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O CÍRCULO DOS NOVE PONTOS E ALGUMAS DE SUAS POTENCIALIDADES DIDÁTICAS

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●✐❧❞❡r ❞❛ ❙✐❧✈❛ ▼❡sq✉✐t❛

❖ ❝ír❝✉❧♦ ❞♦s ♥♦✈❡ ♣♦♥t♦s ❡ ❛❧❣✉♠❛s ❞❡ s✉❛s

♣♦t❡♥❝✐❛❧✐❞❛❞❡s ❞✐❞át✐❝❛s

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●✐❧❞❡r ❞❛ ❙✐❧✈❛ ▼❡sq✉✐t❛

❖ ❝ír❝✉❧♦ ❞♦s ♥♦✈❡ ♣♦♥t♦s ❡ ❛❧❣✉♠❛s ❞❡ s✉❛s

♣♦t❡♥❝✐❛❧✐❞❛❞❡s ❞✐❞át✐❝❛s

❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ à ❈♦♦r❞❡♥❛çã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ❞❛ ❯♥✐✈❡rs✐✲ ❞❛❞❡ ❋❡❞❡r❛❧ ❘✉r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦✱ ♦r✐❡♥✲ t❛❞❛ ♣❡❧♦ Pr♦❢❡ss♦r ❉♦✉t♦r ❆❞r✐❛♥♦ ❘❡❣✐s ▼❡❧♦ ❘♦❞r✐❣✉❡s ❞❛ ❙✐❧✈❛✱ ♣❡❧♦ ❛❧✉♥♦ ●✐❧❞❡r ❞❛ ❙✐❧✈❛ ▼❡sq✉✐t❛✱ ❝♦♠♦ r❡q✉✐s✐t♦ à ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛✳

❖r✐❡♥t❛❞♦r✿

❆❞r✐❛♥♦ ❘❡❣✐s ❘♦❞r✐❣✉❡s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❘✉r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦

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❉✐ss❡rt❛çã♦ ❞❡ Pr♦❥❡t♦ ❋✐♥❛❧ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ s♦❜ ♦ tít✉❧♦ ✏❖ ❝ír❝✉❧♦ ❞♦s ♥♦✈❡ ♣♦♥t♦s ❡ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣♦t❡♥❝✐❛❧✐❞❛❞❡s ❞✐❞át✐❝❛s✑✱ ❞❡❢❡♥❞✐❞❛ ♣♦r ●✐❧❞❡r ❞❛ ❙✐❧✈❛ ▼❡sq✉✐t❛ ❡ ❛♣r♦✈❛❞❛ ❡♠ ✷✽ ❞❡ ❛❣♦st♦ ❞❡ ✷✵✶✸✱ ❡♠ ❘❡❝✐❢❡✱ ❊st❛❞♦ ❞❡ P❡r♥❛♠❜✉❝♦✱ ♣❡❧❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❝♦♥st✐t✉í❞❛ ♣❡❧♦s ♣r♦❢❡ss♦r❡s✿

Pr♦❢✳ ❉r✳ ❆❞r✐❛♥♦ ❘❡❣✐s ❘♦❞r✐❣✉❡s ❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r✳ ❏♦r❣❡ ❆♥t♦♥✐♦ ❍✐♥♦❥♦s❛ ❱❡r❛ ❯❋❘P❊

Pr♦❢✳ ❉r✳ P❛✉❧♦ ❘♦❜❡rt♦ ❙❛♥t✐❛❣♦ ❯❋❘P❊

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❆♣r❡s❡♥t❛çã♦

❆s q✉❡stõ❡s r♦t✉❧❛❞❛s ❝♦♠♦ ❞✐❢í❝❡✐s s❡♠♣r❡ ❞❡s♣❡rt❛r❛♠ ♠✐♥❤❛ ❝✉r✐♦s✐❞❛❞❡✱ ❞❡s❞❡ ♦s ❥á ❞✐st❛♥t❡s ❛♥♦s ❞❛ ❛❞♦❧❡s❝ê♥❝✐❛✱ é♣♦❝❛ ❡♠ q✉❡ ❡r❛ ❛tr❛í❞♦ ♣❡❧❛s ♣r♦✈❛s ❞❡ ▼❛t❡♠át✐❝❛ ❞♦s ❝♦♥❝✉rs♦s ♠✐❧✐t❛r❡s ❞❡ ❡♥tã♦✱ ♦s q✉❛✐s ♠❡ ❡r❛♠ ♠❛✐s ❝♦♥❤❡❝✐❞♦s ❞♦ q✉❡ ❛s ♦❧✐♠♣í❛❞❛s ❞❡ ▼❛t❡♠át✐❝❛✳

❖ q✉❡ ❝♦♠❡ç♦✉ ❡♠ ❢♦r♠❛ ❞❡ ❝✉r✐♦s✐❞❛❞❡ t❡r♠✐♥♦✉ s❡ tr❛♥s❢♦r♠❛♥❞♦ ❡♠ ♣r❛③❡r✱ ❞❡ ❢♦r♠❛ q✉❡✱ ❛t✉❛❧♠❡♥t❡✱ ♥ã♦ ♠❡ é ♣♦ssí✈❡❧ ♠❛✐s ❞✐❢❡r✐r ❡ss❛s s❡♥s❛çõ❡s✱ t❛♠♣♦✉❝♦ ❞❡❧✐♠✐t❛r ✕ ♦ q✉❡ é ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ❢r✉str❛♥t❡✱ ♣♦ré♠✱ ❡s♣✐r✐t✉❛❧♠❡♥t❡ ❣r❛t✐✜❝❛♥t❡✦ P♦r ✈❡③❡s✱ ♣♦❞❡ ✈✐r ❛ s❡r ❡♥❧❡✈❛♥t❡ ♦❜s❡ssã♦✱ ❝♦♠♦ q✉❛♥❞♦ ✐♥s✐st✐✱ ♣♦r ♥♦✈❡ ♠❡s❡s✱ ♥❛ ❜✉s❝❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ➪❧❣❡❜r❛✳

◆♦ ❡♥t❛♥t♦✱ ❛ ●❡♦♠❡tr✐❛✶ s❡♠♣r❡ ❢♦✐ ❛ q✉✐♥t❡ssê♥❝✐❛ ❞♦s ♠❡✉s ❞❡s❧✉♠❜r❛♠❡♥t♦s ♠❛✲

t❡♠át✐❝♦s✳ ▼❛r❛✈✐❧❤❛✈❛✲♠❡ ❝♦♠ ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s q✉❡ ❡r❛♠ r❡s♦❧✈✐❞♦s ♣♦r tr❛ç❛❞♦s ♦♣♦rt✉♥♦s ❡✴♦✉ ♣r♦♣r✐❡❞❛❞❡s s✉r♣r❡❡♥❞❡♥t❡s ❡♠ s❡✉s ❡❢❡✐t♦s✳

❉❡ ❝❡rt♦ ♠♦❞♦✱ ❡ss❛ ❡①♣❡r✐ê♥❝✐❛ ✐♥s♣✐r❛❞♦r❛ r❡♣❡t✐✉✲s❡ ♥❛ ❝❛❞❡✐r❛ ▼❆✷✶✱ q✉❛♥❞♦ ♦ ♣r♦❢❡ss♦r ❆❞r✐❛♥♦ ❘❡❣✐s ♣r♦♣ôs ✉♠❛ ❧✐st❛ ❞❡ ❡①❡r❝í❝✐♦s ❞❡s❛✜❛❞♦r❡s✱ ♦s q✉❛✐s r❡♣r❡s❡♥t❛✲ ✈❛♠ ✉♠❛ ♣❛r❝❡❧❛ ❞❛ ●❡♦♠❡tr✐❛ r❛r❛♠❡♥t❡ ✈✐st❛ ♥❛ ❡s❝♦❧❛✱ q✉❡ ❞✐rá ❡①♣❧♦r❛❞❛✱ t❛✐s ❝♦♠♦ ✐s♦❣♦♥❛✐s✱ s✐♠❡❞✐❛♥❛s✱ q✉á❞r✉♣❧❛ ❤❛r♠ô♥✐❝❛✱ ❝ír❝✉❧♦ ❞❡ ❆♣♦❧ô♥✐♦✱ t❡♦r❡♠❛ ❞❡ ▼❡♥❡❧❛✉s✱ t❡♦r❡♠❛ ❞❡ ❈❡✈❛✱ ♣♦♥t♦ ❞❡ ▲❡♠♦✐♥❡✱ r❡t❛ ❞❡ ❊✉❧❡r✱ r❡t❛ ❞❡ ❙✐♠♣s♦♥✱ ♣♦♥t♦ ❞❡ ●❡r❣♦♥♥❡✱ ♣♦♥t♦ ❞❡ ◆❛❣❡❧✱ ❡t❝✳ ▼❛s✱ ♦ q✉❡ ♠❛✐s ❝❛♣t♦✉ ♠✐♥❤❛ ❛t❡♥çã♦ ❢♦✐ ♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥✲ t♦s✳ ❆❝❤❡✐ ♣♦r ❞❡♠❛✐s ❜❡❧♦ três t❡r♥♦s ❞❡ ♣♦♥t♦s ❞❡ ♥❛t✉r❡③❛s ❞✐❢❡r❡♥t❡s s❡r❡♠ ✐♥t❡❣r❛♥t❡s ❞❡ ✉♠ ♠❡s♠♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ tã♦ s✐♠♣❧❡s✳

❈♦♥t✉❞♦✱ ❛♦ t❡♥t❛r ♠❡ ❞❡❜r✉ç❛r s♦❜r❡ ♦ t❡♠❛✱ t✐✈❡ ❝❡rt❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ ❡♥❝♦♥tr❛r ❢♦♥t❡s ❞❡ ❝♦♥s✉❧t❛✳ ◆♦r♠❛❧♠❡♥t❡✱ ♦ t❡♠❛ é ❞✐s❝✉t✐❞♦ ❡♠ r❡✈✐st❛s ❡s♣❡❝✐❛❧✐③❛❞❛s ❡♠ ▼❛t❡✲ ♠át✐❝❛✱ ❡♠ ❧✐✈r♦s ❞❡ ●❡♦♠❡tr✐❛ ❵♠❛✐s ❛✈❛♥ç❛❞❛✬ ♦✉ ♠❛t❡r✐❛✐s ❞❡ ❝♦♠♣❡t✐çõ❡s ♠❛t❡♠át✐❝❛s✳ ■ss♦✱ t❛❧✈❡③✱ ♦❝❛s✐♦♥♦✉ ♦ ♣❛r❛❞✐❣♠❛ ❞❡ q✉❡✱ ♣♦r ❡st❛r ♥♦ ✜♥❛❧ ❞♦ ❧✐✈r♦✱ ❥á ♣r❡ss✉♣✉♥❤❛ ❝♦✲ ♥❤❡❝✐♠❡♥t♦ ♣ré✈✐♦ ❞♦ ❧❡✐t♦r✳ ❆❧é♠ ❞♦ ♠❛✐s✱ ❣❡r❛❧♠❡♥t❡✱ ♣❛r❛ ❡①♣♦s✐çã♦ ❞❡ss❡s t❡♠❛s ♠❛✐s ✐♥❝♦♠✉♥s✱ r❡s❡r✈❛✲s❡ ❞❡♠♦♥str❛çõ❡s ❝✉rt❛s✱ ❝♦♥t❡♥❞♦ ❛♣❡♥❛s ♦ q✉❡ s❡ ❝♦♥s✐❞❡r❛ ✏❡ss❡♥❝✐❛❧✑ ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦✳

▼❛s✱ q✉❡ ❜♦♠ q✉❡ ♦ ❡♥❝❛♥t♦ ❢♦✐ ♠❛✐♦r q✉❡ ❛ ❞✐✜❝✉❧❞❛❞❡✦ ❊✱ ❛♣ós ❞✐✈❡rs❛s ❧❡✐t✉r❛s✱ ❥✉❧❣✉❡✐ ✈✐s❧✉♠❜r❛r ✉♠❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞✐❞át✐❝❛ q✉❡ s❡ tr❛❞✉③ ❝♦♠♦ ❡①❝❡❧❡♥t❡ ❛♣❧✐❝❛çã♦ ❞❡ ✈ár✐♦s ❝♦♥❝❡✐t♦s ❞❡ ●❡♦♠❡tr✐❛ ♥✉♠ só t❡♠❛✳

◆❛ ✈❡r❞❛❞❡✱ ♦ q✉❡ ♠❡ ♣r♦♣✉s ❢♦✐ r❡t✐r❛r ❞❡t❛❧❤❡s ❞❡ ❝❛❞❛ ❛❜♦r❞❛❣❡♠ ❡ ❥✉♥tá✲❧❛s ♥✉♠❛ ♥♦✈❛ ♣r♦♣♦st❛✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ ❝✐t♦ ❡ ❞❡st❛❝♦ ❛ r❡❧❡✈â♥❝✐❛ ❞♦ ✏❈ír❝✉❧♦ ❞♦s ❙❡✐s P♦♥t♦s✑✱ ♦ q✉❛❧ ♠❡ ❡r❛ ❞❡s❝♦♥❤❡❝✐❞♦✳

P❡♥s❛♥❞♦ ♥❛ ❝♦♥❥✉♥t✉r❛ ❛t✉❛❧ ❞❡ ♠✉✐t♦s ♣r♦❢❡ss♦r❡s ♥❡ss❡ ♣❛ís✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ ❞❛s s✉❛s ❞✐✈❡rs❛s ❞✐✜❝✉❧❞❛❞❡s✱ t❡♥❝✐♦♥❡✐ ❝♦♥❞❡♥s❛r ♥✉♠❛ só ❢♦♥t❡ s✉❜sí❞✐♦s ♣❛r❛ ♣r♦♣♦r❝✐♦♥❛r✲

(5)

❧❤❡s ✉♠ ❝♦♥s✐❞❡rá✈❡❧ s✉♣♦rt❡ ❞❡ ❝♦♥s✉❧t❛ ♣❛r❛ ✉♠❛ ❛t✐✈✐❞❛❞❡ ❡♥r✐q✉❡❝❡❞♦r❛ ♣❛r❛ s❡✉s ❛❧✉♥♦s✱ ❞❡s❞❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❧❡✐t✉r❛ ❡ ❛q✉✐s✐çã♦ ❞❛ ♣r♦♣♦st❛ ❛té ❛ ♣r❡t❡♥❞✐❞❛ ❛♣❧✐❝❛çã♦✳

❉❡✈✐❞♦ ❛ ✐ss♦✱ ♦s ❝♦♠❡♥tár✐♦s s♦❜r❡ ❛s ❛❜♦r❞❛❣❡♥s ❡♥❝♦♥tr❛❞❛s✱ ❜❡♠ ❝♦♠♦ ❛ ♣r♦♣♦st❛ ❞✐❞át✐❝❛ ❞❡ ❛t✐✈✐❞❛❞❡ ❝♦♠ ♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s✱ ❢♦r❛♠ ❞❡s❝r✐t♦s ❞❡t❛❧❤❛❞❛♠❡♥t❡✱ ✈✐s❛♥❞♦ ♣r❡❡♥❝❤❡r ❛s ❧✐♠✐t❛çõ❡s t✐♣♦❣rá✜❝❛s ❞♦s ❧✐✈r♦s q✉❡ ❡✈❡♥t✉❛❧♠❡♥t❡ ❛❜♦r❞❛♠ ♦ t❡♠❛✳ ❖✉tr♦ss✐♠✱ ❛ ❢✉♥❝✐♦♥❛❧✐❞❛❞❡ ❞❡st❡ tr❛❜❛❧❤♦ r❡s✐❞✐✉ ♥❛ ♣r❡♠✐ss❛ ❞❡ q✉❡ é ♣♦ssí✈❡❧✱ s✐♠✱ ❞❡s♣❡rt❛r ♦ ❡❞✉❝❛♥❞♦ ♣❛r❛ ❛t✐✈✐❞❛❞❡s ✐♥st✐❣❛♥t❡s ❡✴♦✉ ❞❡s❛✜❛❞♦r❛s✱ ✉s❛♥❞♦ ♣❛r❛ t❛❧ ❛♣❡♥❛s ❝♦♥❤❡❝✐♠❡♥t♦s ❣❡♦♠étr✐❝♦s ❜ás✐❝♦s ✭♥♦ ❝❛s♦✱ ❞♦ ♦✐t❛✈♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✮ ❡✱ q✉❡♠ s❛❜❡✱ ♣❛r❛ s✉❛ ❝❛♣❛❝✐❞❛❞❡ ✐♥✈❡♥t✐✈❛✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❡s♣❡r❛✲s❡ t❛♠❜é♠ q✉❡ ♣r♦✈♦q✉❡ ♥♦ ❛❧✉♥♦ ❛ ❝❡rt❡③❛ ❞❛ s✉❛ ❝❛♣❛❝✐❞❛❞❡✳

(6)

❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ♣r♦♣✐❝✐❛r ❛♦ ♣r♦❢❡ss♦r✱ ❞❡s❞❡ ♦ ♦✐t❛✈♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ s✉♣♦rt❡ ♠❛t❡♠át✐❝♦ ❡ s✉❣❡stõ❡s ❞✐❞át✐❝❛s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❛t✐✈✐❞❛❞❡ ❡♠ ❝❧❛ss❡✱ ♣❛r❛ ✈❡r✐✜❝❛çã♦ ❞❡ ❝♦♥❝❡✐t♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ♣❧❛♥❛✱ ✈✐sã♦ ❣❡♦♠étr✐❝❛✱ ❡①❡r❝í❝✐♦ ❡ ❡stí♠✉❧♦ à ❝r✐❛t✐✈✐❞❛❞❡ ❞♦ ❛❧✉♥♦✳

❊♠ ❝♦♥s♦♥â♥❝✐❛ ❝♦♠ ❛♠♣❧♦s ♦❜❥❡t✐✈♦s✱ ❢♦✐ ❡s❝♦❧❤✐❞♦ ♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ❝♦♠♦ ❝♦♥t❡ú❞♦ ❛ s❡r ❞✐s❝✉t✐❞♦✱ t❡♠❛ ❛❜♦r❞❛❞♦ q✉❛s❡ q✉❡ ❡①❝❧✉s✐✈❛♠❡♥t❡ ❡♠ ❧✐t❡r❛t✉r❛ ♠❛t❡♠á✲ t✐❝❛ ♥♦r♠❛❧♠❡♥t❡ ♥ã♦ ✉s❛❞❛ ♣❡❧❛s ✐♥st✐t✉✐çõ❡s ❞❡ ❡♥s✐♥♦ ❡♠ ❣❡r❛❧✳ ◆♦ ❡♥t❛♥t♦✱ ❡st❡ tó♣✐❝♦ s❡ r❡✈❡st❡ ❞❡ ❝♦♥s✐❞❡rá✈❡❧ ✈❛r✐❡❞❛❞❡ ❞❡ ✉s♦ ❞❡ ❝♦♥❝❡✐t♦s ❣❡♦♠étr✐❝♦s✱ ♦ q✉❡ s❡ ♣♦❞❡ tr❛❞✉✲ ③✐r ♥✉♠❛ ❛t✐✈✐❞❛❞❡ ♣❡❞❛❣ó❣✐❝❛ ❜❛st❛♥t❡ ❞✐✈❡rs✐✜❝❛❞❛✱ ❝♦♠ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ s❡ ❡①♣❧♦r❛r ✈ár✐♦s ❛s♣❡❝t♦s ❞✐❞át✐❝♦s✳

P❛r❛ ♠❛t❡r✐❛❧✐③❛r ❡ss❛s ✐❞é✐❛s✱ ❢♦✐ ♣r♦♣♦st♦ ✉♠ ❡st✉❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♦♥❞❡✱ ♣❛r❛ t❛♥t♦✱ sã♦ ❢❡✐t❛s ❞❡♠♦♥str❛çõ❡s ❝♦♠ ê♥❢❛s❡ ❡♠ ❣❡♦♠❡tr✐❛ ❛♥❣✉❧❛r ✭✐❞❡♥t✐✜❝❛çã♦ ❡ ♠❡❞✐çã♦ ❞❡ â♥❣✉❧♦s✮✱ ❛♦ ♣❛ss♦ q✉❡✱ ♥❛ ♠étr✐❝❛ ✭♠❡❞✐çã♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦s✮✱ ❡①❝❧✉✐✉✲s❡ ✉♠ ❛s♣❡❝t♦ ❝♦♥st❛♥t❡ ❡♠ ❞❡♠♦♥str❛çõ❡s tr❛❞✐❝✐♦♥❛✐s✱ ❛ s❛❜❡r ✕ ❛ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ ✉♠❛ ✈❡③ q✉❡ ❡st❛r✐❛ ❛♣r♦♣r✐❛♥❞♦✲s❡ ❞❡ ❝♦♥t❡ú❞♦ ❢♦r❛ ❞❛ ♣r♦♣♦st❛ ❜ás✐❝❛ ❞♦ ♣ú❜❧✐❝♦✲❛❧✈♦ ♣r❡t❡♥❞✐❞♦✳

◗✉❛tr♦ ♥♦✈✐❞❛❞❡s sã♦ ❛♣r❡s❡♥t❛❞❛s ♥❡st❡ t❡①t♦✱ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s✱ ❝♦♥st✐t✉❡♠ ❛ ❡ssê♥❝✐❛ ❞❡ss❡ tr❛❜❛❧❤♦✿

✕ P❛rt✐♥❞♦✲s❡ ❞♦ ❝ír❝✉❧♦ q✉❡ ❝♦♥té♠ ♦s três ♣♦♥t♦s ♠é❞✐♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❛❝✉tâ♥❣✉❧♦❀ ✕ ❆ ♥ã♦ ✉t✐❧✐③❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ❝♦♠♦ ♣ré✲r❡q✉✐s✐t♦ ♣❛r❛ ❛ ❛❜♦r❞❛❣❡♠ ❞♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s❀

✕ ✉♠❛ ❢♦r♠❛ ❞❡ ❝♦♥❝❧✉✐r q✉❡ ♦ r❡❢❡r✐❞♦ ❝ír❝✉❧♦ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ❞❡ ❊✉❧❡r ♥✉♠ tr✐â♥❣✉❧♦❀

❡♥q✉❛♥t♦ ❛ q✉❛rt❛✱

✕ ❆ ❛✉sê♥❝✐❛ ❞❛ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s ♥♦ ♣r♦❝❡ss♦ ❞❡❞✉t✐✈♦✳

(7)

❆❜str❛❝t

❚❤✐s ♣r❡s❡♥t ✇♦r❦ ❤❛✈❡ ❛✐♠❡❞ t♦ ♣r♦✈✐❞❡ t❤❡ t❡❛❝❤❡r s✐♥❝❡ t❤❡ ❡✐❣❤t❤ ❣r❛❞❡ ❧❡✈❡❧✱ ♠❛t❤❡♠❛t✐❝❛❧ s✉♣♣♦rt ❛♥❞ t❡❛❝❤✐♥❣ s✉❣❣❡st✐♦♥s ❢♦r t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❝❧❛ssr♦♦♠ ❛❝t✐✈✐t② ❢♦r ✈❡r✐✜❝❛t✐♦♥ ❝♦♥❝❡♣ts ♦❢ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ●❡♦♠❡tr②✱ ❣❡♦♠❡tr✐❝ ✈✐s✐♦♥✱ ❡①❡r❝✐s❡ ❛♥❞ st✐♠✉❧❛t❡ st✉❞❡♥t✬s ❝r❡❛t✐✈✐t②✳

■♥ ❝♦♥s♦♥❛♥❝❡ ✇✐t❤ ❜r♦❛❞ ♦❜❥❡❝t✐✈❡s✱ ✇❛s ♥❛♠❡❞ t❤❡ ❈✐r❝❧❡ ♦❢ ◆✐♥❡✲P♦✐♥t ❛s ❝♦♥t❡♥t t♦ ❜❡ ❞✐s❝✉ss❡❞✱ t♦♣✐❝ ❞✐s❝✉ss❡❞ ❛❧♠♦st ❡①❝❧✉s✐✈❡❧② ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ❧✐t❡r❛t✉r❡ ♥♦t ♥♦r♠❛❧❧② ✉s❡❞ ❜② ❡❞✉❝❛t✐♦♥❛❧ ✐♥st✐t✉t✐♦♥s ✐♥ ❣❡♥❡r❛❧✳ ❍♦✇❡✈❡r✱ t❤✐s t♦♣✐❝ r❡✈❡❛❧s ✐ts ❝♦♥s✐❞❡r❛❜❧❡ ❝✉st♦♠❛r② ✈❛r✐❡t② ♦❢ ❣❡♦♠❡tr✐❝ ❝♦♥❝❡♣ts✱ ✇❤✐❝❤ ❝❛♥ tr❛♥s❧❛t❡ ✐♥t♦ ♣❡❞❛❣♦❣✐❝❛❧ ❛❝t✐✈✐t② q✉✐t❡ ❞✐✈❡rs❡✱ ✇✐t❤ t❤❡ ♣♦ss✐❜✐❧✐t② t♦ ❡①♣❧♦r❡ ✈❛r✐♦✉s ❛s♣❡❝ts ♦❢ t❡❛❝❤✐♥❣✳

❚♦ ♠❛t❡r✐❛❧✐③❡ t❤❡s❡ ✐❞❡❛s✱ ✇❛s ♣r♦♣♦s❡❞ ❛ st✉❞② ♦❢ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ◆✐♥❡✲P♦✐♥t ❈✐r❝❧❡ ✇❤✐❝❤✱ t❤❡r❡❢♦r❡✱ ❛r❡ ♠❛❞❡ ❞❡♠♦♥str❛t✐♦♥s ✇✐t❤ ❡♠♣❤❛s✐s ♦♥ ❛♥❣✉❧❛r ❣❡♦♠❡tr② ✭✐❞❡♥t✐✜❝❛t✐♦♥ ❛♥❞ ♠❡❛s✉r❡♠❡♥t ♦❢ ❛♥❣❧❡s✮✱ ✇❤❡r❡❛s✱ ✐♥ t❤❡ ♠❡tr✐❝ ✭♠❡❛s✉r✐♥❣ ❧❡♥❣t❤✮ ❡①❝❧✉❞❡❞ ✐❢ ❛ ❝♦♥st❛♥t ❢❡❛t✉r❡ ✐♥ tr❛❞✐t✐♦♥❛❧ ❞❡♠♦♥str❛t✐♦♥s✱ ♥❛♠❡❧② ✲ t❤❡ s✐♠✐❧❛r✐t② ♦❢ tr✐❛♥❣❧❡s✱ s✐♥❝❡ ✐t ✇♦✉❧❞ ❜❡ ❛♣♣r♦♣r✐❛t✐♥❣ ❝♦♥t❡♥t ♦✉ts✐❞❡ ♦❢ t❤❡ ❜❛s✐❝ ♣r♦♣♦s❛❧ ♦❢ t❤❡ ✐♥t❡♥❞❡❞ ❛✉❞✐❡♥❝❡✳

❋♦✉r ♥❡✇ ❢❡❛t✉r❡s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ♣❛♣❡r✱ ✐♥ t❤❡ st❛t❡♠❡♥t ♦❢ ❡①✐st❡♥❝❡ t❤❡ ◆✐♥❡✲ P♦✐♥t ❈✐r❝❧❡✱ ❝♦♥st✐t✉t❡ t❤❡ ❡ss❡♥❝❡ ♦❢ t❤✐s ✇♦r❦✿

✕ ❙t❛rt✐♥❣ ❢r♦♠ t❤❡ ❝✐r❝❧❡ ❝♦♥t❛✐♥✐♥❣ t❤❡ t❤r❡❡ ♠✐❞♣♦✐♥ts ♦❢ ❛♥ ❛❝✉t❡ tr✐❛♥❣❧❡❀

✕ ❋❛✐❧✉r❡ t♦ ✉s❡ t❤❡ ❊✉❧❡r ▲✐♥❡ ❛s ❛ ♣r❡r❡q✉✐s✐t❡ ❢♦r t❤❡ ❛♣♣r♦❛❝❤ ♦❢ t❤❡ ◆✐♥❡✲P♦✐♥t ❈✐r❝❧❡❀

✕ ❆ ✇❛② t♦ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❝✐t❡❞ ❝✐r❝❧❡ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ P♦✐♥ts ❊✉❧❡r ♦❢ ❛ tr✐❛♥❣❧❡❀

t❤❡♥ t❤❡ ❢♦✉rt❤✱

✕ ❚❤❡ ❛❜s❡♥❝❡ ♦❢ s✐♠✐❧❛r tr✐❛♥❣❧❡s ✐♥ t❤❡ ❞❡❞✉❝t✐✈❡ ♣r♦❝❡ss✳

(8)

❉❡❞✐❝❛tór✐❛

(9)

❆❣r❛❞❡❝✐♠❡♥t♦s

❊♠ ♣r✐♠❡✐ríss✐♠♦ ❧✉❣❛r✱ ✏❇❡♥❞✐t♦ s❡❥❛ ♦ ❉❡✉s ❡ P❛✐ ❞❡ ♥♦ss♦ ❙❡♥❤♦r ❏❡s✉s ❈r✐st♦✱ ♦ P❛✐ ❞❛s ♠✐s❡r✐❝ór❞✐❛s ❡ ♦ ❉❡✉s ❞❡ t♦❞❛ ❝♦♥s♦❧❛çã♦✳✑ ✭■■ ❈♦rí♥t✐♦s ✶✿✸✮ q✉❡ s❡♠♣r❡ ❛❝r❡❞✐t♦✉ ❡♠ ♠✐♠ ❡ ✏♥ã♦ ✈ê ❝♦♠♦ ✈ê ♦ ❤♦♠❡♠✱ ♣♦✐s ♦ ❤♦♠❡♠ ✈ê ♦ q✉❡ ❡stá ❞✐❛♥t❡ ❞♦s ♦❧❤♦s✱ ♣♦ré♠ ♦ ❙❡♥❤♦r ♦❧❤❛ ♣❛r❛ ♦ ❝♦r❛çã♦✳✑ ✭■ ❙❛♠✉❡❧ ✶✻✿✼❜✮✳

✏❆♦ ❘❡✐ ❞♦s sé❝✉❧♦s✱ ✐♠♦rt❛❧✱ ✐♥✈✐sí✈❡❧✱ ❛♦ ú♥✐❝♦ ❉❡✉s s❡❥❛ ❤♦♥r❛ ❡ ❣❧ór✐❛ ♣❛r❛ t♦❞♦ ♦ s❡♠♣r❡✳ ❆♠é♠✳✑ ✭■ ❚✐♠ót❡♦ ✶✿✶✼✮✳

❆♦s ❝♦❧❡❣❛s ❇r✉♥♦ ❙❛❧❣❛❞♦ ❈♦❧❡ ❡ ❲❛❣♥❡r ❏♦sé ❞♦s ❙❛♥t♦s✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❞❡ três s❡♠❛♥❛s ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✷✱ q✉❛♥❞♦ ♥♦s ❡♠♣❡♥❤á✈❛♠♦s ♣❡❧❛ q✉❛❧✐✜❝❛çã♦ ♥♦ Pr♦❢♠❛t✳ ❆♦ ú❧t✐♠♦✱ ❛✐♥❞❛ ❛❝r❡s❝❡♥t❛r✐❛✿ ❛ ♣❡rs♦♥✐✜❝❛çã♦ ❞♦ ❞✐t❛❞♦ ✕ ✏❛♠✐❣♦ s❡ ❝♦♥❤❡❝❡ ♥❛s ❤♦r❛s ❞✐❢í❝❡✐s✑✳

❆♦s ❞✐✈❡rs♦s ❝♦❧❡❣❛s ❞♦ ✏❣r✉♣♦ ❞♦ ■♣s❡♣✑✱ ♣❡❧❛s ❞✐✈❡rs❛s r❡✉♥✐õ❡s ❞❡ ❡st✉❞♦ ❡ ♣❡❧♦s ♠♦♠❡♥t♦s ❢r❛t❡r♥♦s ✐♥❡sq✉❡❝í✈❡✐s q✉❡ t✐✈❡♠♦s✱ ♦s q✉❛✐s s❛❜❡♠ ♦ q✉❡ ♠❡ s✐❣♥✐✜❝❛ ❛ ❢r❛s❡ ✏s❡♥t✐r❡✐ s❛✉❞❛❞❡s✑✿ ❢♦✐ ❣r❛t✐✜❝❛♥t❡ s❛❜❡r q✉❛♥t❛ ❣❡♥t❡ ❝❛♣❛❝✐t❛❞❛ ❤á ♥❡ss❡ ♣❛ís✦

❆♦ ♠❡✉ ❝♦❧❡❣❛ ❞❡ tr❛❜❛❧❤♦✱ ▼♦❛❝② ❱❛s❝♦♥❝❡❧♦s ❈❛❜r❛❧✿ ♣♦r s❡ ❞✐s♣♦r ❛ r❡✈✐s❛r ❛s ✈❡rsõ❡s ✜♥❛✐s ❞♦ t❡①t♦✱ ❛❜❞✐❝❛♥❞♦ ❞♦s s❡✉s ❤♦rár✐♦s ❞❡ ❧❛③❡r ❡ ❞❡s❝❛♥s♦✳

❆♦ ♦r✐❡♥t❛❞♦r✱ ❆❞r✐❛♥♦ ❘❡❣✐s ▼❡❧♦ ❘♦❞r✐❣✉❡s ❞❛ ❙✐❧✈❛✱ ♣❡❧❛ r❡s✐❣♥❛çã♦ ❛♦ s❡ ❞✐❣♥❛r ❞✐s♣❡♥s❛r ❛t❡♥çã♦ ❛♦ t❡♠❛ q✉❡ ❛♣r❡s❡♥t❡✐✳

❊①tr❛ ❯♥✐✈❡rs✐❞❛❞❡✱ t❡♥❤♦ ❞í✈✐❞❛s ♣❛r❛ ❝♦♠✿

(10)

❙✉♠ár✐♦

▲✐st❛ ❞❡ ❋✐❣✉r❛s

■◆❚❘❖❉❯➬➹❖ ♣✳ ✶✹

✶ ❆❜♦r❞❛❣❡♥s tr❛❞✐❝✐♦♥❛✐s ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ♣✳ ✶✻

✶✳✶ ✶❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✻

✶✳✶✳✶ ❈♦♠❡♥tár✐♦s à ✶❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✽

✶✳✷ ✷❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✽

✶✳✷✳✶ ✷❛ ❞❡♠♦♥str❛çã♦ ✲ ❖ ❚r✐â♥❣✉❧♦ ✏▼❡❞✐❛❧✑ ❡ ❛ ❘❡t❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ♣✳ ✶✽

✶✳✷✳✷ ❈♦♠❡♥tár✐♦s à ✷❛ ❞❡♠♦♥str❛çã♦ ✲ ❖ ❚r✐â♥❣✉❧♦ ✏▼❡❞✐❛❧✑ ❡ ❛ ❘❡t❛

❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✵ ✶✳✸ ❈♦♥s✐❞❡r❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✸

✷ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♣✳ ✷✹

✷✳✶ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ✲ ✶❛ ❞❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✹

✷✳✶✳✶ ❈♦♠❡♥tár✐♦s à ✶❛ ❞❡♠♦♥str❛çã♦ ✲ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ✳ ✳ ✳ ♣✳ ✷✼

✷✳✶✳✷ ❘❡✢❡①õ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✵ ✷✳✷ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ✲ ✷❛ ❞❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✶

✷✳✷✳✶ ❈♦♠❡♥tár✐♦s à ✷❛ ❞❡♠♦♥str❛çã♦ ✲ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ✳ ✳ ✳ ♣✳ ✸✸

✷✳✷✳✶✳✶ ❞❡s♠❡♠❜r❛♥❞♦ ❛ ✜❣✉r❛ ✷✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✹ ✷✳✷✳✶✳✷ ❞❡s♠❡♠❜r❛♥❞♦ ❛ ✜❣✉r❛ ✷✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✺ ✷✳✷✳✷ ❘❡✢❡①õ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✻

✸ Pr♦♣♦st❛s ♣✳ ✸✼

(11)

✸✳✷✳✷ ❖s P♦♥t♦s ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✻

❈♦♥❝❧✉sõ❡s ♣✳ ✺✵

❆♣ê♥❞✐❝❡ ❆ ✲✲ P♦♥t♦ ❞❡ ❡♥❝♦♥tr♦ ❞❛s ♠❡❞✐❛tr✐③❡s ♣✳ ✺✶

❆♣ê♥❞✐❝❡ ❇ ✲✲ ❚♦❞♦ tr❛♣é③✐♦ ✐♥s❝r✐t♦ ♥✉♠ ❝ír❝✉❧♦ é ✐sós❝❡❧❡s ♣✳ ✺✸

❆♣ê♥❞✐❝❡ ❈ ✲✲ Pr♦♣r✐❡❞❛❞❡s ❞❛ ▼❡❞✐❛♥❛ r❡❧❛t✐✈❛ à ❍✐♣♦t❡♥✉s❛ ♣✳ ✺✺

❆♣ê♥❞✐❝❡ ❉ ✲✲ ❖s ♣♦♥t♦s E ❡ F sã♦ ♣és ❞❛s ❛❧t✉r❛s BE ❡ CF ♣✳ ✺✼

❆♣ê♥❞✐❝❡ ❊ ✲✲ ❖ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ▲ ❡ ▼ ♣✳ ✺✾

❆♣ê♥❞✐❝❡ ❋ ✲✲ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ✕ ❛♣❧✐❝❛❞♦ ❛♦ ❚r✐â♥❣✉❧♦ ❘❡tâ♥✲

❣✉❧♦ ♣✳ ✻✶

❆♣ê♥❞✐❝❡ ● ✲✲ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ✕ ❛♣❧✐❝❛❞♦ ❛♦ ❚r✐â♥❣✉❧♦ ❖❜t✉✲

sâ♥❣✉❧♦ ♣✳ ✻✷

●✳✶ ❊♥❝♦♥tr❛♥❞♦ ♦s ♣és ❞❛s ❛❧t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✷ ●✳✷ ❊♥❝♦♥tr❛♥❞♦ ♦s P♦♥t♦s ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✺ ●✳✸ ❆ ❘❡t❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✸

❆♣ê♥❞✐❝❡ ❍ ✲✲ P♦♥t♦ ❞❡ ❡♥❝♦♥tr♦ ❞❛s ❛❧t✉r❛s ♣✳ ✼✾

❆♣ê♥❞✐❝❡ ■ ✲✲ ❛ r❡❧❛çã♦ ❡♥tr❡ ✈ért✐❝❡s ❡ ♦rt♦❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ♣✳ ✽✵

❆♣ê♥❞✐❝❡ ❏ ✲✲ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ✕ ❛♣❧✐❝❛❞♦ ❛♦ ❚r✐â♥❣✉❧♦s ❊q✉✐✲

❧át❡r♦ ♣✳ ✽✷

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▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶ ❘❡t❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✼ ✷ ❉❡♠♦♥str❛çã♦ r❡❧❛t✐✈❛ à ❘❡t❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✼ ✸ ❚r✐â♥❣✉❧♦ ♠❡❞✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✾ ✹ ❆s ♠❡❞✐❛♥❛s ❞♦ △ABC❡stã♦ ❝♦♥t✐❞❛s ♥❛s ♠❡❞✐❛♥❛s ❞♦ ABC ✳ ✳ ✳ ♣✳ ✷✶

✺ N é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ HO ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✷

✻ r = R

2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✷ ✼ ❈♦♥str✉✐♥❞♦ ✉♠ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♣❡❧♦s ♣♦♥t♦s ♠é❞✐♦s ❡ ❞❡ ❊✉❧❡r ♣✳ ✷✺ ✽ N é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ HO ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✻

✾ C′B=LM ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✼

✶✵ C′L=BM ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✼

✶✶ B′CLM é ♣❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✼

✶✷ B′CLM é r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✽

✶✸ A′BKL é r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✽

✶✹ C′AM K é r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✽

✶✺ ❈♦♥str✉çã♦ ❞♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♣♦r r❡tâ♥❣✉❧♦s ✐♥s❝r✐t♦s ✳ ✳ ✳ ✳ ♣✳ ✷✾ ✶✻ ❖ ♣é ❞❛ ❛❧t✉r❛ ♣❡rt❡♥❝❡ ❛♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✵ ✶✼ P♦♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✶ ✶✽ ❖ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✶ ✶✾ λ ♣❛ss❛ ♣❡❧♦ ♣é ❞❛ ❛❧t✉r❛ AK ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✷

✷✵ ▲ ♣❡rt❡♥❝❡ ❛λ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✷

✷✶ M é ♦ ❝❡♥tr♦ ❞❡ λ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✷

✷✷ ■♥s❝r✐t✐❜✐❧✐❞❛❞❡ ❞❡KF LE ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✹

✷✸ F L é ❜❛s❡ ♠é❞✐❛ ❞❡ AHC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✹

✷✹ EF L é ✉♠ â♥❣✉❧♦ r❡t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✹

✷✺ KF LE é ✉♠ q✉❛❞r✐❧át❡r♦ ✐♥s❝r✐tí✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✹

✷✻ △LF E é r❡tâ♥❣✉❧♦ ❡♠ F ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✺

(13)

✷✽ ❚r✐â♥❣✉❧♦ ❛❝✉tâ♥❣✉❧♦ ❡ s❡✉s ♣♦♥t♦s ♠é❞✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✽ ✷✾ ❊♥❝♦♥tr❛♥❞♦ ♦ ❝❡♥tr♦ ❞♦ ❝ír❝✉❧♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛s ♠❡❞✐❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✾ ✸✵ ◗✉❛❞r✐❧át❡r♦ ✐♥s❝r✐t♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦rA′B C D ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✾

✸✶ ❊st✉❞❛♥❞♦ ❛ ♥❛t✉r❡③❛ ❞♦ s❡❣♠❡♥t♦AD ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✵

✸✷ AD é ✉♠❛ ❛❧t✉r❛ ❞♦ △ABC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✶

✸✸ ❚r✐â♥❣✉❧♦ Órt✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✷ ✸✹ ❖s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦△BEA ❡ ❞♦ △CF Asã♦ ❝♦♥❣r✉❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✷

✸✺ ■♥s❝r✐t✐❜✐❧✐❞❛❞❡ ❞♦ q✉❛❞r✐❧át❡r♦BF HD ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✸

✸✻ ❊q✉✐✈❛❧ê♥❝✐❛ ❞♦s â♥❣✉❧♦s FBHˆ ❡FDHˆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✸

✸✼ ■♥s❝r✐t✐❜✐❧✐❞❛❞❡ ❞♦ q✉❛❞r✐❧át❡r♦CEHD ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✹

✸✽ ❊q✉✐✈❛❧ê♥❝✐❛ ❞♦s â♥❣✉❧♦s ECHˆ ❡ EDHˆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✹

✸✾ ❆❧t✉r❛AD é ❜✐ss❡tr✐③ ❞♦ ✈ért✐❝❡ D ❞♦ ❚r✐â♥❣✉❧♦ Órt✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✺

✹✵ ❆s ❛❧t✉r❛s ❞♦△ABC sã♦ ❜✐ss❡tr✐③❡s ❞♦ s❡✉ ❚r✐â♥❣✉❧♦ Órt✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✺

✹✶ EF Kˆ ❡ EDKˆ tê♠ ♠❡s♠❛ ♠❡❞✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✻

✹✷ ▼❛r❝❛♥❞♦ ♦s â♥❣✉❧♦s ❝♦♠ ♠❡❞✐❞❛γ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✼

✹✸ DHCˆ ❡ FHAˆ tê♠ ♠❡s♠❛ ♠❡❞✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✽

✹✹ ✉♥✐❝✐❞❛❞❡ ❞♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s ♠❡❞✐❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✶ ✹✺ ❖s â♥❣✉❧♦s ❛❞❥❛❝❡♥t❡s ❛♦s ❧❛❞♦s ♣❛r❛❧❡❧♦s sã♦ s✉♣❧❡♠❡♥t❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✸ ✹✻ ❙❡ ✉♠ q✉❛❞r✐❧át❡r♦ é ✐♥s❝r✐tí✈❡❧✱ ♦s â♥❣✉❧♦s ♦♣♦st♦s sã♦ s✉♣❧❡♠❡♥t❛r❡s ✳ ♣✳ ✺✹ ✹✼ ▼❡❞✐❛♥❛ r❡❧❛t✐✈❛ à ❤✐♣♦t❡♥✉s❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✺ ✹✽ ▼❡❞✐❛♥❛ r❡❧❛t✐✈❛ à ❤✐♣♦t❡♥✉s❛ ❞❡✈✐❞♦ à ✐♥s❝r✐t✐❜✐❧✐❞❛❞❡ ♥♦ ❝ír❝✉❧♦ ✳ ✳ ✳ ♣✳ ✺✻ ✹✾ ◗✉❛❞r✐❧át❡r♦ ✐♥s❝r✐t♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦rA′B C F ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✼

✺✵ ◗✉❛❞r✐❧át❡r♦ ✐♥s❝r✐t♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦rA′B E C✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✽

✺✶ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ ▲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✾ ✺✷ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ ▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✵ ✺✸ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♥✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✶ ✺✹ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♥♦ ❚r✐â♥❣✉❧♦ ❖❜t✉sâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✷ ✺✺ ❖ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♣❛ss❛ ♣❡❧♦ ♣é ❞❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛♦ ❧❛❞♦BC ♣✳ ✻✸

✺✻ ❖ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♣❛ss❛ ♣❡❧♦ ♣é ❞❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛♦ ❧❛❞♦AB ♣✳ ✻✸

✺✼ ❖ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♣❛ss❛ ♣❡❧♦ ♣é ❞❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛♦ ❧❛❞♦AC ♣✳ ✻✹

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✻✵ ▼♦str❛♥❞♦ q✉❡M é ♣♦♥t♦ ♠é❞✐♦ ❞❡HC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✼

✻✶ EM é ♠❡❞✐❛♥❛ r❡❧❛t✐✈❛ à ❤✐♣♦t❡♥✉s❛HC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✽

✻✷ ▼♦str❛♥❞♦ q✉❡K é ♣♦♥t♦ ♠é❞✐♦ ❞❡ AH ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✻✾

✻✸ F K é ♠❡❞✐❛♥❛ r❡❧❛t✐✈❛ à ❤✐♣♦t❡♥✉s❛AH ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✵

✻✹ ▼♦str❛♥❞♦ q✉❡L é ♣♦♥t♦ ♠é❞✐♦ ❞❡ BH ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✶

✻✺ F K é ♠❡❞✐❛♥❛ r❡❧❛t✐✈❛ à ❤✐♣♦t❡♥✉s❛BH ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✷

✻✻ ❖ ❝❡♥tr♦ ✭N✮ ❞♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡s P♦♥t♦s ♣❡rt❡♥❝❡ à ❘❡t❛ ❞❡ ❊✉❧❡r ✳ ♣✳ ✼✸

✻✼ ▼♦str❛♥❞♦ q✉❡M N//OC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✹

✻✽ ❖s â♥❣✉❧♦sCAOˆ ❡ ACOˆ sã♦ ❝♦♥❣r✉❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✺

✻✾ M N é ❜❛s❡ ♠é❞✐❛ ❞♦ △OHC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✻

✼✵ ▼♦str❛♥❞♦ q✉❡N é ♣♦♥t♦ ♠é❞✐♦ ❞❡ HO ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✼

✼✶ ❯♥✐❝✐❞❛❞❡ ❞♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s ❛❧t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✼✾ ✼✷ ❆s ❛❧t✉r❛s ❞♦△ABC ❢♦r♠❛♠ três tr✐â♥❣✉❧♦s ♦❜t✉sâ♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✽✵

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✶✹

■◆❚❘❖❉❯➬➹❖

❉❡s❡❥❛✲s❡ ♣r♦♠♦✈❡r ❛ ✐♥t❡❣r❛çã♦ ❞♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ❝♦♠♦ ❛t✐✈✐❞❛❞❡ ♣❡❞❛❣ó✲ ❣✐❝❛ ✈✐á✈❡❧ ❛ ♣❛rt✐r ❞♦ ♦✐t❛✈♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❛ q✉❛❧ ♣♦ss✐❜✐❧✐t❡ ❛♦s ♣r♦❢❡ss♦r❡s ❞❡ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ r❡✈✐s❛r ❡ ❡①❡r❝✐t❛r✱ ♥✉♠❛ ú♥✐❝❛ ❛t✐✈✐❞❛❞❡✱ ❞✐✈❡rs♦s ❝♦♥❝❡✐t♦s ♠❛✲ t❡♠át✐❝♦s ❛té ❛q✉❡❧❡ ♣❡rí♦❞♦ ❞❡ ❡st✉❞♦s✱ t❛✐s ❝♦♠♦✿

• ❚r✐â♥❣✉❧♦s✿ â♥❣✉❧♦s ✭✐♥t❡r♥♦s ❡ ❡①t❡r♥♦s✮ ❡ ♣r♦♣r✐❡❞❛❞❡s ♠étr✐❝❛s❀

• ❈ír❝✉❧♦✿ â♥❣✉❧♦s ✭✐♥s❝r✐t♦✱ ❡①❝ê♥tr✐❝♦ ✐♥t❡r♥♦ ❡ ❡①❝ê♥tr✐❝♦ ❡①t❡r♥♦✮ ❡ ❛r❝♦ ❝❛♣❛③❀

• ◗✉❛❞r✐❧át❡r♦s ♥♦tá✈❡✐s✿ ♣r♦♣r✐❡❞❛❞❡s ❛♥❣✉❧❛r❡s ❡ ♠étr✐❝❛s✳

❉❛❞♦ ✉♠ tr✐â♥❣✉❧♦✱ ♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ❝♦♥s✐st❡ ♥♦ ❝ír❝✉❧♦ q✉❡ ❝♦♥té♠ ♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s✱ ♦s ♣és ❞❛s ❛❧t✉r❛s ❡ ♦s ♣♦♥t♦s ♠é❞✐♦s ✭P♦♥t♦s ❞❡ ❊✉❧❡r✮ ❞♦s s❡❣♠❡♥t♦s ❝♦♠♣r❡❡♥❞✐❞♦s ❡♥tr❡ ♦ ♦rt♦❝❡♥tr♦ ❡ ♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦✳

❆ ♣r✐♥❝✐♣❛❧ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ✐♥♦✈❛çã♦ ❝♦♥st✐t✉✐✲s❡ ❡♠ s❡ ♣❛rt✐r✱ à ♣r✐♦r✐✱ ❞❡ ✉♠ ❝ír❝✉❧♦ q✉❡ ♣❛ss❡ ♣❡❧♦s ♣♦♥t♦s ♠é❞✐♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ❛♦ ✐♥✈és ❞❡ ❝♦♥str✉✐r ♦ r❡❢❡r✐❞♦ ❝ír❝✉❧♦ só ♥♦ ✜♥❛❧ ❞❛ ❞❡♠♦♥str❛çã♦✳ ◆♦ ❝❛s♦✱ ♣r♦❝✉r❛r✲s❡✲á ❞❡♠♦♥str❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❞❡ss❡ ❝ír❝✉❧♦ ❝♦♠ ♦s ❧❛❞♦s ❡ ❛❧t✉r❛s ❞♦ tr✐â♥❣✉❧♦✱ ❞♦♥❞❡ ✈❡r✲s❡✲á q✉❡ ❡ss❛s ✐♥t❡rs❡çõ❡s sã♦ ❡♠ ♥ú♠❡r♦ ❞❡ ♥♦✈❡ ✕ ♣♦r ✐ss♦✱ ♦ ♥♦♠❡ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s✳

P♦❞❡r✲s❡✲✐❛ ❞❡s❡♥✈♦❧✈❡r t❛❧ ❛t✐✈✐❞❛❞❡ ❝♦♠ s✉♣♦rt❡ ✐♥❢♦r♠át✐❝♦✱ ♣♦ré♠✱ ❛ ✐♥t❡♥çã♦ ❞❡st❛ ♣r♦♣♦st❛ é ❢❛③❡r ✉s♦ ❞❡ ✐♥str✉♠❡♥t♦s ❝♦♥❝r❡t♦s ♣❛r❛ ♦ ❞❡s❡♥❤♦ ❣❡♦♠étr✐❝♦ ✕ ❡sq✉❛❞r♦✱ ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ✕ ♣❡❧♦s q✉❛✐s s❡ ❞❡s❡❥❛ q✉❡ ♦ ❛❧✉♥♦ ♣❡r❝❡❜❛ ❛s tr❛♥s❢♦r♠❛çõ❡s✱ ❜❡♠ ❝♦♠♦ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❣❡♦♠étr✐❝❛s✳

❖ tr❛❜❛❧❤♦ s❡❣✉❡ ♦ s❡❣✉✐♥t❡ r♦t❡✐r♦✿

✕ P❛r❛ ❝❛❞❛ ✉♠ ❞♦s ❞♦✐s ♣r✐♠❡✐r♦s ❝❛♣ít✉❧♦s✱ ❢♦r❛♠ s❡❧❡❝✐♦♥❛❞♦s ❞✉❛s ❞❡♠♦♥str❛çõ❡s ✭❝♦♥❢♦r♠❡ ❝♦♥st❛♠ ♥❛s r❡❢❡rê♥❝✐❛s✮✱ s❡❣✉✐❞❛s ❞❡ ❝♦♠❡♥tár✐♦s✱ ❛❧é♠ ❞❡ ❝♦♥s✐❞❡r❛çõ❡s ❛♦ ✜♥❛❧ ❞❡ ❛♠❜♦s ♦s ❝❛♣ít✉❧♦s❀

✕ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ❢❛r✲s❡✲á ❛ ♣r♦♣♦st❛ ♣r❡❝í♣✉❛ ❞❡st❡ tr❛❜❛❧❤♦❀ ❡✱

✕ ❊♠ ❛❧❣✉♥s ♠♦♠❡♥t♦s✱ ❢❛r✲s❡✲á ❛❧✉sõ❡s ❛ ❞❡t❛❧❤❡s ❝♦♥st❛♥t❡s ♥♦s ❛♣ê♥❞✐❝❡s✳

➱ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ t❡r ❡♠ ♠❡♥t❡ q✉❡ ♦ ♣ú❜❧✐❝♦ ♣r❡❢❡r❡♥❝✐❛❧ q✉❡ s❡ ♣r❡t❡♥❞❡ ❛❧❝❛♥ç❛r sã♦ ♦s ❛❧✉♥♦s ❞♦ ♦✐t❛✈♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✳

❉❡✈✐❞♦ à ❡s❝❛ss❡③ ❞❡ r❡❢❡rê♥❝✐❛s✱ ♦s ❝❛♣ít✉❧♦s ✶ ❡ ✷ ❢♦r❛♠ ✐♥❝❧✉í❞♦s ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ s❡r❡♠ ♠❛t❡r✐❛✐s ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ❝♦♥s✉❧t❛✱ t❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ s❡✉s ❝♦♥t❡ú❞♦s ♥ã♦ sã♦ ❞❡ ❡♥t❡♥❞✐♠❡♥t♦ ✐♠❡❞✐❛t♦ ♣❛r❛ ♦s ❛❧✉♥♦s s✉♣r❛❝✐t❛❞♦s✳

❆❧❣✉♠❛s ❝♦♥✈❡♥çõ❡s ♣❛r❛ ♥♦t❛çõ❡s ♠❛t❡♠át✐❝❛s✿

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■◆❚❘❖❉❯➬➹❖ ✶✺

ABCˆ →â♥❣✉❧♦ ❝✉❥♦ ✈ért✐❝❡ é ♦ ♣♦♥t♦ B ❀ 6 ABC →♠❡❞✐❞❛ ❞❡ ABCˆ ✳

d

ABC →t❛♥t♦ r❡♣r❡s❡♥t❛rá ♦ ❛r❝♦✱ ❝♦♠♦ ❛ ♠❡❞✐❞❛ ❞♦ ❛r❝♦✳

❖s tr✐â♥❣✉❧♦s q✉❡ ❢♦r❛♠ ❝♦♥s✐❞❡r❛❞♦s ♥♦s três ❝❛♣ít✉❧♦s sã♦ ❛❝✉tâ♥❣✉❧♦s ❡ ❡s❝❛✲ ❧❡♥♦s✶✳ ❉♦ ❝♦♥trár✐♦✱ ❛❧❣✉♥s ❞♦s ♣♦♥t♦s ♥♦tá✈❡✐s✱ ❝♦♠ ♦s q✉❛✐s tr❛❜❛❧❤❛r❡♠♦s✱ s❡rã♦

❝♦✐♥❝✐❞❡♥t❡s✱ ❡ ♥ã♦ t❡rí❛♠♦s ♦s ✏♥♦✈❡ ♣♦♥t♦s✑ ❞✐st✐♥t♦s ✕ ♣❛r❛ ♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ ♣♦♥t♦s✳

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✶✻

✶ ❆❜♦r❞❛❣❡♥s tr❛❞✐❝✐♦♥❛✐s ❞❛ ❘❡t❛

❞❡ ❊✉❧❡r

▼♦str❛r✲s❡✲á ❛ s❡❣✉✐r q✉❡✱ ❡♠ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✱ ♦ ♦rt♦❝❡♥tr♦✱ ♦ ❜❛r✐❝❡♥tr♦ ❡ ♦ ❝✐r❝✉♥❝❡♥tr♦ sã♦ ❝♦❧✐♥❡❛r❡s✳ ❆ r❡t❛ q✉❡ ❝♦♥té♠ ❡ss❡s três ♣♦♥t♦s é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❘❡t❛ ❞❡ ❊✉❧❡r✳

❉❛s s❡✐s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✱ ❛♣❡♥❛s ❬❄❪ ♥ã♦ ♣r✐♠❛ ♣♦r ❛♣r❡s❡♥t❛r ❛ ❘❡t❛ ❞❡ ❊✉❧❡r ❝♦♠♦ ♣ré✲r❡q✉✐s✐t♦ ❛♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s✳ P❛r❡❝❡ q✉❡✱ ♣❡❧♦ ❢❛t♦ ❞♦ ♦❜❥❡t♦ ❡♠ q✉❡stã♦ s❡r ✉♠ ❝ír❝✉❧♦✱ ❛s ❛❜♦r❞❛❣❡♥s ♠❛✐s ❝♦♠✉♥s tê♠ ❝♦♠♦ ❡✐①♦ ♣r✐♥❝✐♣❛❧ ❞❡ r❛❝✐♦❝í♥✐♦ ❜✉s❝❛r ♣r❡✈✐❛♠❡♥t❡ ♦ ❝❡♥tr♦ ❞♦ s✉♣♦st♦ ❝ír❝✉❧♦✳ ◆❛ ✈❡r❞❛❞❡✱ ✐ss♦ s❡ ❞❡✈❡ à ♣❛rt✐❝✉❧❛r✐❞❛❞❡ ❞♦ ❝❡♥tr♦ ❞♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s ♣❡rt❡♥❝❡r ❛ ❡ss❛ r❡t❛✱ ❛❧é♠ ❞❡ ❝♦✐♥❝✐❞✐r ❝♦♠ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ ♦ ♦rt♦❝❡♥tr♦ ❡ ❝✐r❝✉♥❝❡♥tr♦✳

◆♦ ❡♥t❛♥t♦✱ ❛ ❛❜♦r❞❛❣❡♠ tr❛❞✐❝✐♦♥❛❧ ❞❡ss❛ r❡t❛✱ t❛❧✈❡③ ♣❛r❛ s❡r ♠❛✐s ❛tr❛t✐✈❛✱ s✉✲ ❣❡r❡ ♦❜s❡r✈❛r q✉❡ três ♣♦♥t♦s ♥♦tá✈❡✐s ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ❛ s❛❜❡r✱ ♦rt♦❝❡♥tr♦✱ ❜❛r✐❝❡♥tr♦ ❡ ❝✐r❝✉♥❝❡♥tr♦ sã♦ ❝♦❧✐♥❡❛r❡s✳ ◆♦ ❝❛s♦✱ ❛ r❡t❛ q✉❡ ❝♦♥té♠ ❡ss❡s três ♣♦♥t♦s é ❞❡♥♦♠✐✲ ♥❛❞❛ ❞❡ ❘❡t❛ ❞❡ ❊✉❧❡r✱ ♠✉✐t♦ ❡♠❜♦r❛✱ ♥❛ ♣rát✐❝❛✱ ♦s ❡st✉❞♦s s❡ ❝♦♥❝❡♥tr❡♠ ♥♦ s❡❣♠❡♥t♦ ❞❡❧✐♠✐t❛❞♦ ♣❡❧♦ ♦rt♦❝❡♥tr♦ ❡ ❝✐r❝✉♥❝❡♥tr♦✳

❖ ❝♦♥t❡ú❞♦ ❞❡st❡ ❝❛♣ít✉❧♦ ♥ã♦ é ♣❛r❛ ❛♣♦✐❛r ♦✉ r❡❢♦rç❛r ❛ ♥❡❝❡ss✐❞❛❞❡ ❞✐❞át✐❝❛ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ❛♥t❡❝❡❞❡r ❛ ❛❜♦r❞❛❣❡♠ ❞♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s✳ ❚❛♥t♦ é ✈❡r❞❛❞❡ q✉❡✱ ♥♦ ❝❛♣ít✉❧♦ ✸ ✭❝♦rr❡s♣♦♥❞❡♥t❡ à ♣r♦♣♦st❛ ❞❡st❡ tr❛❜❛❧❤♦✮✱ s❡q✉❡r s❡ ❢❛③ ♠❡♥çã♦ ❛♦ ❜❛r✐❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦✳ ❖ ✐♥t✉✐t♦ é ❡①♣♦r ❛s ❧✐♥❤❛s ♠❛✐s tr❛❞✐❝✐♦♥❛✐s ❞❡ ❛♣r❡s❡♥t❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r✱ ❜❡♠ ❝♦♠♦

✶✳✶ ✶

❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r

❖ ❡♥✉♥❝✐❛❞♦ ❞♦ ❚❡♦r❡♠❛ ❛ s❡❣✉✐r✱ ❜❡♠ ❝♦♠♦ s✉❛ ❞❡♠♦♥str❛çã♦✱ ❡♥❝♦♥tr❛✲s❡ ❡♠ ❬❄❪✱ ♣á❣✐♥❛ ✶✾✻✳

❚❡♦r❡♠❛ ✶✳✶✳✶✳ ❊♠ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✱ ♦ ❖rt♦❝❡♥tr♦H✱ ♦ ❝❡♥tr♦G❡ ♦ ❈✐r❝✉♥❝❡♥tr♦ O ❡stã♦ s♦❜r❡ ✉♠❛ ♠❡s♠❛ ❧✐♥❤❛ r❡t❛ ✭❝♦♥❤❡❝✐❞❛ ♣♦r ❘❡t❛ ❞❡ ❊✉❧❡r✮ ❡ ❛ ❞✐stâ♥❝✐❛ ❞♦

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✶✳✶ ✶❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ✶✼

❋✐❣✉r❛ ✶✿ ❘❡t❛ ❞❡ ❊✉❧❡r

❙❡❥❛♠✱ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ ✷✱ ♦ tr✐â♥❣✉❧♦ ABC✱ ✉♠❛ ❞❡ s✉❛s ♠❡❞✐❛♥❛s✱ ♣♦r ❡①❡♠♣❧♦ AA′✱ r❡❧❛t✐✈❛ ❛♦ ❧❛❞♦ BC✱ ♦ ♣♦♥t♦ G s♦❜r❡ ❡❧❛ ❝♦♠AG = 2·GA′ ❡ ♦ ❝✐r❝✉♥❝❡♥tr♦ O

❧♦❝❛❧✐③❛❞♦ s♦❜r❡ ❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛BC q✉❡ ♣❛ss❛ ♣♦r A′

❋✐❣✉r❛ ✷✿ ❉❡♠♦♥str❛çã♦ r❡❧❛t✐✈❛ à ❘❡t❛ ❞❡ ❊✉❧❡r

❚r❛❝❡ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦rO ❡G❡✱ s♦❜r❡ ❡❧❛✱ ♥❛ s❡♠✐rr❡t❛ ♦♣♦st❛ à ❞♦ s❡❣♠❡♥t♦GO✱

t♦♠❡ ♦ ♣♦♥t♦ ❨ t❛❧ q✉❡ GY = 2·GO

❖s â♥❣✉❧♦s OGAˆ ′ AGYˆ sã♦ ❝♦♥❣r✉❡♥t❡s✱ ♣♦r s❡r❡♠ ♦♣♦st♦s ♣❡❧♦ ✈ért✐❝❡✳ ❈♦♠♦ ♦s

s❡❣♠❡♥t♦s AG ❡ GY sã♦ ♣r♦♣♦r❝✐♦♥❛✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛♦s s❡❣♠❡♥t♦s A′G OG✱ ♦s

tr✐â♥❣✉❧♦s AGY ❡A′GO sã♦ s❡♠❡❧❤❛♥t❡s✳ ▲♦❣♦✱ ❛s r❡t❛s AY AO sã♦ ♣❛r❛❧❡❧❛s✳ ❈♦♠♦

A′O é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ BCAY t❛♠❜é♠ ♦ é✳ ▼❛s ❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛BC tr❛ç❛❞❛ ❛ ♣❛rt✐r

❞❡Aé ❛ r❡t❛ q✉❡ ❝♦♥té♠ ❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ r❡❧❛t✐✈❛ àq✉❡❧❡ ❧❛❞♦✳ P♦rt❛♥t♦✱ ♦ ♣♦♥t♦ Y

❡stá s♦❜r❡ ❛ r❡t❛ ❞❡ ✉♠❛ ❞❛s ❛❧t✉r❛s ❞♦ tr✐â♥❣✉❧♦✳ P♦r r❛❝✐♦❝í♥✐♦ ❛♥á❧♦❣♦✱ ♣r♦✈❛✲s❡ q✉❡ ❨ ❡stá s♦❜r❡ ❛s r❡t❛s ❞❛s ❞✉❛s ♦✉tr❛s ❛❧t✉r❛s✳ ▲♦❣♦✱Y é ♣♦♥t♦ ❞❡ ❡♥❝♦♥tr♦ ❞❛q✉❡❧❛s três

r❡t❛s✱ ♦✉ s❡❥❛✱ Y é ❡①❛t❛♠❡♥t❡ ♦ ♦rt♦❝❡♥tr♦ ❍✳ ❊ ❝♦♠♦✱ ♣♦r ❝♦♥str✉çã♦✱ Y ❡stá s♦❜r❡ ❛

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✶✳✷ ✷❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ✶✽

✶✳✶✳✶ ❈♦♠❡♥tár✐♦s à ✶

❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r

P❛r❛ ✉♠ ❛❧✉♥♦ ❞♦ ♦✐t❛✈♦ ❛♥♦✱ é ♣♦✉❝♦ ✐♥t✉✐t✐✈♦ ❛s♣❡❝t♦s ❞♦ t✐♣♦✿ ✶◦✮ ◗✉❛❧ ❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ❝r✐❛r ♦ ♣♦♥t♦ ❨ ❄

✷◦✮P♦r q✉❡ GY ♥ã♦ ♣♦❞❡r✐❛ s❡r t❛❧ q✉❡ GY = 3·GO✱ ♣♦r ❡①❡♠♣❧♦❄

◆❛ ✈❡r❞❛❞❡✱ ♦ ✐♥t✉✐t♦ é q✉❡ s❡ ❞❡s❡❥❛ ♣r♦✈❛r q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞♦ ❇❛r✐❝❡♥tr♦ ❛♦ ❖rt♦✲ ❝❡♥tr♦ é ♦ ❞♦❜r♦ ❞❛ ❞✐stâ♥❝✐❛ ❞♦ ❇❛r✐❝❡♥tr♦ ❛♦ ❈✐r❝✉♥❝❡♥tr♦✳ ❈♦♥t✉❞♦✱ ❛♦ s❡ ❞✐③❡r ✏t♦♠❡ ♦ ♣♦♥t♦ Y t❛❧ q✉❡ GY = 2·GO✑✱ ❡stá✲s❡ ❡❢❡t✉❛♥❞♦ ✉♠ t✐♣♦ ❞❡ ❝♦♥str✉çã♦ ♣♦✉❝♦ ♥❛t✉r❛❧

♣❛r❛ ✉♠ ❛❧✉♥♦ ❞❡ss❡ ♥í✈❡❧ ❡s❝♦❧❛r✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ❝♦♥t❡ú❞♦ s❡♠❡❧❤❛♥ç❛ ♥ã♦ s❡r ❞♦ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦✳

❖✉tr♦ ❞❡t❛❧❤❡ é✿

✏❊ ❝♦♠♦✱ ♣♦r ❝♦♥str✉çã♦✱ ❨ ❡stá s♦❜r❡ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ❜❛r✐❝❡♥tr♦ ❡ ♣❡❧♦ ❝✐r✲ ❝✉♥❝❡♥tr♦✱ ❡♥tã♦ ❛q✉❡❧❡s três ♣♦♥t♦s ❡stã♦ s♦❜r❡ ✉♠❛ ♠❡s♠❛ r❡t❛✳ P❡❧❛ s❡♠❡❧❤❛♥ç❛ ❞♦s tr✐â♥❣✉❧♦s✱ HG= 2·GO ❝✳q✳❞✑

❙❡r✐❛ ♠❡❧❤♦r ❝♦♥❝❧✉✐r q✉❡✿

❝♦♠♦ Y =H ❡✱ ♣♦r ❝♦♥str✉çã♦✱ GY = 2·GO ✭✳✳✳✮ ❡♥tã♦ GH = 2·GO

❆ss✐♠✱ ♥ã♦ s❡ ♣r❡❝✐s❛r✐❛ ❡✈♦❝❛r ❛ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s ♥♦ ✜♥❛❧ ❞❛ ❞❡♠♦♥str❛çã♦✳

✶✳✷ ✷

❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r

❖ t❡①t♦ ❛ s❡❣✉✐r✱ r❡t✐r❛❞♦ ❞❡ ❬❄❪✱ ♣á❣✐♥❛s ✶✽✲✶✾✱ ❢♦✐ ✉♠❛ ✐♥s♣✐r❛çã♦ ♣r♦♣✉❧s♦r❛ ♣❛r❛ ♦ t❡♠❛ ♣r✐♥❝✐♣❛❧✳ ❉❛❞❛ s✉❛ r❡❧❡✈â♥❝✐❛✱ ❛❝❤♦✉✲s❡ ♣♦r ❜❡♠ ✐♥❝❧✉í✲❧♦✱ ❛♦ ♠❡s♠♦ t❡♠♣♦ q✉❡✱ ❡s♣❡r❛✲s❡✱ s❡❥❛ ❡♥r✐q✉❡❝❡❞♦r ♣❛r❛ ♦ ❧❡✐t♦r✳

❆ r❡❢❡rê♥❝✐❛ é ❡①tr❡♠❛♠❡♥t❡ r❡❝♦♠❡♥❞á✈❡❧ ♣❛r❛ ♦s q✉❡ ❞❡s❡❥❛♠ ✏✐r ✉♠ ♣♦✉❝♦ ♠❛✐s ❛❧é♠✑ ❞❛ ●❡♦♠❡tr✐❛ tr❛❞✐❝✐♦♥❛❧ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳ ■♥❝❧✉s✐✈❡✱ ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ❡①✲ ♣♦♠♦s s✉❛ ❜♦❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦ ❈ír❝✉❧♦ ❞♦s ◆♦✈❡ P♦♥t♦s✳ P❛r❛ ♦ ♥í✈❡❧ ❞❡ss❛ ♦❜r❛✱ s✉❛s ❡①♣♦s✐çõ❡s ❡ ❞❡♠♦♥str❛çõ❡s sã♦ ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡ ❞✐❞át✐❝❛s✳

◆♦ q✉❡ t❛♥❣❡ ❛ ❡st❡ tr❛❜❛❧❤♦✱ ❛ ú♥✐❝❛ r❡ss❛❧✈❛ é✱ ♥♦✈❛♠❡♥t❡✱ s♦❜r❡ ❛ ❧✐♠✐t❛çã♦ t✐♣♦✲ ❣rá✜❝❛ ✕ ❛❧é♠ ❞❛ ✜❣✉r❛ ✶✸ s❡r ❛ ú♥✐❝❛ ❛♣r❡s❡♥t❛❞❛ ♣❛r❛ ❡st❛ ❞❡♠♦♥str❛çã♦✱ ❡ ❡♠ ❡s❝❛❧❛ ♠❡♥♦r q✉❡ ♦ ❡①♣♦st♦ ❛ s❡❣✉✐r✱ ♦ ❞❡s❡♥❤♦ ♦r✐❣✐♥❛❧ é ❞❡s♣r♦✈✐❞♦ ❞❡ ❝♦r❡s✳

✶✳✷✳✶ ✷

❞❡♠♦♥str❛çã♦ ✲ ❖ ❚r✐â♥❣✉❧♦ ✏▼❡❞✐❛❧✑ ❡ ❛ ❘❡t❛ ❞❡ ❊✉❧❡r

❖ t❡①t♦ ❛ s❡❣✉✐r✱ ❜❡♠ ❝♦♠♦ s✉❛ ❞❡♠♦♥str❛çã♦✱ ❡♥❝♦♥tr❛✲s❡ ❡♠ ❬❄❪✱ ♣á❣✐♥❛ ✶✽✳

❏✉♥t❛♥❞♦✲s❡ ♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❞❛❞♦✱ ♦ tr✐â♥❣✉❧♦ ❢♦r♠❛❞♦ s❡rá ❝❤❛♠❛❞♦ ❞❡ tr✐â♥❣✉❧♦ ♠❡❞✐❛❧✳ ◆❛ ✜❣✉r❛ ✸✱△ABCé ✉♠ tr✐â♥❣✉❧♦ ♠❡❞✐❛❧ ❞♦ABC

❋♦r❛♠ tr❛ç❛❞❛s ❞✉❛s ♠❡❞✐❛♥❛s (AA′BB)✱ ❛s q✉❛✐s s❡ ❡♥❝♦♥tr❛♠ ❡♠ G✱ ❞✉❛s ❛❧t✉r❛s

(20)

✶✳✷ ✷❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ✶✾

❋✐❣✉r❛ ✸✿ ❚r✐â♥❣✉❧♦ ♠❡❞✐❛❧

❉❡ ❛♥t❡♠ã♦✱ △ABCt❡♠ ♦s ❧❛❞♦s ♣❛r❛❧❡❧♦s ❛♦s ❞♦ ABC✳ ▲♦❣♦✱ sã♦ s❡♠❡❧❤❛♥t❡s✳

C′B= 1

2 ·BC✳ ❊♠ s❡❣✉✐❞❛✱ ✈ê✲s❡ q✉❡ AC

ABé ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ ♦♥❞❡ ♦ ♣♦♥t♦ ❞❡

✐♥t❡rs❡çã♦ ❡♥tr❡AA✬ ❡B′Cé ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦s ❞♦✐s s❡❣♠❡♥t♦s✳ P♦rt❛♥t♦✱ ❛s ♠❡❞✐❛♥❛s ❞❡ABC❡stã♦ ❝♦♥t✐❞❛s ♥❛s ♠❡❞✐❛♥❛s ❞♦ABC✱ ♦ q✉❡ ❛❝❛rr❡t❛ q✉❡ ❛♠❜♦s ♦s tr✐â♥❣✉❧♦s

tê♠ ♦ ♠❡s♠♦ ❜❛r✐❝❡♥tr♦✱G✳ P♦r ❢❛❧❛r ♥✐ss♦✱ ♦ ♣♦♥t♦ ♠é❞✐♦P ❞❡B′Cé t❛♠❜é♠ ♦ ♣♦♥t♦

♠é❞✐♦ ❞❡ AA′✳ ❆❣♦r❛✱ ❛s ❛❧t✉r❛s ❞♦ ABCsã♦ ♠❡❞✐❛tr✐③❡s ❞♦s ❧❛❞♦s AB BC ❞♦ABC✳ ❈♦♥❝❧✉✐✲s❡ q✉❡O✱ ♦ ♦rt♦❝❡♥tr♦ ❞♦△ABC✱ é✱ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ♦ ❝✐r❝✉♥❝❡♥tr♦

❞♦△ABC✳ ❙❡♥❞♦H ♦ ♦rt♦❝❡♥tr♦ ❞♦ △ABC✱ ❡✱ O ♦ ♦rt♦❝❡♥tr♦ ❞♦ tr✐â♥❣✉❧♦ s❡♠❡❧❤❛♥t❡ △ABC AH = 2·OA′✳ ❉♦ ❚❡♦r❡♠❛ ✶✳✸✷✶✱ r❡❝♦r❞❛♠♦s q✉❡ AG= 2·AG✳ ❋✐♥❛❧♠❡♥t❡✱

❞❡s❞❡ q✉❡ AD ❡ A′O sã♦ ❛♠❜♦s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛♦ ❧❛❞♦ BC✱ ❡❧❡s sã♦ ♣❛r❛❧❡❧♦s ❡♥tr❡ s✐✳

❉❛í

6 HAG=6 OAG ,HAG∼ △OAG

e

6 AGH =6 AGO

■st♦ ♠♦str❛ q✉❡ ♦s ♣♦♥t♦s O✱G✱ H sã♦ ❝♦❧✐♥❡❛r❡s✱ ❡ HG= 2·GO

❚❡♦r❡♠❛ ✶✳✷✳✶✳ ❖ ♦rt♦❝❡♥tr♦✱ ❜❛r✐❝❡♥tr♦ ❡ ❝✐r❝✉♥❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ sã♦ ❝♦❧✐♥❡❛r❡s✳ ❖ ❜❛r✐❝❡♥tr♦ ❞✐✈✐❞❡ ❛ ❞✐stâ♥❝✐❛ ❞♦ ♦rt♦❝❡♥tr♦ ❛♦ ❝✐r❝✉♥❝❡♥tr♦ ♥❛ r❛③ã♦ ✷✿✶✳

(21)

✶✳✷ ✷❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ✷✵

❊st✉❞❡✲s❡ ❛ ✜❣✉r❛ ✸ ♠❛✐s ❛t❡♥t❛♠❡♥t❡✳ ▼❛r❝❛r❛✲s❡ ♦ ♣♦♥t♦ N ♦♥❞❡ ❛ ❘❡t❛ ❞❡ ❊✉❧❡r HO ❡♥❝♦♥tr❛ ❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ B′C❜❛✐①❛❞❛ ♣♦r P✳ ❆s três r❡t❛sAH P NAO✱ t♦❞❛s

♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛B′C✱ sã♦ ♣❛r❛❧❡❧❛s ❡♥tr❡ s✐✳ ❈♦♠♦ AP =P A′✱ ❛s r❡t❛sAHP NAO

sã♦ ❡q✉✐t❛t✐✈❛♠❡♥t❡ ❡s♣❛ç❛❞♦s✿ P N ❡q✉✐❞✐st❛ ❞❡ AH ❡A′O✳ ❉❛í✱ N é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦

s❡❣♠❡♥t♦HO✳

❆té ❛q✉✐✱ ❝♦♥❞✉③✐✉✲s❡ ❛s ❞✐s❝✉ssõ❡s ❝♦♠ r❡❧❛çã♦ ❛♦ ❧❛❞♦B′C❞♦ABC✳ ❙❡✱ ❡♠ ✈❡③

❞✐ss♦✱ ❛♣❧✐❝❛r✲s❡ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ♣❛r❛ q✉❛❧q✉❡r ✉♠ ❞♦s ♦✉tr♦s ❧❛❞♦s✱ ♦ s❡❣♠❡♥t♦HO

♣❡r♠❛♥❡❝❡ ✜①♦❀ ❡ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ HO s❡rá ❛ ✐♥t❡rs❡çã♦ ❝♦♠ ❛ ♠❡❞✐❛tr✐③ ❞♦ ♥♦✈♦ ❧❛❞♦✳

❈♦♠♦ HO t❡♠ s♦♠❡♥t❡ ✉♠ ♣♦♥t♦ ♠é❞✐♦✱ ♣♦❞❡✲s❡ ❛✜r♠❛r q✉❡ ❛s ♠❡❞✐❛tr✐③❡s ❞♦s três

❧❛❞♦s ❞❡△ABC♣❛ss❛rã♦ ♣❡❧♦ ♣♦♥t♦N✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱N ❞❡✈❡ s❡r ♦ ❝✐r❝✉♥❝❡♥tr♦

❞❡△ABC

❊♠ r❡s✉♠♦✱ ♦ ❝✐r❝✉♥❝❡♥tr♦ ❞♦ tr✐â♥❣✉❧♦ ♠❡❞✐❛❧ ❡stá ♥♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ ❍❖ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ❞♦ △ABC✳ ❆❧é♠ ❞✐ss♦✱ ✉♠❛ ✈❡③ q✉❡ △ABC∼ △ABC✱ ♦ r❛✐♦ ❞♦

❝ír❝✉❧♦ ❝✐♥❝✉♥s❝r✐t♦ ❞♦ tr✐â♥❣✉❧♦ ♠❡❞✐❛❧ é ✐❣✉❛❧ à ♠❡t❛❞❡ ❞♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ❞♦ △ABC

✶✳✷✳✷ ❈♦♠❡♥tár✐♦s à ✷

❞❡♠♦♥str❛çã♦ ✲ ❖ ❚r✐â♥❣✉❧♦ ✏▼❡❞✐❛❧✑ ❡ ❛

❘❡t❛ ❞❡ ❊✉❧❡r

❊♠❜♦r❛ ❡ss❛ ❛❜♦r❞❛❣❡♠ ♣♦ss❛ s❡r ✉♠❛ ❞❛s ♠❛✐s ❡✈✐❞❡♥t❡s q✉❛♥t♦ à ❝❧❛r❡③❛✱ ❛ q✉❛♥t✐✲ ❞❛❞❡ ❞❡ tr❛ç❛❞♦s ♥❛ ✜❣✉r❛ ✸✱ ♣♦r s✐ só✱ ❥á é s✉✜❝✐❡♥t❡ ♣❛r❛ ❞✐✜❝✉❧t❛r ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❝♦♠♣r❡❡♥sã♦✱ ✉♠❛ ✈❡③ q✉❡ t❡♥❞❡ ❛ ❞✐s♣❡rs❛r ❛ ❝♦♥❝❡♥tr❛çã♦ ❞♦ ❧❡✐t♦r ❡♠ ❣❡r❛❧✱ ♥♦ t❡♠❛ ❡♠ ❛♣r❡ç♦✳

❙❡♥❞♦ ❛ss✐♠✱ é ❞❡ ❜♦♠ s✐s♦ q✉❡ s❡❥❛♠ ❛❝r❡s❝❡♥t❛❞❛s ❛❧❣✉♠❛s ✜❣✉r❛s q✉❡ ✐❧✉str❡♠ ♦ ♣❛ss♦ ❛ ♣❛ss♦ ❞❛ ❞❡♠♦♥str❛çã♦✱ ❜❡♠ ❝♦♠♦ ❝✐♥❝♦ ❞❡t❛❧❤❡s sã♦ r❡❝♦♠❡♥❞á✈❡✐s ♣❛r❛ ♦ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦✳

✶♦ ❞❡t❛❧❤❡✿ ✏✈❡♠♦s q✉❡ ❆❈✬❆✬❇✬ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡r✲

s❡çã♦ ❡♥tr❡ ❆❆✬ ❡ ❇✬❈✬ é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦s ❞♦✐s s❡❣♠❡♥t♦s✑✳

❈♦♠ ❡❢❡✐t♦✱ s❡♥❞♦A′C♣♦♥t♦s ♠é❞✐♦s ❞❡BCAB

  

A′C//AB

A′C= 1

2·AC =AB

=CB

■❣✉❛❧♠❡♥t❡✱ ♣❛r❛A′B♣♦♥t♦s ♠é❞✐♦s ❞❡BCAC

  

A′B//AC

A′B= 1

2·AB =AC

=BC

❖r❛✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ q✉❡ ❛s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ (AC′AB) ✐♥t❡rs❡❝t❛♠✲

s❡ ❡♠ s❡✉ ♣♦♥t♦ ♠é❞✐♦ ❢♦r♥❡❝❡✱ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ q✉❡AP =A′PBP =CP✳ ▲♦❣♦✱ P

(22)

✶✳✷ ✷❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ✷✶

❋✐❣✉r❛ ✹✿ ❆s ♠❡❞✐❛♥❛s ❞♦ △ABC❡stã♦ ❝♦♥t✐❞❛s ♥❛s ♠❡❞✐❛♥❛s ❞♦ ABC

❉❡ss❛ ❢♦r♠❛✱ A′P é ♠❡❞✐❛♥❛ ❞♦ ABC

A′P = 1

2·AA

′ ✳

✷♦ ❞❡t❛❧❤❡✿ ✏❛s ♠❡❞✐❛♥❛s ❞❡ABC❡stã♦ ❝♦♥t✐❞❛s ♥❛s ♠❡❞✐❛♥❛s ❞♦ABC✱ ♦ q✉❡

❛❝❛rr❡t❛ q✉❡ ❛♠❜♦s ♦s tr✐â♥❣✉❧♦s tê♠ ♦ ♠❡s♠♦ ❜❛r✐❝❡♥tr♦✱ ●✳✑

❈♦♠♦A′P ❡stá ❝♦♥t✐❞❛ ♥❛ ❞✐❛❣♦♥❛❧AA❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ACAB✱ ❡AAé ♠❡❞✐❛♥❛

❞♦ △ABCAP ❡stá ❝♦♥t✐❞❛ ♥❛ ♠❡❞✐❛♥❛ AA✳ ❆ss✐♠✱ ♣♦r ❛♥❛❧♦❣✐❛✱ é ♣♦ssí✈❡❧ ❝♦♥❝❧✉✐r

q✉❡ ❛s ♠❡❞✐❛♥❛s ❞♦△ABC❡stã♦ ❝♦♥t✐❞❛s ♥❛s ♠❡❞✐❛♥❛s ❤♦♠ó❧♦❣❛s ❞♦ABC✳ ❆ss✐♠✱

♣♦ss✉❡♠ ♦ ♠❡s♠♦ ❜❛r✐❝❡♥tr♦G✳

✸♦ ❞❡t❛❧❤❡✿ ✏▼❛r❝❛r❛✲s❡ ♦ ♣♦♥t♦N ♦♥❞❡ ❛ ❘❡t❛ ❞❡ ❊✉❧❡rHO❡♥❝♦♥tr❛ ❛ ♣❡r♣❡♥❞✐❝✉❧❛r

❛ B′C❜❛✐①❛❞❛ ♣♦r P✳ ❆s três r❡t❛s AH P N AO✱ t♦❞❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛ BC✱ sã♦

♣❛r❛❧❡❧❛s ❡♥tr❡ s✐✳ ❈♦♠♦ AP = P A′✱ ❛s r❡t❛s AHP NAO sã♦ ❡q✉✐t❛t✐✈❛♠❡♥t❡

❡s♣❛ç❛❞♦s✿ P N ❡q✉✐❞✐st❛ ❞❡ AH ❡ A′O✳ ❉❛í✱ N é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ HO✳✑

❖✉tr❛ ♣❛ss❛❣❡♠ ♠❡r❡❝❡❞♦r❛ ❞❡ ✉♠❛ ✐❧✉str❛çã♦ ❡①tr❛✳ P❛r❛ t❛❧✱ r❡t✐r❡✲s❡ ♦ △ABCABC ❞❛ ✜❣✉r❛ ✸✱ ♦❜t❡♥❤❛✲s❡ ❛ ✜❣✉r❛ ✺✿

P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❚❛❧❡s✱ r❡t❛s ♣❛r❛❧❡❧❛s ❞❡t❡r♠✐♥❛♠✱ s♦❜r❡ r❡t❛s tr❛♥s✈❡rs❛✐s✱ s❡❣♠❡♥✲ t♦s ♣r♦♣♦r❝✐♦♥❛✐s✳ ❖r❛✱ ❛s r❡t❛s s✉♣♦rt❡s ❞❡AH✱P N ❡OA′ ❞❡t❡r♠✐♥❛♠ s♦❜r❡ ♦ s❡❣♠❡♥t♦

AA′ ❛ r❛③ã♦ AP

A′P✱ ❛ss✐♠ ❝♦♠♦ ❞❡t❡r♠✐♥❛♠ s♦❜r❡ ♦ s❡❣♠❡♥t♦HO ❛ r❛③ã♦

HN

N O✱ ❞❡ ♠❛♥❡✐r❛

q✉❡ HN

N O = AP

A′P✳ ▼❛s✱ ❥á s❡ s❛❜❡ q✉❡

AP

A′P = 1✳ ▲♦❣♦✱

HN

N O = 1 ⇒HN =N O✱ ♦✉ s❡❥❛✱

(23)

✶✳✷ ✷❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ✷✷

❋✐❣✉r❛ ✺✿ N é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ HO

✹♦ ❞❡t❛❧❤❡✿ ✏♦ ❝✐r❝✉♥❝❡♥tr♦ ❞♦ tr✐â♥❣✉❧♦ ♠❡❞✐❛❧ ❡stá ♥♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ ❍❖

❞❛ ❘❡t❛ ❞❡ ❊✉❧❡r ❞♦ △ABC✳ ❆❧é♠ ❞✐ss♦✱ ✉♠❛ ✈❡③ q✉❡ △ABC∼ △ABC✱ ♦ r❛✐♦ ❞♦

❝ír❝✉❧♦ ❝✐♥❝✉♥s❝r✐t♦ ❞♦ tr✐â♥❣✉❧♦ ♠❡❞✐❛❧ é ✐❣✉❛❧ à ♠❡t❛❞❡ ❞♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ❞♦△ABC✳✑

◆♦✈❛♠❡♥t❡ ❡✈♦❝❛✲s❡ ♦ ❛s♣❡❝t♦ s✉♣r❛❝✐t❛❞♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ❡♥tr❡ tr✐â♥❣✉❧♦s✳ ▼❛✐s ❡s✲ ♣❡❝✐✜❝❛♠❡♥t❡✱ ❥á s❡ ❧❡✈❛ ❡♠ ❝♦♥t❛ q✉❡ ♦s s❡❣♠❡♥t♦s ✭r❛✐♦s ❞♦s ❝ír❝✉❧♦s ❝✐r❝✉♥s❝r✐t♦s ❛♦

ABC ABC✮ ❝✉❥❛s ❡①tr❡♠✐❞❛❞❡s sã♦ ♦ ❝✐r❝✉♥❝❡♥tr♦ ✭◆ ❡ ❖✮ ❡ ✉♠ ❞♦s ✈ért✐❝❡s

❝♦rr❡s♣♦♥❞❡♥t❡s sã♦ ❤♦♠ó❧♦❣♦s ✭✜❣✉r❛ ✻✮✱ ♦✉ s❡❥❛✱ ♠❛♥tê♠ ❛ r❛③ã♦

1 2

❞❡ s❡♠❡❧❤❛♥ç❛ ❡♥tr❡ ♦△ABC ABC✳ P♦rt❛♥t♦✱ ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ❝✐♥❝✉♥s❝r✐t♦ ❛♦ ABCé ✐❣✉❛❧ à ♠❡t❛❞❡ ❞♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ❝✐r❝✉♥s❝r✐t♦ ❛♦ △ABC

Referências

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