✐❞❡r❛r∆OAn₋3An₋4

❡ ∆OAn₋2An₋1✮An ✭❜❛st❛ ❝♦♥s✐❞❡r❛r ∆OAn₋1An₋2 ❡∆OAnA1✮ ✐♥s❝r✐t♦ ♥❛ ❝✐r❝✉♥✲

❢❡rê♥❝✐❛ C✭✈❡❥❛ ❬✻❪✮✳

❉❛ ✉♥✐❝✐❞❛❞❡ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ♣❛ss❛ ♣♦r A1A2A3 s❛✐ ❛ ✉♥✐❝✐❞❛❞❡ ❞❡C ♣♦r

A1A2A3A4· · ·An₋1An✳

❚❡♦r❡♠❛ ✷✳✼ ❚♦❞♦ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r é ❝✐r❝✉♥s❝r✐tí✈❡❧ ❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳

❋✐❣✉r❛ ✷✳✼✿

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ A1A2· · ·An₋1An ♦ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r✳ ❊♠ ✈✐st❛ ❞♦ ❚❡♦r❡♠❛ ✷✳✻✱ ❡❧❡ é ✐♥s❝r✐t♦ ♥✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛C ✭✈❡❥❛ ✜❣✉r❛ ✷✳✼✮✳ ❙❡❥❛ O ♦ ❝❡♥tr♦ ❞❡ss❛ ❝✐r✲

❝✉♥❢❡rê♥❝✐❛✳ ❖s ❧❛❞♦sA1A2✱A2A3✱· · ·✱An₋1An✱ AnA1 sã♦ ❝♦r❞❛s ❝♦♥❣r✉❡♥t❡s ❞❡C✱

♣♦r ✐ss♦ ❞✐st❛♠ ✐❣✉❛❧♠❡♥t❡ ❞♦ ❝❡♥tr♦ ❖✳ ❙❡♥❞♦A′

1✱A′2✱ · · ·✱A′n₋1✱A′n ♦s r❡s♣❡❝t✐✈♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s A1A2✱ A2A3✱· · ·✱ An₋1An✱AnA1✱ t❡♠♦s

OA′

1 ≡ OA′2 ≡ · · · ≡ OA′n₋1 ≡ OA′n ✭❞✐stâ♥❝✐❛ ❞♦ ❝❡♥tr♦ ❛ ❝♦r❞❛s ❝♦♥❣r✉❡♥t❡s✮ ❞♦♥❞❡ s❡ ❝♦♥❝❧✉✐ q✉❡ ❖ é ♦ ❝❡♥tr♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ C′ _{q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s}

A′

1✱A′2✱· · ·✱ A′n₋1✱A′n✳ ❊ ❛✐♥❞❛ s❡♥❞♦

OA′

1⊥A1A2✱OA′2⊥A2A3✱· · ·✱OA′n₋1⊥An₋1An❡OA′n⊥AnA1✱ t❡♠♦s q✉❡A1A2· · ·An₋1An t❡♠ ❧❛❞♦s t❛♥❣❡♥t❡s ❛ C′✳ ❈♦♥❧✉í♠♦s q✉❡ ♦ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r _{A1A2A3A4· · ·An}

−1An é ❝✐r❝✉♥s❝r✐t♦ à ❝✐r❝✉♥❢❡rê♥❝✐❛ C′✳

❯♥✐❝✐❞❛❞❡ ❞❡C′_{✱ s❡ ❡①✐st✐ss❡ ♦✉tr❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ♥♦ ♣♦❧í❣♦♥♦A1A2A3A4· · ·An} −1An✱ ❡❧❛ ♣❛ss❛r✐❛ ♣❡❧♦s ♣♦♥t♦sA′

1✱A′2✱A′3 · · ·✱ ❡ s❡r✐❛✱ ❡♥tã♦ ❝♦✐♥❝✐❞❡♥t❡ ❝♦♠C′✭✈❡❥❛ ❬✻❪✮✳

✷✳✸ ❈♦♥str✉çõ❡s ❞♦ P❡♥tá❣♦♥♦ ❘❡❣✉❧❛r

❚❡♦r❡♠❛ ✷✳✽ ❙❡ ✉♠ q✉❛❞r✐❧át❡r♦ t❡♠ ❞♦✐s ❧❛❞♦s ♣❛r❛❧❡❧♦s ❡ ❝♦♥❣r✉❡♥t❡s ❡♥tr❡ s✐ ❡❧❡ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳

❋✐❣✉r❛ ✷✳✽✿

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ♦ q✉❛❞r✐❧át❡r♦ ❆❇❈❉✱ ❝♦♠ ❆❇ ❂ ❈❉ ❡ ❆❇✴✴❈❉ ✭✈❡❥❛ ✜❣✉r❛ ✷✳✸✮✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ ❆❉ ❂ ❇❈ ❡ ❇❈✴✴❆❉✳ ❯♥✐♥❞♦✲s❡ ❇ ❛ ❉✱ ✜❝❛♠ ❢♦r♠❛❞♦s ♦s tr✐â♥❣✉❧♦s ❆❇❉ ❡ ❇❈❉✳ ❚❛✐s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝r✐tér✐♦ ▲❆▲✱ ♣♦✐s ❇❉ é ✉♠ ❧❛❞♦ ❝♦♠✉♠✱ ♦∠❆❇❉ ❂∠❇❈❉ ✭❛❧t❡r♥♦s✲✐♥t❡r♥♦s✮ ❡ ❆❇ ❂ ❈❉✳ ▲♦❣♦✱ ❇❈

❂ ❆❉ ❡ ♦ ∠❆❉❇ ❂ ∠❈❇❉✳ ▲♦❣♦✱ ❇❈✴✴❆❉ ✭❬✺❪✮✳

❚❡♦r❡♠❛ ✷✳✾ ❙❡ ♣❡❧♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ✉♠ ❧❛❞♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❢♦r tr❛ç❛❞❛ ❛ ♣❛r❛❧❡❧❛ ❛ ♦✉tr♦ ❧❛❞♦✱ ❡ss❛ ♣❛r❛❧❡❧❛ ❝♦rt❛rá ♦ t❡r❝❡✐r♦ ❧❛❞♦ ❡♠ s❡✉ ♣♦♥t♦ ♠é❞✐♦✳ ❘❡❝✐♣r♦❝❛✲ ♠❡♥t❡✱ ❛ r❡t❛ q✉❡ ✉♥❡ ♦s ♣♦♥t♦s ♠é❞✐♦s ❞❡ ❞♦✐s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ♣❛r❛❧❡❧❛ ❛♦ t❡r❝❡✐r♦ ❧❛❞♦✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ♦ tr✐â♥❣✉❧♦ ❆❇❈ ❡ ♦ ♣♦♥t♦ ▼✱ ♠é❞✐♦ ❞❡ ❆❇ ✭✈❡❥❛ ✜❣✉r❛ ✷✳✾✮ P♦r ▼ tr❛ç❛♠♦s ✉♠❛ ♣❛r❛❧❡❧❛ ❛ ❇❈✱ ❝♦rt❛♥❞♦ ♦ ❧❛❞♦ ❆❈ ♥♦ ♣♦♥t♦ ◆✳ ◗✉❡r❡♠♦s✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ♣r♦✈❛r q✉❡ ◆ é ♣♦♥t♦ ♠é❞✐♦ ❞❡ ❆❈✱ ♦✉ s❡❥❛✱ ◆❆ ❂ ◆❈✳ P♦r ❈ tr❛❝❡♠♦s ❛ ♣❛r❛❧❡❧❛ ❛ ❆❇✱ ❝♦rt❛♥❞♦ ❛ r❡t❛ ▼◆ ♥♦ ♣♦♥t♦ ❉✳

❖ q✉❛❞r✐❧át❡r♦ ❇❈❉▼ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ✭❧❛❞♦s ♦♣♦st♦s ♣❛r❛❧❡❧♦s✱ ♣♦r ❝♦♥s✲ tr✉çã♦✮✳ ▲♦❣♦✱ ❇▼ ❂ ❈❉ ❡ ▼❉ ❂ ❇❈✳ ❈♦♠♦ ❇▼ ❂ ❈❉ ❡ ❇▼ ❂ ❆▼ ✭▼ é ♦ ♣♦♥t♦

❋✐❣✉r❛ ✷✳✾✿

♠é❞✐♦ ❞❡ ❆❇✮✱ ❆▼ ❂ ❈❉✳ ❖s tr✐â♥❣✉❧♦s ❆▼◆ ❡ ❈❉◆ sã♦ ❝♦♥❣r✉❡♥t❡s ✭❆▲❆✮ ♣♦✐s ❆▼ ❂ ❈❉✱∠❆▼◆ ❂ ∠❈❉◆ ✭❛❧t❡r♥♦s ✐♥t❡r♥♦s✮ ❡ ∠▼❆◆ ❂∠❉❈◆ ✭❛❧t❡r♥♦s ✐♥t❡r✲

♥♦s✮✳ ▲♦❣♦ ◆❆ ❂ ◆❈✳

Pr♦✈❡♠♦s ❛❣♦r❛ ♦ r❡❝í♣r♦❝♦✱ ♦✉ s❡❥❛✱ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ♠é❞✐♦s ▼ ❡ ◆ ❞❡ ❆❇ ❡ ❆❈ é ♣❛r❛❧❡❧❛ ❛ ❇❈✳ ❆s ❤✐♣ót❡s❡s ❛❣♦r❛ sã♦ ❆▼ ❂ ❇▼ ❡ ◆❆ ❂ ◆❈✳ ◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡ ▼◆✴✴❇❈✳ P♦r ❈ tr❛❝❡♠♦s ❛ ♣❛r❛❧❡❧❛ ❛ ❆❇✱ ❝♦rt❛♥❞♦ ▼◆ ♥♦ ♣♦♥t♦ ❉✳ ◆♦✈❛♠❡♥t❡ ♦s tr✐â♥❣✉❧♦s ❆▼◆ ❡ ❈◆❉ sã♦ ❝♦♥❣r✉❡♥t❡s ✭❆▲❆✮✱ ♣♦✐s∠▼❆◆

❂ ∠❉❈◆ ✭❛❧t❡r♥♦s ✐♥t❡r♥♦s✮✱ ◆❆ ❂ ◆❈ ❡ ∠❆◆▼ ❂ ∠❈◆❉ ✭♦♣♦st♦s ♣❡❧♦ ✈ért✐❝❡✮✳

▲♦❣♦✱ ❈❉ ❂ ❆▼✳ ▼❛s ❆▼ ❂❇▼ ✭♣♦r ❤✐♣ót❡s❡✮✳ ▲♦❣♦ ❇▼ ❂ ❈❉✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✽✱ ♦ q✉❛❞r✐❧át❡r♦ ❇❈❉▼ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ ♣♦rq✉❡ t❡♠ ❞♦✐s ❧❛❞♦s ❝♦♥❣r✉❡♥t❡s ✭❇▼ ❡ ❈❉✮✳ ▲♦❣♦ ▼❉✴✴❇❈✱ ♦✉ s❡❥❛✱ ▼◆✴✴❇❈✳ ❆✐♥❞❛ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ ❞♦s tr✐â♥❣✉❧♦s ❆▼◆ ❡ ❈◆❉ t❡♠✲s❡ ▼◆ ❂ ❈❉ ❡✱ ❝♦♠♦ ▼◆ ✰ ◆❉ ❂ ▼❉✱ t❡♠✲s❡ ▼❉ ❂ ✷▼◆✳ ▼❛s ▼❉ ❂ ❇❈✳ ▲♦❣♦✱ ❇❈ ❂ ✷▼◆✱ ♦✉ s❡❥❛✱ ♦ s❡❣♠❡♥t♦ q✉❡ ✉♥❡ ♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ❝♦♥❣r✉❡♥t❡ ❝♦♠ ❛ ♠❡t❛❞❡ ❞♦ t❡r❝❡✐r♦ ❧❛❞♦ ♣❛r❛❧❡❧♦ ❛ ❡❧❡✳

❚❡♦r❡♠❛ ✷✳✶✵ ❚♦❞♦ tr✐â♥❣✉❧♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❞❡ ♠❡s♠❛ ❜❛s❡ ❡ ❛❧t✉r❛ ❝♦♥❣r✉❡♥t❡ ❝♦♠ ❛ ♠❡t❛❞❡ ❞❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦✳

❊q✉✐✈❛❧❡♥t❡✱ ♥❡st❡ ❝♦♥t❡①t♦✱ s✐❣♥✐✜❝❛ q✉❡ ❛s ✜❣✉r❛s ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ár❡❛✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ♦ tr✐â♥❣✉❧♦ ❆❇❈ q✉❛❧q✉❡r ✭✈❡❥❛ ✜❣✉r❛ ✷✳✶✵✮✳ ❚♦♠❡♠♦s ❝♦♠♦ ❜❛s❡ ✉♠ ❞♦s ❧❛❞♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❧❛❞♦ ❇❈ ❡ s♦❜r❡ ✉♠ ❞♦s ♦✉tr♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❧❛❞♦ ❆❇✱ ❝♦♥s✐❞❡r❡ ♦ ♣♦♥t♦ ♠é❞✐♦ ▼✳ P♦r ▼ tr❛❝❡♠♦s ❛ ♣❛r❛❧❡❧❛ ❛ ❇❈✱ q✉❡ ❝♦rt❛rá ♦ ❧❛❞♦

❋✐❣✉r❛ ✷✳✶✵✿

❆❈ ♥♦ s❡✉ ♣♦♥t♦ ♠é❞✐♦ ◆✱ ✭❚❡♦r❡♠❛ ✷✳✾✮✳ ❙❡❥❛ ❉ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❡♥tr❡ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♥♦s ♣♦♥t♦s ▼ ❡ ◆ ❡ ❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛ ❆❇ ♣♦rq✉❡ ♣❛ss❛ ♥♦ ♣♦♥t♦ ❈✳ ❖s tr✐â♥❣✉❧♦s ❆▼◆ ❡ ❈❉◆ sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❆ss✐♠✱ ♦ tr✐â♥❣✉❧♦ ❆❇❈ ❢♦✐ ❞❡❝♦♠♣♦st♦ ♥♦ q✉❛❞r✐❧át❡r♦ ❇❈◆▼ ❡ ♥♦ tr✐â♥❣✉❧♦ ❆▼◆✱ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❇❈❉▼ ❢♦✐ ❞❡❝♦♠♣♦st♦ ♥♦ q✉❛❞r✐❧át❡r♦ ❇❈◆▼ ❡ ♥♦ tr✐â♥❣✉❧♦ ❈❉◆ q✉❡ é ❝♦♥❣r✉❡♥t❡ ❝♦♠ ♦ tr✐â♥❣✉❧♦ ❆▼◆✳ ▲♦❣♦✱ ♦ tr✐â♥❣✉❧♦ ❆❇❈ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❇❈❉▼✳ ❘❡st❛ ♣r♦✈❛r q✉❡ ❛ ❛❧t✉r❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❇❈❉▼ é ❛ ♠❡t❛❞❡ ❞❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ ❆❇❈✳ P❡❧♦ ♣♦♥t♦ ❆ tr❛❝❡♠♦s ❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❆❍ ❛♦ ❧❛❞♦ ❇❈✱ ❝♦rt❛♥❞♦ ❛ r❡t❛ ▼❉ ♥♦ ♣♦♥t♦ P✳ ◆♦t❡ q✉❡ ❆❍ é ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ ❆❇❈ ❡ P❍ é ❛ ❛❧t✉r❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❇❈❉▼✱ ♥♦ tr✐â♥❣✉❧♦ ❆❇❍ ❛ r❡t❛ ▼P é ♣❛r❛❧❡❧❛ ❛ r❡t❛ ❇❍ ❡ ▼ é ♣♦♥t♦ ♠é❞✐♦ ❞♦ ❧❛❞♦ ❆❇✳ ▲♦❣♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✾✱ P é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ❆❍✳ ❆ss✐♠✱ ❛ ❛❧t✉r❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❇❈❉▼ é ❛ ♠❡t❛❞❡ ❞❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ ❆❇❈✭ ✈❡❥❛ ❬✺❪✮✳

❉❡ ♣♦ss❡ ❞❡st❡s três t❡♦r❡♠❛s✱ ❛❣♦r❛ ♣♦❞❡♠♦s ❞❡♠♦♥str❛r ♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦✲ r❛s✳ ❊ss❛ ❞❡♠♦♥str❛çã♦ ❢♦✐ ❢❡✐t❛ ♣♦r ❊✉❝❧✐❞❡s✳

❚❡♦r❡♠❛ ✷✳✶✶ ❊♠ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✱ ❛ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❝♦♥str✉í❞♦s s♦❜r❡ ♦s ❝❛t❡t♦s é ❡q✉✐✈❛❧❡♥t❡ ❛♦ q✉❛❞r❛❞♦ ❝♦♥str✉í❞♦ s♦❜r❡ ❛ ❤✐♣♦t❡♥✉s❛✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠✱ ❡♥tã♦ ✉♠ tr✐â♥❣✉❧♦ ❆❇❈ ❡ ♦s q✉❛❞r❛❞♦s ❇❈❊❉✱ ❆❇❋● ❡ ❆❈❑❍✱ ❝♦♥str✉í❞♦s s♦❜r❡ ❛ ❤✐♣♦t❡♥✉s❛ ❡ s❡✉s ❝❛t❡t♦s ✭✈❡❥❛ ✜❣✉r❛ ✷✳✶✶✮✳ ◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡✱ s♦♠❛❞♦s✱ ♦s q✉❛❞r❛❞♦s ❆❇❋● ❡ ❆❈❑❍✱ s♦❜r❡ ♦s ❝❛t❡t♦s✱ sã♦ ❡q✉✐✈❛❧❡♥t❡s ❛♦ q✉❛❞r❛❞♦ ❇❈❊❉✱ s♦❜r❡ ❛ ❤✐♣♦t❡♥✉s❛✳

❋✐❣✉r❛ ✷✳✶✶✿

❙❡❥❛ ❆▲ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ❤✐♣♦t❡♥✉s❛ ✭♣♦rt❛♥t♦ ♣❛r❛❧❡❧❛ ❛ ❇❉ ❡ ❈❊✮✱ ❝♦rt❛♥❞♦✲❛ ♥♦ ♣♦♥t♦ ▼✳ ❯♥✐♥❞♦ ❋ ❛ ❈ ❡ ❆ ❛ ❉✱ ✜❝❛♠ ❢♦r♠❛❞♦s ♦s tr✐â♥❣✉❧♦s ❇❈❋ ❡ ❆❇❉✳ ❊❧❡s sã♦ ❝♦♥❣r✉❡♥t❡s ✭▲❆▲✮ ♣♦rq✉❡ ❇❋ ❂ ❆❇ ✭❧❛❞♦s ❞❡ ✉♠ q✉❛❞r❛❞♦✮✱ ❇❈ ❂ ❇❉ ✭❧❛❞♦s ❞❡ ♦✉tr♦ q✉❛❞r❛❞♦✮ ❡∠❋❇❈ ❂∠❆❇❉✱ ♣♦✐s ❛♠❜♦s sã♦ ❛ s♦♠❛ ❞❡ ✉♠ r❡t♦ ❝♦♠ ♦∠❆❇❈✳

❆ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ ❇❈❋ ❡♠ r❡❧❛çã♦ à ❜❛s❡ ❇❋ é ♦ ❧❛❞♦ ❋● ❞♦ q✉❛❞r❛❞♦ ❆❇❋●✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✶✵✱ ♦ tr✐â♥❣✉❧♦ ❇❈❋ é ❡q✉✐✈❛❧❡♥t❡ à ♠❡t❛❞❡ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❞❡ ❜❛s❡ ❇❋ ❡ ❛❧t✉r❛ ❆❇✱ ♦✉ s❡❥❛✱ à ♠❡t❛❞❡ ❞♦ q✉❛❞r❛❞♦ ❆❇❋●✳ ❆ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ ❆❇❉✱ ❡♠ r❡❧❛çã♦ à ❜❛s❡ ❇❉✱ é ♦ s❡❣♠❡♥t♦ ❇▼✳ ▲♦❣♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✶✵✱ ♦ tr✐â♥❣✉❧♦ ❆❇❉ é ❡q✉✐✈❛❧❡♥t❡ à ♠❡t❛❞❡ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❞❡ ❜❛s❡ ❇❉ ❡ ❛❧t✉r❛ ❇▼✱ ♦✉ s❡❥❛✱ à ♠❡t❛❞❡ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❇❉▲▼✳ ❊♥tã♦✱ ❝♦♠♦ ♦s tr✐â♥❣✉❧♦s ❇❈❋ ❡ ❆❇❉ sã♦ ❝♦♥❣r✉❡♥t❡s✱ ❛ ♠❡t❛❞❡ ❞♦ q✉❛❞r❛❞♦ ❆❇❋● é ❡q✉✐✈❛❧❡♥t❡ à ♠❡t❛❞❡ ❞♦ ♣❛❛❧❡❧♦❣r❛♠♦ ❇❉▲▼✳ ❖✉ s❡❥❛✱ ♦ q✉❛❞r❛❞♦ ❆❇❋● é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ✭q✉❡ t❛♠❜é♠ é ✉♠ r❡tâ♥❣✉❧♦✮ ❇❉▲▼✳

❯♥❛✲s❡✱ ❛❣♦r❛✱ ♦ ✈ért✐❝❡ ❇ ❛♦ ✈ért✐❝❡ ❑ ❡ ♦ ✈ért✐❝❡ ❆ ❛♦ ✈ért✐❝❡ ❊✳ ❋✐❝❛♠ ❢♦r✲ ♠❛❞♦s ♦s tr✐â♥❣✉❧♦s ❇❈❑ ❡ ❆❈❊✳ ❈♦♠ r❛❝✐♦❝í♥✐♦ ❛♥á❧♦❣♦ ❛♦ ❡♠♣r❡❣❛❞♦ ❝♦♠ ♦s tr✐â♥❣✉❧♦s ❇❈❋ ❡ ❆❇❉✱ ♣r♦✈❛✲s❡ q✉❡ ♦s tr✐â♥❣✉❧♦s ❇❈❑ ❡ ❆❈❊ sã♦ ❝♦♥❣r✉❡♥t❡s ❡ q✉❡ ♦ q✉❛❞r❛❞♦ ❆❈❑❍ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ r❡tâ♥❣✉❧♦ ❈❊▲▼✳

❙✐❣♥✐✜❝❛ q✉❡ ♦s q✉❛❞r❛❞♦s ❆❇❋● ❡ ❆❈❑❍✱ s♦♠❛❞♦s✱ sã♦ ❡q✉✐✈❛❧❡♥t❡s ❛♦s r❡✲ tâ♥❣✉❧♦s ❇❉▲▼ ❡ ❈❊▲▼ s♦♠❛❞♦s✳ ▲♦❣♦✱ ❛ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❝♦♥str✉í❞♦s s♦❜r❡

♦s ❝❛t❡t♦s é ❡q✉✐✈❛❧❡♥t❡ ❛♦ q✉❛❞r❛❞♦ ❝♦♥str✉í❞♦ s♦❜r❡ ❛ ❤✐♣♦t❡♥✉s❛✭❬✹❪✱ ❬✺❪✳

❊ss❛ ❞❡♠♦♥str❛çã♦ é ♥❡❝❡ssár✐❛ ♣♦✐s ❡❧❛ s❡r✈❡ ❞❡ ❡♠❜❛s❛♠❡♥t♦ ♣❛r❛ ❛s ♣ró①✐♠❛s ❞❡♠♦♥str❛çõ❡s✱ ❝♦♥str✉çõ❡s ❡ ❛♣❧✐❝❛çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r✳

❚❡♦r❡♠❛ ✷✳✶✷ ❙❡❥❛ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ AB ❞✐✈✐❞✐❞♦ ♣❡❧♦ ♣♦♥t♦ ♠é❞✐♦ C ❡ ♣r♦❧♦✲

❣❛♥❞❛ ❛té ♦ ♣♦♥t♦ D✱ ❡♥tã♦ AD_·DB+BC2 =CD2✳

❊ss❡ t❡♦r❡♠❛ ❢♦✐ ❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ♥♦ ❧✐✈r♦ ❞❡ ❊✉❝❧✐❞❡s ✲ ❙❡ ✉♠❛ ❧✐♥❤❛ r❡t❛ ❢♦r ❞✐✈✐❞✐❞❛ ❡♠ ❞✉❛s ♣❛rt❡s ✐❣✉❛✐s ❡ ❢♦r ♣r♦❧♦♥❣❛❞❛✱ ♦ r❡tâ♥❣✉❧♦ ❝♦♠♣r❡❡♥❞✐❞♦ ♣❡❧❛ r❡t❛ t♦❞❛ ❡ ♠❛✐s ❛ ❛❞❥✉♥t❛✱ ❡ ♣❡❧❛ ♠❡s♠❛ ❛❞❥✉♥t❛✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❛ ♠❡t❛❞❡ ❞❛ ♣r✐♠❡✐r❛ r❡t❛ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ ❛ r❡t❛ ❢♦r♠❛❞❛ ♣♦r ♠❡t❛❞❡ ❞❛ r❡t❛ ♠❛✐s ❛ ❛❞❥✉♥t❛✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ ❈ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ❆❇ ❡ ❉✱ s♦❜r❡ ♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❡ ❆❇ ✭✈❡❥❛ ✜❣✉r❛ ✷✳✶✷✮

❋✐❣✉r❛ ✷✳✶✷✿

❈♦♠ ♦ ❧❛❞♦ ❈❉ ❝♦♥str✉❛ ♦ q✉❛❞r❛❞♦ ❈❊❋❉✳ ❯♥❛ ❉ ❛ ❊✳ P♦r ❇ tr❛❝❡ ●❍❇ ♣❛r❛❧❡❧❛ ❛ ❈❊ ♦✉ ❛ ❉❋✳ P❡❧♦ ♣♦♥t♦ ❍✱ tr❛❝❡ ❑▲▼ ♣❛r❛❧❡❧❛ ❛ ❆❉ ♦✉ ❛ ❊❋ ❡ ♣♦r ❆ tr❛❝❡ ❆❑ ♣❛r❛❧❡❧❛ ❛ ❈▲ ♦✉ ❛ ❉▼✳ ❊♥tã♦✱ ❝♦♠♦ ❆❈ é ✐❣✉❛❧ ❛ ❈❇✱ ♦ r❡tâ♥❣✉❧♦ ❆❑▲❈ é ✐❣✉❛❧ ❛♦ r❡tâ♥❣✉❧♦ ❈▲❍❇✳ ▼❛s ❈▲❍❇ é ✐❣✉❛❧ ❛ ❍●❋▼✱ ❡ ♣♦rt❛♥t♦ t❛♠❜é♠ ❆❑❈▲ é ✐❣✉❛❧ ❛ ❍●❋▼✳ ❆ ❝❛❞❛ ✉♠ ❞❡ss❡s r❡tâ♥❣✉❧♦s ❛❞✐❝✐♦♥❡ ❈▲▼❉✳ ❆ss✐♠✱ ❆❑▼❉ s❡rá ✐❣✉❛❧ ❛ ❈▲❍●❋❉✳ ▼❛s ❆❑▼❉ é ♦ r❡tâ♥❣✉❧♦ ❢♦r♠❛❞♦ ♣♦r ❆❉ ❡ ❉❇✱ ♣♦✐s ❉▼ é ✐❣✉❛❧ ❛ ❉❇✳

❊♥tã♦✱ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s ❆❉ ❡ ❉❇ é ✐❣✉❛❧ ❛ ❈▲❍●❋❉✳ ❆ ❝❛❞❛ ✉♠✱ ❛❞✐❝✐♦♥❡ ▲❊●❍✱ q✉❡ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ ❈❇✳ P♦rt❛♥t♦ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s ❆❉ ❡

❉❇✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❈❇✱ é ✐❣✉❛❧ ❛ ❈▲❍●❋❉ ❡ ❛ ▲❊●❍✳ ▼❛s ❈▲❍●❋❉✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ▲❊●❍✱ ❢♦r♠❛♠ ❈❊❋❉✱ q✉❡ é ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❈❉✳ P♦rt❛♥t♦✱ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s ❆❉ ❡ ❉❇✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❈❇✱ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ ❈❉ ✭❬✽❪✮✳

❇❡♠✱ ♣❛r❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦✱ ♠♦str❛r❡♠♦s ❡ss❛ ❞❡♠♦♥str❛çã♦ ❛❧❣❡❜r✐❝❛✲ ♠❡♥t❡✳

❋✐❣✉r❛ ✷✳✶✸✿

P❡❧❛ ✜❣✉r❛ ✷✳✶✸✱ t❡♠♦sAC =CB ♠❡❞✐❞❛ ❞❡ AC =CB =a ❡ ♦ ♣r♦❧♦♥❣❛♠❡♥t♦

❞❡ AB, BD=x✳ ❊♥tã♦ AD= (2a+x), BC =a ❡CD = (a+x)✳

AD_·DB+BC2 =CD2

(2a+x)_·x+a2 =CD2

2ax+x2+a2 =CD2

a2+ 2ax+x2 =CD2

(a+x)2

=CD2

CD = (a+x)

❯t✐❧✐③❛r❡♠♦s ❛ s❡❣✉✐♥t❡ ❝♦♥str✉çã♦ q✉❡ s❡rá ♥❡❝❡ssár✐❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❝♦♥str✉çã♦ ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r❀ q✉❡ é ❞✐✈✐❞✐r ✉♠❛ r❡t❛ ❞❛❞❛ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♦ r❡tâ♥❣✉❧♦ ❢♦r♠❛❞♦ ♣❡❧♦ t♦❞♦ ❡ ♣♦r ✉♠❛ ❞❛s ♣❛rt❡s s❡❥❛ ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ ❛ ♦✉tr❛ ♣❛rt❡✳ ❊ss❛ ❝♦♥str✉çã♦ ❞❛rá ♦r✐❣❡♠ ❛♦ s❡❣♠❡♥t♦ á✉r❡♦✭❬✽❪✮✳

◗✉❡r❡♠♦s ❞❡t❡r♠✐♥❛r ✉♠ ♣♦♥t♦ ❍ s♦❜r❡ ❆❇ t❛❧ q✉❡ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s ❆❇ ❡ ❍❇ s❡❥❛ ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ ❆❍ ✭✈❡❥❛ ✜❣✉r❛ ✷✳✶✹✮✳

❋✐❣✉r❛ ✷✳✶✹✿

❈♦♥str✉❛ ♦ q✉❛❞r❛❞♦ ❆❇❈❉ ❡ s❡❥❛ ❊ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ❆❈✳ ❈♦♠ ❝❡♥tr♦ ❡♠ ❊ ❡ r❛✐♦ ❊❇✱ ❞❡s❝r❡✈❛ ♦ ❛r❝♦ ❞❡ ❝ír❝✉❧♦ q✉❡ ❝♦rt❛ ♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❡ ❈❆ ❡♠ ❋✳ ❈♦♠ ❝❡♥tr♦ ❡♠ ❊ ❡ r❛✐♦ ❊❇✱ ❞❡s❝r❡✈❛ ♦ ❛r❝♦ ❞❡ ❝ír❝✉❧♦ q✉❡ ❝♦rt❛ ♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❡ ❈❆ ❡♠ ❋✳ ❈♦♠ ❝❡♥tr♦ ❞❡ ❆ ❡ r❛✐♦ ❆❋✱ ❞❡s❝r❡✈❛ ♦ ❛r❝♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ❝♦rt❛ ❆❇ ❡♠ ❍✳ ❆✜r♠❛♠♦s q✉❡ ♦ ♣♦♥t♦ ❍ é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✳ ◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡

AH2

=HB_·AB✳

❆♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ✷✳✶✷ à r❡t❛ AC✱ ❞✐✈✐❞✐❞❛ ❛♦ ♠❡✐♦ ♣♦rC✱ ♣r♦❧♦♥❣❛❞❛ ❝♦♠ AF

t❡♠♦s q✉❡

F C_·F G+AE2 =EF2

♠❛s

EF =EB

❡♥tã♦

F C_·F G+AE2 =EB2 _⇒F C_·F G=EB2₋AE2(1)

♣❡❧♦ t❡♦r❡♠❛ ✷✳✶✶

EB2

=AE2

+AB2

❧♦❣♦✱

EB2

−AE2

=AB2

(2)

❉❡ ✭✶✮ ❡ ✭✷✮ t❡♠♦s q✉❡ F C _· F G = AB2

✱ s✉❜tr❛✐♥❞♦ ❞❡ ❝❛❞❛ ❧❛❞♦ ❞❛ ✐❣✉❛❧❞❛❞❡

AC_·AH✱

F C _·F G₋AC_·AH =AB2₋AC_·AH

F G2 =HB_·BD

♠❛s

F G=AH

❡

BD=AB

❧♦❣♦✱

AH2

=HB_·AB

❈♦♠ ✐ss♦ t❡♠♦s q✉❡ ♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦AH é ✐❣✉❛❧ ❛♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦sHB ❡ AB✳

❚❡♦r❡♠❛ ✷✳✶✸ ❙❡❥❛ P ✉♠ ♣♦♥t♦ ❡①t❡r✐♦r ❛ ✉♠ ❝ír❝✉❧♦C✱ s❡❥❛r ✉♠❛ r❡t❛ t❛♥❣❡♥t❡

❛ C ♥♦ ♣♦♥t♦ T _∈ C✱ ♣❛ss❛♥❞♦ ♣♦r P ❡ s❡❥❛ s ✉♠❛ r❡t❛ ❛r❜r✐tár✐❛ q✉❡ ♣❛ss❛ ♣♦r P

❡ ❝♦rt❛ C ♥♦s ♣♦♥t♦s R ❡ S✳ ❊♥tã♦ P R_·P S =P T2✳

❊①✐st❡♠ ❞♦✐s ❝❛s♦s ♣❛r❛ ❞❡♠♦♥str❛r✱ ✉♠ ❡♠ q✉❡ ❛ r❡t❛ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞♦ ❝ír❝✉❧♦ ❡ ♦✉tr❛ ❡♠ q✉❡ ❡❧❛ ♥ã♦ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦✳

❉❡♠♦♥str❛çã♦✿ ✭❆ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P✱ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞♦ ❝ír❝✉❧♦ ✭✈❡❥❛ ✜❣✉r❛ ✷✳✶✺✮✳

❋✐❣✉r❛ ✷✳✶✺✿

❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✷✳✶✷ à r❡t❛ ❘❙✱ ❞✐✈✐❞❛ ❛♦ ♠❡✐♦ ♣♦r ❖ ❡ ♣r♦❧♦♥❣❛❞❛ ❝♦♠ P❘✳ ❚❡♠♦s ❡♥tã♦ q✉❡ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s P❘ ❡ P❙✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❘❖✱ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ P❖✳ ❈♦♠♦ ❘❖ ❡ ❚❖ sã♦ r❛✐♦s ❞♦ ♠❡s♠♦ ❝ír❝✉❧♦✱ r❡t✐r❛♥❞♦ ❞♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s P❘ ❡ P❙✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❘❖ ❡ ❞♦ q✉❛❞r❛❞♦ s♦❜r❡ P❖ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❘❖ ❡ s♦❜r❡ ❚❖✱ t❡♠♦s q✉❡ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❘❖ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ ❚❖✳ ❙❡❣✉❡✲s❡ ❡♥tã♦ q✉❡ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s P❘ ❡ P❙ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ P❖ ♠❡♥♦s ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❚❖✳ ❖ ❚❡♦r❡♠❛ ✷✳✶✷ ♠♦str❛ q✉❡ ♦ q✉❛❞r❛❞♦ s♦❜r❡ P❖ ♠❡♥♦s ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❚❖ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ P❚✳ ❆ss✐♠✱ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s P❘ ❡ P❙ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ P❚✱ ❞❡st❛ ❢♦r♠❛ ♣r♦✈❛♠♦s ♦ ❝❛s♦ ❡♠ q✉❡ ❛ r❡t❛ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞♦ ❝ír❝✉❧♦✳

Pr♦✈❛r❡♠♦s ❛❣♦r❛ ♦ ❝❛s♦ ♦♥❞❡✱ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P✱ ♥ã♦ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞♦ ❝ír❝✉❧♦ ✭✈❡❥❛ ✜❣✉r❛ ✷✳✶✻✮✳

❙❡❥❛ ▼ ♦ ♣é ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❜❛✐①❛❞❛ ❞❡ ❖ s♦❜r❡ P❙✳ ❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✷✳✶✷ ❛ ❘❙ ♣r♦❧♦♥❣❛❞❛ ♣♦r P❘✱ t❡♠♦s q✉❡ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s P❘ ❡ P❙✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❘▼✱ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ P▼✳ ❆❞✐❝✐♦♥❛♥❞♦ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❖▼ ❛♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s P❘ ❡ P❙✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❘▼ ❡ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ P▼✱ t❡♠♦s q✉❡ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s P❘ ❡ P❙✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦s q✉❛❞r❛❞♦s s♦❜r❡ ❘▼ ❡ ❖▼✱ é ✐❣✉❛❧ ❛♦s q✉❛❞r❛❞♦s s♦❜r❡ P▼ ❡ ❖▼✳ ▼❛s✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✶✷✱ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❘▼✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❖▼✱ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ ❘❖ ❡ ♦ q✉❛❞r❛❞♦ s♦❜r❡ P▼✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ q✉❛❞r❛❞♦ ❖▼✱ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ P❖✳ ▼❛s ❡st❛♠♦s ♥❛ s✐t✉❛çã♦ ❞♦ ♣r✐♠❡✐r♦ ❝❛s♦✳ ❈♦♠♦ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❘❖ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ s♦❜r❡ ❚❖✱ t❡♠♦s q✉❡ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s

❋✐❣✉r❛ ✷✳✶✻✿

P❘ ❡ P❙ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ P❖ ♠❡♥♦s ♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ❚❖✱ ♦✉ s❡❥❛✱ ♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ P❚ ✭❬✽❪✮✳

❚❡♦r❡♠❛ ✷✳✶✹ ❙❡❥❛ ✉♠ ♣♦♥t♦ ❆ ❢♦r❛ ❞♦ ❝ír❝✉❧♦✱ ❢♦r❡♠ tr❛ç❛❞❛s ❞✉❛s r❡t❛s ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ ❆✱ ✉♠❛ ❞❡❧❛s ✐♥t❡r❝❡♣t❛♥❞♦ ♦ ❝ír❝✉❧♦ ❡♠ ❇ ❡ ❋✱ ❡ ❛ ♦✉tr❛ ✐♥t❡r❝❡♣t❛♥❞♦ ♦ ❝ír❝✉❧♦ ❡♠ ❉✱ ❡♥tã♦ s❡ AB_·AF =AD2✱ ❡♥tã♦ AD é t❛♥❣❡♥t❡ ♥♦ ♣♦♥t♦ ❉✳

❋✐❣✉r❛ ✷✳✶✼✿

❉❡♠♦♥str❛çã♦✿ ♣♦r ❆ tr❛ç❛♠♦sAT ✭✈❡❥❛ ✜❣✉r❛ ✷✳✶✼✮ t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ ♥♦ ♣♦♥t♦

❚✳ ❊♥tã♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✶✸✱ s❛❜❡♠♦s q✉❡ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s AB ❡AF é ✐❣✉❛❧ ❛♦

q✉❛❞r❛❞♦ s♦❜r❡ AD✳ ❙❡ ❞♦✐s q✉❛❞r❛❞♦s sã♦ ✐❣✉❛✐s✱ s❡✉s ❧❛❞♦s s❡rã♦ ✐❣✉❛✐s✱ ♦✉ s❡❥❛✱ AT =AD✳ ▼❛s ❝♦♠ ❚ ❡ ❉ sã♦ s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ à r❡t❛ q✉❡ ♣❛ss❛ ♣♦r ❆ ❡ ❚✱

❉ s❡rá ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛✱ ♦✉ s❡❥❛✱ AD é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦✭❬✽❪✮✳

❚❡♦r❡♠❛ ✷✳✶✺ ❖s â♥❣✉❧♦s ♦♣♦st♦s ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ ❝ír❝✉❧♦ sã♦ ✐❣✉❛✐s✱ ❝♦♥❥✉♥t❛♠❡♥t❡✱ ❛ ❞♦✐s r❡t♦s✳

❋✐❣✉r❛ ✷✳✶✽✿

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ♦ ❝ír❝✉❧♦ ✭✈❡❥❛ ✜❣✉r❛ ✷✳✶✽✮ ❡ ♥❡❧❡ ❡st❡❥❛ ♦ q✉❛❞r✐❧át❡r♦✱ ❆❇❈❉ q✉❡r❡♠♦s ❞❡♠♦♥str❛r q✉❡ ♦s â♥❣✉❧♦s ♦♣♦st♦s sã♦ ✐❣✉❛✐s ❛ ❞♦✐s r❡t♦s✳ ❚r❛❝❡ ♦s s❡❣✲ ♠❡♥t♦s ❆❈✱ ❇❉✳ ❈♦♠♦ s❡ s❛❜❡✱ ♦s três â♥❣✉❧♦s ❞❡ t♦❞♦ tr✐â♥❣✉❧♦ sã♦ ✐❣✉❛✐s ❛ ❞♦✐s r❡t♦s✱ ♣♦rt❛♥t♦✱ ♦s três â♥❣✉❧♦s∠❈❆❇✱∠❆❇❈ ❡∠❇❈❆ ❞♦ tr✐â♥❣✉❧♦ ❆❇❈ sã♦ ✐❣✉❛✐s

❛ ❞♦✐s r❡t♦s✳ ▼❛s✱ ♣♦r ✉♠ ❧❛❞♦✱ ♦ ∠❈❆❇ é ✐❣✉❛❧ ❛♦ ∠❇❉❈✱ ♣♦✐s ❡stã♦ ♥♦ ♠❡s♠♦

❛r❝♦ ❇❆❉❈✱ ❡ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ∠❆❈❇ é ✐❣✉❛❧ ❛♦∠❆❉❇✱ ♣♦✐s ❡stã♦ ♥♦ ♠❡s♠♦ ❛r❝♦

❆❉❈❇✱ ♣♦rt❛♥t♦✱ ♦ ∠❆❉❈ é ✐❣✉❛❧ ❛ s♦♠❛ ❞♦s∠❇❆❈ ❡∠❆❈❇✳ ❆❞✐❝✐♦♥❡ ♦∠❆❇❈ ❛♦

â♥❣✉❧♦ ∠❆❉❈ ❡ ❛♦s â♥❣✉❧♦s∠❇❆❈ ❡ ∠❆❈❇✱ t❡♠♦s ❡♥tã♦ q✉❡ ∠❆❉❈ ❡ ∠❆❇❈ sã♦

✐❣✉❛✐s ❛ ∠❆❇❈✱ ∠❇❆❈ ❡∠❆❈❇✱ q✉❡ sã♦ ✐❣✉❛✐s ❛ ❞♦✐s r❡t♦s✳ ▲♦❣♦✱ ∠❆❉❈ ❡∠❆❇❈

sã♦ ✐❣✉❛✐s ❛ ❞♦✐s r❡t♦s✳ ❉♦ ♠❡s♠♦ ♠♦❞♦✱ ♣r♦✈❛✲s❡ q✉❡ ∠❇❆❉ ❡ ∠❉❈❇ sã♦ ✐❣✉❛✐s

❛ ❞♦✐s r❡t♦s✳ P♦rt❛♥t♦✱ ✉♠ q✉❛❞r✐❧át❡r♦ ✐♥s❝r✐t♦ ♥♦ ❝ír❝✉❧♦✱ ♦s â♥❣✉❧♦s ♦♣♦st♦s sã♦ ✐❣✉❛✐s ❛ ❞♦✐s r❡t♦s✭❬✺❪✮✳

❚❡♦r❡♠❛ ✷✳✶✻ ❊♠ ✉♠ ❝ír❝✉❧♦✱ ♦ â♥❣✉❧♦ s✉❜t❡♥❞✐❞♦ ♣♦r ✉♠ ❞✐â♠❡tr♦ é r❡t♦❀ ♦ ❝♦♥t✐❞♦ ❡♠ ✉♠ s❡❣♠❡♥t♦ ❞❡ ❝ír❝✉❧♦ ♠❡♥♦r ❞♦ q✉❡ ✉♠ s❡♠✐✲❝ír❝✉❧♦ s❡rá ♠❛✐♦r ❞♦ q✉❡ ✉♠ r❡t♦❀ ❡ ♦ ❝♦♥t✐❞♦ ❡♠ ✉♠ s❡❣♠❡♥t♦ ❞❡ ❝ír❝✉❧♦ ♠❛✐♦r q✉❡ ✉♠ s❡♠✐✲❝ír❝✉❧♦ s❡rá ♠❡♥♦r q✉❡ ✉♠ r❡t♦✳

❋✐❣✉r❛ ✷✳✶✾✿

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ♦ ❝ír❝✉❧♦ ❆❇❈❉ ✭✈❡❥❛ ✜❣✉r❛ ✷✳✶✾✮ ❡ s❡❥❛♠ ♦ ❇❈ ✉♠ ❞✐â♠❡✲ tr♦ ❞❡❧❡ ❡ ❊ ♦ ❝❡♥tr♦✱ ❡ ✜q✉❡♠ ❢♦r♠❛❞♦s ♦s s❡❣♠❡♥t♦s ❇❆✱ ❆❈✱ ❆❉✱ ❉❈✳ ❉✐❣♦ q✉❡

∠BAC ♥♦ s❡♠✐❝ír❝✉❧♦ ❇❆❈ é r❡t♦✱ ♦ ∠ABC q✉❡ ❡stá ♥♦ ❛r❝♦ ❆❇❈ é ♠❛✐♦r q✉❡ ♦

s❡♠✐❝ír❝✉❧♦ ❇❆❈ é ♠❡♥♦r q✉❡ ✉♠ r❡t♦✱ ❡♥q✉❛♥t♦ angleADC ♥♦ ❛r❝♦ ❆❉❈ é ♠❡♥♦r

q✉❡ ♦ s❡♠✐❝ír❝✉❧♦ ❇❆❈ é ♠❛✐♦r q✉❡ ✉♠ r❡t♦✳ ▲✐❣✉❡ ❆❊ ❡ tr❛❝❡ ❇❆ ❛té ❋✳

BE =EA _⇒∠ABE =∠BAE

❞♦ △AEB ❡

CE =EA_⇒∠ACE =∠CAE

❞♦ △CEA✱ ♣♦rt❛♥t♦

∠BAC =∠ABC+∠ACB

♠❛s

∠F AC =∠ABC+∠ACB

â♥❣✉❧♦ ❡①t❡r♥♦ ❞♦ △ABC✳ P♦rt❛♥t♦∠F AC =∠BAC✱ ❝❛❞❛ ✉♠ é ✉♠ â♥❣✉❧♦ r❡t♦✳

❊ ❝♦♠♦ ♦s â♥❣✉❧♦s∠ACB,∠BAC❞♦_△ABC sã♦ ♠❡♥♦r❡s q✉❡ ❞♦✐s r❡t♦s✱ ❡ ♦∠BAC

é r❡t♦✱ ❧♦❣♦ ∠ABC é ♠❡♥♦r q✉❡ ✉♠ r❡t♦ ❡ ❡stá ♥♦ ❛r❝♦ ❆❇❈✱ q✉❡ é ♠❛✐♦r q✉❡ ♦

s❡♠✐❝ír❝✉❧♦✳

❈♦♠♦ ❆❇❈❉ é q✉❛❞r✐❧át❡r♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ ❝ír❝✉❧♦✱ ❧♦❣♦ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✶✺✱ ♦s

∠ABC ❡ ∠ADC ❂ ❞♦✐s r❡t♦s ❡ ∠ABC é ♠❡♥♦r q✉❡ ✉♠ r❡t♦✱ ♣♦rt❛♥t♦ ∠ADC é

♠❛✐♦r q✉❡ ✉♠ r❡t♦ ❡ ❡stá ♥♦ ❛r❝♦ ❆❉❈ q✉❡ é ♠❡♥♦r ❞♦ q✉❡ ♦ s❡♠✐❝ír❝✉❧♦✳ ❖❜s❡r✈❛♥❞♦ ❛ ✜❣✉r❛ ✷✳✶✾ ♦ ∠ADC s✉❜t❡♥❞✐❞♦ ♣❡❧♦ ❛r❝♦ ❆❇❈ é ♠❛✐♦r q✉❡ ✉♠ â♥❣✉❧♦ r❡t♦✱ ♣♦r

♦✉tr♦ ❧❛❞♦✱ ♦ ∠ABC s✉❜t❡♥❞✐❞♦ ♣❡❧♦ ❛r❝♦ ❆❉❈ é ♠❡♥♦r q✉❡ ✉♠ r❡t♦✳ ❖ â♥❣✉❧♦

❢♦r♠❛❞♦ ♣❡❧♦s s❡❣♠❡♥t♦s ❇❆ ❡ ❆❈ é r❡t♦✳ P♦rt❛♥t♦✱ ♦ â♥❣✉❧♦ s✉❜t❡♥❞✐❞♦ ♣❡❧♦ ❛r❝♦ ❆❇❈ é ♠❛✐♦r q✉❡ ✉♠ r❡t♦✳ ❉❡ ♥♦✈♦✱ ❝♦♠♦ ♣❡❧♦s s❡❣♠❡♥t♦s ❆❈ ❡ ❆❋ é r❡t♦✱ ♦ â♥❣✉❧♦ s✉❜t❡♥❞✐❞♦ ♣❡❧♦ ❛r❝♦ ❆❉❈ é ♠❡♥♦r q✉❡ ✉♠ r❡t♦✳

▲♦❣♦✱ ❡♠ ✉♠ ❝ír❝✉❧♦✱ ♦ â♥❣✉❧♦ ♥♦ s❡♠✐❝ír❝✉❧♦ é r❡t♦✱ ♦ â♥❣✉❧♦ q✉❡ s❡ ❡♥❝♦♥tr❛ ♥♦ ❛r❝♦ ♠❡♥♦r é ♠❛✐♦r q✉❡ ✉♠ r❡t♦✱ ❡ ♦ â♥❣✉❧♦ q✉❡ s❡ ❡♥❝♦♥tr❛ ♥♦ ❛r❝♦ ♠❡♥♦r é ♠❛✐♦r q✉❡ ✉♠ r❡t♦✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ♦ â♥❣✉❧♦ ❞♦ ❛r❝♦ ♠❛✐♦r é ♠❛✐♦r q✉❡ ✉♠ r❡t♦✱ ❡ ♦ â♥❣✉❧♦ ❞♦ ❛r❝♦ ♠❡♥♦r é ♠❡♥♦r q✉❡ ✉♠ r❡t♦✭❬✹❪✮✳

❚❡♦r❡♠❛ ✷✳✶✼ ❖ â♥❣✉❧♦ ❡♥tr❡ ✉♠❛ t❛♥❣❡♥t❡ ❡ ✉♠❛ ❝♦r❞❛ ❞❡ ✉♠ ❝ír❝✉❧♦ é ✐❣✉❛❧ ❛♦ â♥❣✉❧♦ s✉❜t❡♥❞✐❞♦ ♣❡❧❛ ❝♦r❞❛✱ ❞♦ ❧❛❞♦ ♦♣♦st♦ ❛ t❛♥❣❡♥t❡✳

❋✐❣✉r❛ ✷✳✷✵✿

❉❡♠♦♥str❛çã♦✿ ❈♦♠ ❡❢❡✐t♦✱ s❡❥❛♠ ❊❋ ❛ r❡t❛ t❛♥❣❡♥t❡ ❇❉ ❛ ❝♦r❞❛✳ ❚r❛❝❡ ♦ ❞✐â♠❡tr♦ ❆❇ ❞♦ ❝ír❝✉❧♦✱ ♣♦r ❇ ✭ ✈❡❥❛ ✜❣✉r❛ ✷✳✷✵✮❀ ❡s❝♦❧❤❛ ❈ q✉❛❧q✉❡r ♣❡rt❡♥❝❡♥t❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ tr❛❝❡ ♦s s❡❣♠❡♥t♦s ❆❉✱ ❉❈ ❡ ❇❈❀ ♦∠❆❉❇✱ q✉❡ s❡ ❡♥❝♦♥tr❛ s♦❜r❡

✉♠❛ s❡♠✐❝✐r❝✉♥❢❡rê♥❝✐❛ é r❡t♦✱ ❞✐ss♦ ❞❡❝♦rr❡ q✉❡ ♦s ∠❇❆❉ ❡ ∠❆❇❉✱ ❥✉♥t♦s✱ sã♦