Open Equações de 2º grau em Geometria Plana

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚

❊q✉❛çõ❡s ❞❡ ✷

♦

❣r❛✉ ❡♠

●❡♦♠❡tr✐❛ P❧❛♥❛

♣♦r

❆❧❞❡❝❦ ▼❡♥❡③❡s ❞❡ ❖❧✐✈❡✐r❛

❉❡③❡♠❜r♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇

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

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚

❊q✉❛çõ❡s ❞❡ ✷

♦

❣r❛✉ ❡♠

●❡♦♠❡tr✐❛ P❧❛♥❛

♣♦r

❆❧❞❡❝❦ ▼❡♥❡③❡s ❞❡ ❖❧✐✈❡✐r❛

❙♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ P❡❞r♦ ❆♥t♦♥✐♦ ❍✐♥♦❥♦s❛ ❱❡r❛

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

❉❡③❡♠❜r♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇

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

●❡♦♠❡tr✐❛ P❧❛♥❛

♣♦r

❆❧❞❡❝❦ ▼❡♥❡③❡s ❞❡ ❖❧✐✈❡✐r❛

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

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ●❡♦♠❡tr✐❛✳ ❆♣r♦✈❛❞❛ ♣♦r✿

Pr♦❢✳ P❡❞r♦ ❆♥t♦♥✐♦ ❍✐♥♦❥♦s❛ ❱❡r❛ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮

Pr♦❢✳ ❋❡r♥❛♥❞♦ ❆♥t♦♥✐♦ ❳❛✈✐❡r ❞❡ ❙♦✉③❛ ✲ ❯❋P❇

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

❆ ❉❡✉s✱ ♣♦r ❡st❛r s❡♠♣r❡ ♣r❡s❡♥t❡✳

❆ ❋❡r♥❛♥❞♦ ❇❛st♦s ❞❡ ❖❧✐✈❡✐r❛✱ ♠❡✉ ♣❛✐✱ ❡ ❛ ❱❡r❛❧✉❝✐❛ ■♥á❝✐♦ ❞❡ ▼❡♥❡③❡s✱ ♠✐♥❤❛ ♠ã❡✱ q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈❛r❛♠ ❛♦ ❡st✉❞♦✳

➚s ♠✐♥❤❛s ✜❧❤❛s✱ ❆♥❞r❡ss❛ ▼❛r✐❛ ▲✐♠❛ ❞❡ ❖❧✐✈❡✐r❛ ❡ ❆❞r✐❡❧② ▲✐♠❛ ❞❡ ❖❧✐✈❡✐r❛✱ q✉❡ ❢♦✐ ♦♥❞❡ ❝♦♥s❡❣✉✐ ❢♦rç❛s ♣❛r❛ ❛ ❞❡❞✐❝❛çã♦ ❛♦ ♠❡✉ tr❛❜❛❧❤♦✳

➚ ♠✐♥❤❛ ❡s♣♦s❛✱ ▼❛r❧❡♥❡ ❋r❡✐r❡s ❞❡ ▲✐♠❛ ❖❧✐✈❡✐r❛✱ ♣❡❧♦ ❛♣♦✐♦ ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✳

❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ♠❛t❡♠á✲ t✐❝❛ q✉❡ ❧❡❝✐♦♥❛r❛♠ ♥♦ P❘❖❋▼❆❚✱ ♣♦✐s ❢♦r❛♠ ❡❧❡s q✉❡♠ ✐♥❝❡♥t✐✈❛r❛♠ ❡ ♦r✐❡♥t❛r❛♠ ♥❛ ❜✉s❝❛ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s✳

❆♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✱ ♣❡❧❛ ❞✐s♣♦s✐çã♦ ❡♠ ❛✈❛❧✐❛r ❡st❡ tr❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ❉r✳ P❡❞r♦ ❆♥t♦♥✐♦ ❍✐♥♦❥♦s❛ ❱❡r❛ q✉❡ ❝♦rr✐❣✐✉ ❡ ♦r✐❡♥t♦✉✱ ❝♦♠ ♠✉✐t❛ ❞❡❞✐❝❛çã♦ ❡ ♣❛s❝✐ê♥❝✐❛ t♦❞♦ ♦ ♠❡✉ tr❛❜❛❧❤♦✳

❆♦s ♠❡✉s ❛♠✐❣♦s✱ ❆❧②ss♦♥ ❊s♣❡❞✐t♦✱ ❋r❛♥❝✐s❝♦ ▲✐♠❛✱ ▼❛r❝❡❧♦ ❉❛♥t❛s✱ ❉✐ê❣♦ ❆②❧❧♦✱ ❘♦❜❡r✈❛❧ ❡ ❘♦♥❛❧❞♦✱ ♣❡❧❛s ✈❡r❞❛❞❡✐r❛s ❛♠✐③❛❞❡s ❝✉❧t✐✈❛❞❛s ❣r❛ç❛s ❛♦ ❛❞✈❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛ ❡♠ ♥♦ss❛s ✈✐❞❛s✳

➚ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭❙❇▼✮ ♣❡❧❛ ❝r✐❛çã♦ ❞♦ ♠❡str❛❞♦ ♣r♦✜ss✐✲ ♦♥❛❧ ❡♠ ♠❛t❡♠át✐❝❛ ❡♠ r❡❞❡ ♥❛❝✐♦♥❛❧ ✭P❘❖❋▼❆❚✮ ❞❛♥❞♦ ♦♣♦rt✉♥✐❞❛❞❡ ♣❛r❛ q✉❡ ♣r♦❢❡ss♦r❡s ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ♣♦ss❛♠ ♠❡❧❤♦r❛r ♦s s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐✲ ❝♦s✱ ❡ ❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ✭❯❋P❇✮✱ ♣♦r ❛❜r❛ç❛r ❡ss❛ ✐❞❡✐❛✳

❈♦♠❡ç❛♠♦s ❡st❡ tr❛❜❛❧❤♦ ❝♦♠ ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞❛ tr✐❣♦♥♦♠❡tr✐❛ ❡ ❞❛ ❡q✉❛✲ çã♦ ❞❡ s❡❣✉♥❞♦ ❣r❛✉ ♣❛r❛ ✜♥❛❧♠❡♥t❡ ❛♣r❡s❡♥t❛r♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s r❡❧❛❝✐♦♥❛♥❞♦ ❛ ❣❡♦♠❡tr✐❛ ❞♦ tr✐â♥❣✉❧♦✱ ❞♦s ♣♦❧í❣♦♥♦s ❝♦♥✈❡①♦s ❡ ❞❛s ❝ô♥✐❝❛s ❝♦♠ ❛s ❡q✉❛çõ❡s ❞❡ s❡❣✉♥❞♦ ❣r❛✉✳

P❛❧❛✈r❛s ❈❤❛✈❡✿ ❚r✐â♥❣✉❧♦✱ ❈ô♥✐❝❛s✱ ❊q✉❛çã♦ ❞❡ s❡❣✉♥❞♦ ❣r❛✉✳

❲❡ st❛rt❡❞ t❤✐s ✇♦r❦ ✇✐t❤ ❛ ❧✐tt❧❡ ❤✐st♦r② ♦❢ tr✐❣♦♥♦♠❡tr② ❛♥❞ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥ t♦ ✜♥❛❧❧② ✐♥tr♦❞✉❝❡ s♦♠❡ r❡s✉❧ts r❡❧❛t✐♥❣ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ tr✐❛♥❣❧❡✱ ♦❢ ❝♦♥✈❡① ♣♦❧②❣♦♥s ❛♥❞ t❤❡ ❝♦♥✐❝❛❧ ✇✐t❤ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳

❑❡② ✇♦r❞s✿ ❚r✐❛♥❣❧❡✱ ❈♦♥✐❝❛❧✱ ◗✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳

✶ Pr❡❧✐♠✐♥❛r❡s ✶ ✶✳✶ ❯♠❛ r❡❧❛çã♦ tr✐❣♦♥♦♠étr✐❝❛ ♥♦ tr✐â♥❣✉❧♦∆✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷

✶✳✷ ❆ ❋ór♠✉❧❛ ❞❡ ❍❡r♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❊q✉❛çõ❡s ❞♦ ✷♦ _{❣r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵}

✶✳✸✳✶ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸✳✷ ❘❛í③❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞♦ ✷♦ _{❣r❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺}

✶✳✹ ❉❡✜♥✐çã♦ ❞❛s ❝ô♥✐❝❛s ❝♦♠♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✷ ❉❡s❧♦❝❛♠❡♥t♦ ♣❛r❛❧❡❧♦ ✷✵

✷✳✶ ❉❡s❧♦❝❛♠❡♥t♦ ♣❛r❛❧❡❧♦ ❛♦ tr✐â♥❣✉❧♦ ∆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✷✳✷ ❉❡s❧♦❝❛♠❡♥t♦ ♣❛r❛❧❡❧♦ ❛ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✸ ❈♦♥❥✉♥t♦s ❡q✉✐❞✐st❛♥t❡s ♥♦ ♣❧❛♥♦ ✷✾ ✸✳✶ ❘❡t❛s ❡ ❝ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ❆s ❝ô♥✐❝❛s ❝♦♠♦ ❝♦♥❥✉♥t♦s ❡q✉✐❞✐st❛♥t❡s ❡ s✉❛s ❡q✉❛çõ❡s ❝❛♥ô♥✐❝❛s ✳ ✸✶

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✶

❊st❡ tr❛❜❛❧❤♦ tr❛t❛✱ ❞❡ ✉♠ ♠♦❞♦ ❣❡r❛❧✱ ❞❛ ♦❜t❡♥çã♦ ❞❡ r❡s✉❧t❛❞♦s q✉❡ r❡❧❛❝✐♦♥❛♠ ❛ ❣❡♦♠❡tr✐❛ ❞♦ tr✐â♥❣✉❧♦✱ ❞♦s ♣♦❧í❣♦♥♦s ❝♦♥✈❡①♦s ❡ ❞❛s ❝ô♥✐❝❛s ❝♦♠ ❛s ❡q✉❛çõ❡s ❞❡ s❡❣✉♥❞♦ ❣r❛✉✳ ❙❡❣✉✐♥❞♦ ❛ ❧✐♥❤❛ ❞❡ ♣❡♥s❛♠❡♥t♦ ❞❡ ❘♦♥❛❧❞♦ ●❛r❝✐❛✱ ❡♠ s❡✉ ❛rt✐❣♦ ✐♥t✐t✉❧❛❞♦ ❞❡ ✑❊q✉❛çõ❡s ❞♦ ❙❡❣✉♥❞♦ ●r❛✉ ❡ ●❡♦♠❡tr✐❛ P❧❛♥❛✑✱ ✈❡r ❬✶❪✱ ❛♣r❡s❡♥t❛♠♦s ❡st❡ tr❛❜❛❧❤♦ ❞❡ ♠❛♥❡✐r❛ ❝❧❛r❛ ❡ ♦❜❥❡t✐✈❛✱ t♦r♥❛♥❞♦✲♦ ❞❡ ❢á❝✐❧ ❝♦♠♣r❡❡♥sã♦ ♣❛r❛ ♦ ❧❡✐t♦r✳

❯♠❛ ✈✐❛❣❡♠ ♥♦ t❡♠♣♦ ♣❛r❛ ❝♦♥❤❡❝❡r ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞❛ ❚r✐❣♦♥♦♠❡tr✐❛ ❡✱ ❜❡♠ ❝♦♠♦✱ ❝♦♥❤❡❝❡r ❛s ♣r✐♠❡✐r❛s ❢ór♠✉❧❛s ♦✉ r❡❣r❛s✱ ❝r✐❛❞❛s ♣♦r ❣r❛♥❞❡s ♠❛✲ t❡♠át✐❝♦s✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞❛s r❛í③❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ ✷♦ _{❣r❛✉✳ ❆ ❞❡❞✉çã♦ ❞❡}

❢ór♠✉❧❛s ❝♦♥❤❡❝✐❞❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ ▲❡✐ ❞♦s ❝♦ss❡♥♦s✱ ❛ ❋ór♠✉❧❛ ❞❡ ❍❡r♦♥✱ ❛ ❢ór♠✉❧❛ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛s r❛í③❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ ✷♦ _{❣r❛✉✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦}

❋ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛✱ ❡ ❛s ❞❡✜♥✐çõ❡s ❞❛s ❝ô♥✐❝❛s ❝♦♠♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❡ ❝♦♥❥✉♥t♦s ❡q✉✐❞✐st❛♥t❡s✳

❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧✱ ❛♣r❡s❡♥t❛r ❛❧❣✉♠❛s s✐t✉❛çõ❡s q✉❡ ❛♣❛r❡❝❡♠ ♥❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛✱ ❡♠ q✉❡ ♠✉✐t❛s ❞❛s ✈❡③❡s ♣❛ss❛♠ ❞❡s♣❡r❝❡❜✐❞❛s ❞❡ ❡st✉❞❛♥t❡s ✭♦✉ ❛té ♠❡s♠♦ ♣r♦❢❡ss♦r❡s✮ ❞❡ ♠❛t❡♠át✐❝❛ ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦ ❛ ♣r❡s❡♥ç❛ ❞❡ ❡①♣r❡ssõ❡s ❞❛ ❢♦r♠❛✿

y=ax2

+bx+c, com a₆= 0, a, b, c_∈R.

❆s ❝♦♥st❛♥t❡sa✱b❡c✱ sã♦ ❝❤❛♠❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ ❝♦❡✜❝✐❡♥t❡ q✉❛❞rát✐❝♦✱ ❝♦❡✜❝✐❡♥t❡ ❧✐♥❡❛r ❡ t❡r♠♦ ✐♥❞❡♣❡♥❞❡♥t❡✳ ❖ t❡r♠♦ ✑q✉❛❞rát✐❝♦✑ ✈❡♠ ❞❡ q✉❛❞r❛t✉s✱ q✉❡ ❡♠ ❧❛t✐♠ s✐❣♥✐✜❝❛ q✉❛❞r❛❞♦✳ ❊q✉❛çõ❡s ❞❡st❡ t✐♣♦✱ ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞❛s ❛tr❛✈és ❞❛ ❢❛t♦r❛çã♦✱ ❞♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ q✉❛❞r❛❞♦s✱ ❞♦ ✉s♦ ❞❡ ❣rá✜❝♦s✱ ❢ór♠✉❧❛s ❡ ♦✉tr♦s ♠ét♦❞♦s✳ ❖ ♠❛t❡♠át✐❝♦ ❇❤❛s❦❛r❛ ❆❦✐r❛ ❛♣r❡s❡♥t♦✉ ❛ s✉❛ ♠❛♥❡✐r❛ ❞❡ tr❛t❛r ✉♠❛ ❡q✉❛çã♦ ❞❡st❛✿ ✉♠❛ ❢ór♠✉❧❛ q✉❡ ❧❡✈❛ s❡✉ ♥♦♠❡✳

P❛r❛ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✱ ❛♣r❡s❡♥t❛♠♦s ❡ ❞❡❞✉③✐♠♦s ❛ ▲❡✐ ❞♦s ❝♦ss❡♥♦s✱ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ ✉♠❛ ❢ór♠✉❧❛ ❣❡♥✐❛❧ q✉❡ ❡♥✈♦❧✈❡ ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❡ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦✳ P❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦✱ ❛ ❋ór♠✉❧❛ ❞❡ ❍❡r♦♥✱ ❞❡ ♣♦✉❝♦ ❞❡st❛q✉❡ ♥♦s ❧✐✈r♦s ❞❡ ❡♥s✐♥♦ ❜ás✐❝♦ ❞❡ ♠❛t❡♠át✐❝❛✱ ♠❛s ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ♥♦ss♦ tr❛❜❛❧❤♦✳

❋✐❣✉r❛ ✶✿ ❈ô♥✐❝❛s

◆♦ ❝❛♣ít✉❧♦ ✶✱ ❛❜♦r❞♦✉✲s❡ ❢ór♠✉❧❛s ❡ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s✱ ♦♥❞❡ ♠✉✐t♦s ❞❡❧❡s ❡♥❝♦♥✲ tr❛❞♦s ❡♠ ❧✐✈r♦s ❞❡ ❡♥s✐♥♦ ❜ás✐❝♦ ❞❡ ♠❛t❡♠át✐❝❛✱ t♦♠❛♥❞♦ ❝♦♠♦ ❛♣♦✐♦ ❞❡ ♣❡sq✉✐s❛✱ ♦s ❧✐✈r♦s ❬✸❪✱ ❬✹❪✱ ❬✺❪✱ ❬✻❪ ❡ ❬✽❪✳ ❙❡❣✉✐♥❞♦ ♥♦ss❛ tr❛❥❡tór✐❛✱ ✈❡♠ ♦ ❝❛♣ít✉❧♦ ✷✱ ❝♦♠ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❞❡ r❡❣✐õ❡s ❣❡r❛❞❛s ❛ ♣❛rt✐r ❞❡ ❝♦♥❥✉♥t♦s ❡q✉✐❞✐st❛♥t❡s ❛ ✉♠❛ ♠❡❞✐❞❛ r ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ❡ ❣❡♥❡r❛❧✐③❛♥❞♦ ♣❛r❛ ♣♦❧í❣♦♥♦s ❝♦♥✈❡①♦s✳ P♦r ✜♠✱ ♥♦ ❝❛♣ít✉❧♦ ✸✱ ❝♦♥❥✉♥t♦s ❡q✉✐❞✐st❛♥t❡s q✉❡ ❞❡✜♥❡♠ r❡t❛s ❝♦♠ ❞❡♥♦♠✐♥❛çõ❡s ❝♦♥❤❡❝✐❞❛s✱ ❝ír❝✉❧♦ ❡ ❝ô♥✐❝❛s✱ s❡♠ ❞❡✐①❛r ❞❡ ❞❡❞✉③✐r s✉❛s ❡q✉❛çõ❡s ❝❛♥ô♥✐❝❛s✳

❊s♣❡r❛♠♦s q✉❡ ❡st❡ tr❛❜❛❧❤♦ s✐r✈❛ ❞❡ ❛♣♦✐♦ ❞❡ ♣❡sq✉✐s❛ ♣❛r❛ ♦✉tr❛s ❞✐ss❡rt❛çõ❡s✱ ♦✉ t❛❧✈❡③ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❞❡ tr❛❜❛❧❤♦ ♥❛ ♣r❡♣❛r❛çã♦ ❞❡ ❛✉❧❛s ♣❛r❛ ♣r♦❢❡ss♦r❡s ❞❡ ♠❛t❡♠át✐❝❛ ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦✳

Pr❡❧✐♠✐♥❛r❡s

❆ ●❡♦♠❡tr✐❛ é ✉♠ r❛♠♦ ❞❛ ♠❛t❡♠át✐❝❛ q✉❡ tr❛t❛ ❞❛s q✉❡stõ❡s r❡❧❛❝✐♦♥❛❞❛s ❛ ❢♦r♠❛✱ t❛♠❛♥❤♦ ❡ ♣♦s✐çõ❡s r❡❧❛t✐✈❛s ❞❡ ✜❣✉r❛s ❡ ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❡s♣❛ç♦✳ ❊❧❛ ❢♦✐ ♣♦st❛ ❡♠ ✉♠❛ ❢♦r♠❛ ❛①✐♦♠át✐❝❛ ♣♦r ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛✱ ♣♦r ✈♦❧t❛ ❞♦ sé❝✉❧♦ ■■■ ❛❈✳ ❆ ❣❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛ é ❝❛r❛❝t❡r✐③❛❞❛ ♣❡❧♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✱ ✐♠✉tá✈❡❧✱ s✐♠étr✐❝♦ ❡ ❣❡♦♠étr✐❝♦✱ ❡ s❡ ♠❛♥t❡✈❡ ✐❧❡s♦ ♥♦ ♣❡♥s❛♠❡♥t♦ ❞♦s ♠❛t❡♠át✐❝♦s ♣♦r ♠✉✐t♦s sé❝✉❧♦s✱ ♦♥❞❡ s♦♠❡♥t❡ ♥♦s t❡♠♣♦s ♠♦❞❡r♥♦s ♣✉❞❡r❛♠ s❡r ❝♦♥str✉í❞♦s ♠♦❞❡❧♦s ❞❡ ❣❡♦♠❡tr✐❛ ♥ã♦✲❡✉❝❧✐❞✐❛♥❛✳

❋✐❣✉r❛ ✶✳✶✿ ❊✉❝❧✐❞❡s

❙❡rá ❡st✉❞❛❞♦ ♥❡st❡ ❝❛♣ít✉❧♦ ✉♠❛ ✐♠♣♦rt❛♥t❡ r❡❧❛çã♦ tr✐❣♦♥♦♠étr✐❝❛ ❞❛❞❛ ❡♠ ❢✉♥çã♦ ❞❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✱ ❝♦♠♦ t❛♠❜é♠ ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞♦ ♠❡s♠♦✳ ❯♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛✱ s❡♠ ❞❡✐①❛r ❞❡ ❞❡❞✉③✐r✱ ❞❡ ♠❛♥❡✐r❛ s✐♠♣❧❡s✱ ❛ ❢❛♠♦s❛ ❢ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛ ♣❛r❛ ❡♥❝♦♥tr❛r ❛s r❛í③❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ ✷♦ _{❣r❛✉✳ ❊ ♣♦r ú❧t✐♠♦ t♦♠❛r ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛s ❞❡✜♥✐çõ❡s}

❞❛s ❝ô♥✐❝❛s ❝♦♠♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✳

✶✳✶ ❯♠❛ r❡❧❛çã♦ tr✐❣♦♥♦♠étr✐❝❛ ♥♦ tr✐â♥❣✉❧♦

∆

❆ tr✐❣♦♥♦♠❡tr✐❛ ❝♦♠❡ç♦✉ ♥❛ ●ré❝✐❛ ❛♥t✐❣❛ s✉r❣✐♥❞♦ ❞❡✈✐❞♦ ás ♥❡❝❡ss✐❞❛❞❡s ❞❛ ❆str♦♥♦♠✐❛✱ ♦♥❞❡ ♦s ❡st✉❞♦s s❡ ❝♦♥❝❡♥tr❛✈❛♠ ♥❛ tr✐❣♦♥♦♠❡tr✐❛ ❡s❢ér✐❝❛✱ ♣❛r❛ ✐ss♦ ❢♦✐ ♥❡❝❡ssár✐♦ ❞❡s❡♥✈♦❧✈❡r ♣❛rt❡s ❞❛ tr✐❣♦♥♦♠❡tr✐❛ ♣❧❛♥❛✳ ❆♣❡s❛r ❞❡ s❡ ❢❛❧❛r ❡♠ ♠✉✐t♦s ❡st✉❞✐♦s♦s ❛♥t✐❣♦s ❡♠ tr✐❣♦♥♦♠❡tr✐❛✱ ❢♦✐ ❞❛❞♦ ♦ tít✉❧♦ ❞❡ ❢✉♥❞❛❞♦r ❞❛ tr✐❣♦♥♦♠❡tr✐❛ ❛ ❍✐♣❛r❝♦ ❞❡ ◆✐❝❡✐❛ ✭✈✐✈❡✉ ❡♠ t♦r♥♦ ❞❡ ✶✷✵ ❛❈✳✮✱ t❡♥❞♦ ❞❡s❡♥✈♦❧✈✐❞♦s tr❛❜❛❧❤♦s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♠ ❆str♦♥♦♠✐❛✳ ➱ ❜❡♠ ♣r♦✈á✈❡❧ q✉❡ ❛ ❞✐✈✐sã♦ ❞♦ ❝ír❝✉❧♦ ❡♠ 360o

t❡♥❤❛ s❡ ♦r✐❣✐♥❛❞♦ ❝♦♠ ❛ t❛❜❡❧❛ ❞❡ ❝♦r❞❛s ❞❡ ❍✐♣❛r❝♦✳

❋✐❣✉r❛ ✶✳✷✿ ❍✐♣❛r❝♦ ❞❡ ◆✐❝é✐❛

◆ã♦ ♣♦❞❡✲s❡ ❞❡✐①❛r ❞❡ ♠❡♥❝✐♦♥❛r ❈❧á✉❞✐♦ Pt♦❧♦♠❡✉ ✭✈✐✈❡✉ ❡♠ t♦r♥♦ ❞❡ ✶✺✵ ❞❈✳✮✱ q✉❡ ❢♦✐ ✉♠ ♦✉tr♦ ❣r❛♥❞❡ ♠❛t❡♠át✐❝♦ q✉❡ ❝♦♥tr✐❜✉✐✉ ♥♦ ❡st✉❞♦ ❞❛ tr✐❣♦✲ ♥♦♠❡tr✐❛✳ ❊❧❡ ❞❡❞✉③✐✉✱ ♦ q✉❡ ❡s❝r✐t❛ ❡♠ ♥♦t❛çã♦ ♠♦❞❡r♥❛✱ ❛s ❡①♣r❡ssõ❡s ♣❛r❛ s❡♥(a_±b) ❡ cos(a _±b)✱ ♦♥❞❡ a ❡ b sã♦ â♥❣✉❧♦s q✉❛✐sq✉❡r✳ ❉❡♠♦♥str♦✉ t❛♠❜é♠ q✉❡ s❡♥2

x+ cos2

x = 1✱ ♦♥❞❡ x é ✉♠ â♥❣✉❧♦ ❛❣✉❞♦✳ ❆ ❚r✐❣♦♥♦♠❡tr✐❛ ❞❛❞❛ ♣♦r Pt♦❧♦♠❡✉ ♥♦ ❆❧♠❛❣❡st♦ ❢♦✐ ♣❛❞rã♦ ❛té ♦ r❡♥❛s❝✐♠❡♥t♦✳

❋✐❣✉r❛ ✶✳✸✿ ❈❧á✉❞✐♦ Pt♦❧♦♠❡✉

❖ ♠❛t❡♠át✐❝♦ ❋r❛♥ç♦✐s ❱✐❡t❛ s✐st❡♠❛t✐③♦✉ ♦ ❡st✉❞♦ ❞❛ tr✐❣♦♥♦♠❡tr✐❛ ❡s❢ér✐❝❛✱ ❞❡❞✉③✐♥❞♦ ❢ór♠✉❧❛s ♣❛r❛ s❡♥(nα)❡ cos(nα)✱ ❞❡♥tr❡ ♦✉tr❛s✳

◆❡st❛ s❡çã♦✱ s❡rã♦ ❞❡❞✉③✐❞❛s ❛ ❧❡✐ ❞♦s ❝♦ss❡♥♦s ❡ ❛ ❢ór♠✉❧❛ ♣❛r❛ ❛ ❝♦t❛♥❣❡♥t❡ ❞❡ α/2 ❡♠ ❢✉♥çã♦ ❞❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦✱ ♦♥❞❡α é ✉♠ â♥❣✉❧♦ ✐♥t❡r♥♦ ❞♦ ♠❡s♠♦✳

Pr♦♣♦s✐çã♦ ✶✳✶ ✭▲❡✐ ❞♦s ❝♦ss❡♥♦s✮ ❊♠ q✉❛❧q✉❡r tr✐â♥❣✉❧♦✱ ♦ q✉❛❞r❛❞♦ ❞❛ ♠❡✲ ❞✐❞❛ ❞❡ ✉♠ ❧❛❞♦ é ✐❣✉❛❧ à s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞❛s ♠❡❞✐❞❛s ❞♦s ♦✉tr♦s ❞♦✐s ❧❛❞♦s ♠❡♥♦s ❞✉❛s ✈❡③❡s ♦ ♣r♦❞✉t♦ ❞❛s ♠❡❞✐❞❛s ❞❡ss❡s ❧❛❞♦s ♣❡❧♦ ❝♦ss❡♥♦ ❞♦ â♥❣✉❧♦ q✉❡ ❡❧❡s ❢♦r♠❛♠✳

❋✐❣✉r❛ ✶✳✹✿ ❚r✐â♥❣✉❧♦ ∆ABC✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ♦ tr✐â♥❣✉❧♦∆ABC ❝♦♠ s✉❛s ♠❡❞✐❞❛sa, b, c❡ â♥❣✉❧♦s ✐♥t❡r♥♦s α, β, γ✳ ❱❡r ✜❣✉r❛ ✶✳✹✳

❚r❛ç❛♥❞♦ ❛ ❛❧t✉r❛ CM r❡❧❛t✐✈❛ ❛♦ ❧❛❞♦ AB✱ ♦❜t❡♠♦s ♦s tr✐â♥❣✉❧♦s ∆BCM ❡

∆ACM✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✺✳

❋✐❣✉r❛ ✶✳✺✿ ❚r✐â♥❣✉❧♦ ∆ABC ❝♦♠ ❛ ❛❧t✉r❛ CM tr❛ç❛❞❛✳

❉♦ tr✐â♥❣✉❧♦ ∆ACM✱ t✐r❛♠♦s✿

AM =b_·cosα ✭✶✳✶✮

❖ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❛♣❧✐❝❛❞♦ ❛♦ tr✐â♥❣✉❧♦∆ACM✱ ♦❜t❡♠♦s✿

b2

=AM2

+CM2

CM2

=b2

−AM2

CM2

=b2

−(b_·cosα)2

CM2

=b2 −b2

·cos2

α ✭✶✳✷✮

❋❛③❡♥❞♦ ♦ ♠❡s♠♦ ❛♦ tr✐â♥❣✉❧♦∆BCM✱ ♦❜t❡♠♦s✿

a2

=CM2

+BM2

a2

=CM2

+ (c₋AM)2

a2

=CM2

+ (c₋b_·cosα)2

❉❛í✱ t❡♠♦s✿

CM2

=a2 −c2

+ 2bc_·cosα₋b2 ·cos2

α ✭✶✳✸✮

❉❡ ✭✶✳✷✮ ❡ ✭✶✳✸✮✱ ♦❜t❡♠♦s✿

b2 −b2

·cos2

α=a2 −c2

+ 2bc_·cosα₋b2 ·cos2

α

b2

=a2 −c2

+ 2bc_·cosα

P♦rt❛♥t♦✱

a2

=b2

+c2

−2bc_·cosα ✭✶✳✹✮

◆♦t❛♠♦s q✉❡ ♥ã♦ ❤á ❛❧t❡r❛çõ❡s ♣❛r❛ tr✐â♥❣✉❧♦s ♦❜t✉sâ♥❣✉❧♦s ✭✈❡r ✜❣✉r❛ ✶✳✻✮✱ ♣♦✐s ❞♦ tr✐â♥❣✉❧♦ ∆ACM✱ t✐r❛♠♦s✿

❋✐❣✉r❛ ✶✳✻✿ ❚r✐â♥❣✉❧♦ ♦❜t✉sâ♥❣✉❧♦ ∆ABC ❝♦♠ ❛ ❛❧t✉r❛CM tr❛ç❛❞❛✳

AM

b = cos(180

o

−α) =₋cosα

AM =₋b_·cosα

❙❡❣✉✐♥❞♦ ♦s ♠❡s♠♦s ♣❛ss♦s✱ ❝❤❡❣❛♠♦s ♥♦✈❛♠❡♥t❡ ❡♠✿

a2

=b2

+c2

−2bc_·cosα,

❛♥❛❧♦❣❛♠❡♥t❡✱

b2

=a2

+c2

−2ac_·cosβ

❡

c2

=a2

+b2

−2ab_·cosγ

❱❛❧❡ r❡ss❛❧t❛r q✉❡✱ ♦ ❢❛♠♦s♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❛ ▲❡✐ ❞♦s ❝♦ss❡♥♦s✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ α= 90o_{✳ ❆ss✐♠ ✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ✭sí♥t❡s❡ ❞❡ ❈❧❛✐r❛✉t✮✿}

α <90o

⇐⇒a2

< b2

+c2

α= 90o

⇐⇒a2

=b2

+c2

α >90o _⇐⇒a2

> b2

+c2_✱

❆ ▲❡✐ ❞♦s ❝♦ss❡♥♦s ♣♦ss✉✐ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s ❡♠ ❣❡♦♠❡tr✐❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s â♥❣✉❧♦s✱ ❛❧t✉r❛s r❡❧❛t✐✈❛s✱ ♠❡❞✐❛♥❛s✱ ❡t❝✳ ❞♦ tr✐â♥❣✉❧♦ ❡♠ ❢✉♥çã♦ ❞❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s✳

Pr♦♣♦s✐çã♦ ✶✳✷ ❈♦♥s✐❞❡r❡ ✉♠ tr✐â♥❣✉❧♦ ∆ABC q✉❛❧q✉❡r ❞❡ ❧❛❞♦s a, b ❡ c✱ s❡♠✐✲ ♣❡rí♠❡tr♦ s = (a +b +c)/2 ❡ ❞❡♥♦t❡ ♣♦r α ♦ â♥❣✉❧♦ ✐♥t❡r♥♦ ♦♣♦st♦ ❛♦ ❧❛❞♦ a. ❊♥tã♦✿

❝♦t❣α

2

=

s

s(s₋a)

(s₋b)(s₋c) ✭✶✳✺✮

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ♦ tr✐â♥❣✉❧♦ ∆ABC ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✼✱ ❧♦❣♦ ❛❜❛✐①♦✿

❋✐❣✉r❛ ✶✳✼✿ ❚r✐â♥❣✉❧♦ ∆ABC ❝♦♠ ❜✐ss❡tr✐③ ♣❛rt✐♥❞♦ ❞♦ ✈ért✐❝❡ A tr❛ç❛❞❛✳

❆ ❜✐ss❡tr✐③ ✐♥t❡r♥❛ ♣❛rt✐♥❞♦ ❞♦ ✈ért✐❝❡A❞♦ tr✐â♥❣✉❧♦∆ABC❞✐✈✐❞❡ ♦ ❧❛❞♦ ♦♣♦st♦ ♥❛ ♠❡s♠❛ r❛③ã♦ ❞♦s ❧❛❞♦s ❛❞❥❛❝❡♥t❡s ✭✈❡r ❬✸❪✮✳ ❈♦♠ ✐ss♦ t❡♠♦s✿

CM BM =

b c ⇒

CM b =

BM c =

a b+c.

❆ss✐♠✱

CM = ab

b+c

❆tr❛✈és ❞♦ ❝á❧❝✉❧♦ ❞❛ ❜✐ss❡tr✐③ ✐♥t❡r♥❛ ✭✈❡r ❬✸❪✮✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

AM = 2

b+c

p

bcs(s₋a)

❆♣❧✐❝❛♥❞♦ ❛ ▲❡✐ ❞♦s ❝♦ss❡♥♦s ✭✶✳✹✮ ♥♦ tr✐â♥❣✉❧♦∆ACM✱ t❡♠♦s✿

CM2

=b2

+AM2

−2_·b_·AM _·cosα 2

ab b+c

2

=b2

+

2

b+c

p

bcs(s₋a)

2

−2b 2 b+c

p

bcs(s₋a)_·cosα 2

a2

b2

(b+c)2 =b 2

+ 4

(b+c)2bcs(s−a)−

4b b+c

p

bcs(s₋a)_·cosα 2

cosα 2

= 4bcs(s−a) +b

2

(b+c)2 −a2

b2

4b(b+c)pbcs(s₋a)

cosα 2

= 4bcs(s−a) +b

2

(b+c+a)(b+c₋a) 4b(b+c)p

bcs(s₋a)

cosα 2

= 4bcs(s−a) +b

2

4s(s₋a) 4b(b+c)p

bcs(s₋a)

cosα 2

= ps(s−a)

bcs(s₋a)

cosα 2

=

r

s(s₋a)

bc . ✭✶✳✻✮

❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ tr✐❣♦♥♦♠❡tr✐❛✱ t❡♠♦s✿

r

s(s₋a)

bc

!2

+s❡♥2α

2

= 1

s❡♥2α

2

= bc−s(s−a)

bc .

❋❛t♦r❛♥❞♦ ♦ ♥✉♠❡r❛❞♦r✱ t❡♠♦s✿

s❡♥α

2

=

r

(s₋b)(s₋c)

bc . ✭✶✳✼✮

❈♦♠ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❞❡ ✭✶✳✻✮ ❡ ✭✶✳✼✮✱ t❡♠♦s✿

❝♦t❣α

2

=

cosα 2

s❡♥α

2

=

r

s(s₋a)

bc

r

(s₋b)(s₋c)

bc

=

s

s(s₋a) (s₋b)(s₋c).

❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ s❡❣✉❡ ❝♦t❣ β

2

=

s

s(s₋b)

(s₋a)(s₋c)❡ ❝♦t❣

γ

2

=

s

s(s₋c) (s₋a)(s₋b)✱

s❡♥❞♦ β ❡ γ â♥❣✉❧♦s ♦♣♦st♦s ❛♦s ❧❛❞♦sb ❡c✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❖❜s❡r✈❛çã♦✿ ❈♦♠♦α é ✉♠ â♥❣✉❧♦ ✐♥t❡r♥♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ♦✉ s❡❥❛✱0< α <180o_✱

❡♥tã♦ 0< α/2<90o_{✱ t❡♥❞♦ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ s❡♥♦✱ ❝♦ss❡♥♦ ❡ ❝♦t❛♥❣❡♥t❡ ♣♦s✐t✐✈♦s✳}

❖ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ β ❡γ✳

✶✳✷ ❆ ❋ór♠✉❧❛ ❞❡ ❍❡r♦♥

❍❡r♦♥ ❞❡ ❆❧❡①❛♥❞r✐❛ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❡ ♠❡❝â♥✐❝♦ q✉❡ ✈✐✈❡✉ ❡♠ ♠❡❛❞♦s ❞♦ sé❝✉❧♦ ■ ✭✶✵ ❞❈✳ ✲ ✼✵ ❞❈✳✮✳

❋✐❣✉r❛ ✶✳✽✿ ❍❡r♦♥

❍♦❥❡ ❡❧❡ é ❝♦♥❤❡❝✐❞♦ ♣❡❧❛ ❢ór♠✉❧❛ q✉❡ ❧❡✈❛ s❡✉ ♥♦♠❡✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❝♦♥❤❡❝✐❞❛s ❛s ♠❡❞✐❞❛s ❞❡ s❡✉s ❧❛❞♦s✳ ❙❡✉ tr❛❜❛❧❤♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ♥❛ ❣❡♦♠❡tr✐❛✱ ▼❡tr✐❝❛✱ ✜❝♦✉ ❞❡s❛♣❛r❡❝✐❞♦ ❛té ✶✽✾✻✳❚♦r♥♦✉✲s❡ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ♣♦r ✐♥✈❡♥t❛r ✉♠ ♠❡❝❛♥✐s♠♦ ♣❛r❛ ♣r♦✈❛r ❛ ♣r❡ssã♦ ❞♦ ❛r s♦❜r❡ ♦s ❝♦r♣♦s✱ ✜❝❛♥❞♦ ♥❛ ❤✐stór✐❛ ❝♦♠♦ ♦ ♣r✐♠❡✐r♦ ♠♦t♦r ❛ ✈❛♣♦r r❡❣✐str❛❞♦✳

Pr♦♣♦s✐çã♦ ✶✳✸ ❙❡❥❛ ♦ tr✐â♥❣✉❧♦ ∆ABC ❝♦♠ ❧❛❞♦s ♠❡❞✐♥❞♦ a, b, c ❡ hc ❛ ❛❧t✉r❛

r❡❧❛t✐✈❛ ❛♦ ❧❛❞♦ AB✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✺✱ ❡ s❡♠✐✲♣❡rí♠❡tr♦ s= (a+b+c)/2✳

❊♥tã♦✱ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ∆ABC é ❞❛❞❛ ♣♦r✿

A=ps(s₋a)(s₋b)(s₋c)

❉❡♠♦♥str❛çã♦✿ ❉❛ ▲❡✐ ❞♦s ❝♦ss❡♥♦s ✭✶✳✹✮ ❡ ❞❡ ✭✶✳✶✮✱ t❡♠♦s

a2

=b2

+c2

−2c_·AM

AM = b

2

+c2 −a2

2c

❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❛♦ tr✐â♥❣✉❧♦ ∆ACM ❡ ❥á ❢❛③❡♥❞♦ ❛ s✉❜st✐✲ t✉✐çã♦ ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ ♦❜t❡♠♦s✿

h2

c = b

2 −

b2

+c2 −a2

2c

2

= (2bc)

2 −(b2

+c2 −a2

)2

4c2

= (2bc+b

2

+c2 −a2

)(2bc₋b2 −c2

+a2

) 4c2

= [(b+c)

2 −a2

][a2

−(b₋c)2

] 4c2

= (b+c+a)(b+c−a)(a+b−c)(a−b+c) 4c2

▲❡♠❜r❛♥❞♦ q✉❡✿ b+c+a= 2s b+c₋a= 2(s₋a)

a+b₋c= 2(s₋c)

a₋b+c= 2(s₋b),

❡ ❢❛③❡♥❞♦ ❛s ❞❡✈✐❞❛s s✉❜st✐t✉✐çõ❡s✱ ♦❜t❡♠♦s✿

h2

c =

16s(s₋a)(s₋b)(s₋c) 4c2

hc =

2

c

p

s(s₋a)(s₋b)(s₋c), ✭✶✳✽✮

❛♥❛❧♦❣❛♠❡♥t❡✱

ha =

2

a

p

s(s₋a)(s₋b)(s₋c)

❡

hb =

2

b

p

s(s₋a)(s₋b)(s₋c)

❈♦♠♦ ❥á s❛❜❡♠♦s q✉❡ ❛ ár❡❛ A ❞♦ tr✐â♥❣✉❧♦∆ABC é ❞❛❞❛ ♣♦r✿

A= a·ha

2 =

b_·hb

2 =

c_·hc

2 ✭✶✳✾✮

❆❣♦r❛ s✉❜st✐t✉✐♥❞♦ ✭✶✳✽✮ ❡♠ ✭✶✳✾✮✱ ♦❜t❡♠♦s✿

A= c·hc

2 =

c

2· 2

c

p

s(s₋a)(s₋b)(s₋c)

A=p

s(s₋a)(s₋b)(s₋c) ✭✶✳✶✵✮

❆♣❡❧✐❞❛❞❛ ❞❡ ❋ór♠✉❧❛ ❞❡ ❍❡r♦♥ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❝♦♥❤❡✲ ❝✐❞❛s ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s✳

✶✳✸ ❊q✉❛çõ❡s ❞♦ ✷

_❣r❛✉

❊st✉❞❛r❡♠♦s ♥❡ss❛ s❡çã♦ ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❝❛ ❞❛ s♦❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❜❡♠ ❝♦♥❤❡❝✐❞❛ ❞❡ t♦❞♦s✱ ❛ ❊q✉❛çã♦ ❞❡ ✷♦ _{❣r❛✉✱ ❡ ♥ã♦ ♣♦❞❡rí❛♠♦s ❞❡✐①❛r ❞❡ ❛♣r❡s❡♥t❛r ❡}

❞❡❞✉③✐r ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❡♥❝♦♥tr❛r s✉❛s r❛í③❡s✳ ◆♦ ❇r❛s✐❧✱ ❡ss❛ ❢ór♠✉❧❛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❋ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛✱ ♠❛s ❡♠ ♦✉tr♦s ♣❛ís❡s é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❛ ❢ór♠✉❧❛ ❣❡r❛❧ ♣❛r❛ r❡s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ s❡❣✉♥❞♦ ❣r❛✉✱ s❡♠ ♥❡♥❤✉♠❛ r❡❢❡rê♥❝✐❛ ❛ ❇❤❛s❦❛r❛✳

✶✳✸✳✶ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛

+bx+c= 0 ♥♦s ❞✐❛s ❛t✉❛✐s✱ é s✐♠♣❧❡s✱ ♣♦✐s

❜❛st❛ ✉t✐❧✐③❛r ❛ ❢ór♠✉❧❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ✧❋ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛✧✿

x= −b±

√

b2 −4ac

2a

❆ ❡q✉❛çã♦ ❞❡ ✷♦ _{❣r❛✉ t❡♠ ✉♠❛ ❧♦♥❣❛ ❤✐stór✐❛ ❡ q✉❡ ♣❛ss♦✉ ♣♦r ♠✉✐t♦s ♠❛t❡♠á✲}

t✐❝♦s ✐♠♣♦rt❛♥t❡s✱ ❞❡ ✈ár✐❛s ❝✐✈✐❧✐③❛çõ❡s✱ ♦s q✉❛✐s s❡ ♣r❡♦❝✉♣❛♥❞♦ ❡♠ ❡♥❝♦♥tr❛r s✉❛s s♦❧✉çõ❡s✳

❖s ♣r♦❝❡ss♦s✱ ♣❛r❛ ❛❝❤❛r ❛s r❛í③❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞♦ ✷♦ _{❣r❛✉✱ ❞♦s ❜❛❜✐❧ô♥✐♦s✱}

❣r❡❣♦s✱ ❤✐♥❞✉s✱ ár❛❜❡s✱ ✐t❛❧✐❛♥♦s ❡r❛♠ ❢♦r♠✉❧❛❞♦s ❝♦♠ ♣❛❧❛✈r❛s✳ ❆ ♥♦t❛çã♦ ❛❧❣é❜r✐❝❛ s✐♠❜ó❧✐❝❛ ✉t✐❧✐③❛❞❛ ♣♦r ♥ós✱ ❤♦❥❡✱ é ❝r✐❛çã♦ r❡❝❡♥t❡ ❞♦s ♠❛t❡♠át✐❝♦s✱ ❝♦♠❡ç❛♥❞♦ ❝♦♠ ❋r❛♥ç♦✐s ❱✐èt❡ ✭✶✺✹✵ ✲ ✶✻✵✸✮ ❡ ❝♦❧♦❝❛❞❛ ♥❛ ❢♦r♠❛ ❛t✉❛❧ ♣♦r ❘❡♥é ❉❡s❝❛rt❡s ✭✶✺✾✻ ✲ ✶✻✺✵✮✳

❖s ❊❣í♣❝✐♦s só tr❛❜❛❧❤❛✈❛♠ ❝♦♠ ❡q✉❛çõ❡s ❞❡ ✷♦ _{❣r❛✉ s✐♠♣❧❡s✳ ❈♦♠♦ ♣♦r ❡①❡♠✲}

♣❧♦✱ ♥♦ ♣❛♣✐r♦ ❞❡ ▼♦s❝♦✉✱ q✉❡ ❞❛t❛✈❛ ❞❡ ✶✽✺✵ ❛❈✳ é ♣❡❞✐❞♦ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ ❜❛s❡ ❞❡

✉♠ r❡tâ♥❣✉❧♦ ❝✉❥❛ ❛❧t✉r❛ lé ✐❣✉❛❧ ❛ 3/4 ❞❛ ♠❡❞✐❞❛ ❞❛ ❜❛s❡ ❡ ❝✉❥❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦

é ✐❣✉❛❧ ❛ ✶✷✳ ❊♠ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ♠♦❞❡r♥❛✱ ❡ss❡ ♣r♦❜❧❡♠❛ é ❡s❝r✐t♦ ❝♦♠♦✿

3 4l

2

= 12

❖s ❇❛❜✐❧ô♥✐♦s ❡s❝r❡✈✐❛♠ ❡♠ t❛❜❧❡t❡s ❞❡ ❛r❣✐❧❛✱ ✉s❛♥❞♦ ❛ ❝❤❛♠❛❞❛ ❡s❝r✐t❛ ❝✉✲ ♥❡✐❢♦r♠❡✳ ❖s ❛rq✉❡ó❧♦❣♦s ❡ ❤✐st♦r✐❛❞♦r❡s ❞❛ ♠❛t❡♠át✐❝❛ ❝♦♥❝❧✉ír❛♠ q✉❡ ❛ q✉❛❧✐❞❛❞❡ ❞❛ ♠❛t❡♠át✐❝❛ ♣r❛t✐❝❛❞❛ ♥❛ ▼❡s♦♣♦tâ♠✐❛ ❡r❛ ❝❧❛r❛♠❡♥t❡ ♠❛✐s ❞❡s❡♥✈♦❧✈✐❞❛ ❞♦ q✉❡ ❛ ♠❛t❡♠át✐❝❛ ❡❣í♣❝✐❛ ♥♦ ✜♠ ❞♦ sé❝✉❧♦ ❳■❳ ❡ ♣r✐♠❡✐r❛ ♠❡t❛❞❡ ❞♦ sé❝✉❧♦ ❳❳✳

❊♥tr❡ ♦s ✐♥ú♠❡r♦s t❛❜❧❡t❡s ❞❡ ❛r❣✐❧❛✱ ✉♠ ♠❡♥❝✐♦♥❛✈❛ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿ ❆❝❤❛r ❛ ♠❡❞✐❞❛ ❞♦ ❧❛❞♦ ❞❡ ✉♠ q✉❛❞r❛❞♦ s❡ s✉❛ ár❡❛ ♠❡♥♦s s❡✉ ❧❛❞♦ é ✐❣✉❛❧ ❛ ✽✼✵✳ ❊♠ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ♠♦❞❡r♥❛ t❡♠♦sx2

−x= 870✱ ♦♥❞❡xé ❛ ♠❡❞✐❞❛ ❞♦ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦✳

❆ s♦❧✉çã♦ r❡❣✐str❛❞❛ ♥♦ t❛❜❧❡t❡ é ❛ s❡❣✉✐♥t❡✿

❚♦♠❡ ❛ ✉♥✐❞❛❞❡✿ ✶

❉✐✈✐❞❛ ❛ ✉♥✐❞❛❞❡ ❡♠ ❞✉❛s ♣❛rt❡s✿ ✶✴✷ ❈r✉③❡ ✭♠✉❧t✐♣❧✐q✉❡✮ ✶✴✷ ♣♦r ✶✴✷✿ ✶✴✹ ❙♦♠❡ ✶✴✹ ❛ ✽✼✵✿ ✸✹✽✶✴✹

➱ ♦ q✉❛❞r❛❞♦ ❞❡ ✺✾✴✷✿ (59/2)2

= 3481/4

❙♦♠❡ ✶✴✷✱ q✉❡ ✈♦❝ê ♠✉❧t✐♣❧✐❝♦✉✱ ❝♦♠ ✺✾✴✷✿ ♦ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦ ♠❡❞❡ ✸✵✳

P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ✐ss♦ é ❡①❛t❛♠❡♥t❡ ❛♣❧✐❝❛r ❛ ❢ór♠✉❧❛✿

x=

r

p

2

2

+q+p 2,

♦♥❞❡ x2

−px=q é ❛ ❡q✉❛çã♦ ❛ s❡r r❡s♦❧✈✐❞❛✳

◆♦s t❡①t♦s ♠❛t❡♠át✐❝♦s ❜❛❜✐❧ô♥✐♦s✱ ❛ ♠❛✐♦r✐❛ ❞♦s ♣r♦❜❧❡♠❛s r❡❧❛t✐✈♦s ❛ ❡q✉❛çõ❡s ❞❡ ✷♦ _{❣r❛✉ sã♦ ❞❛ ❢♦r♠❛✿}

x+y=b, x_·y=a ♦✉

x₋y =b, x_·y=a,

♦♥❞❡ sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❢♦r♠❛s ♥♦r♠❛✐s ❞❛s ❡q✉❛çõ❡s ❞❡ ✷♦ _{❣r❛✉ ❜❛❜✐❧ô♥✐❛s✳ ■ss♦}

s✉❣❡r❡ q✉❡ ♦s ❡s❝r✐❜❛s ❜❛❜✐❧ô♥✐♦s ✐♥✈❡st✐❣❛✈❛♠ ❛ r❡❧❛çã♦ ❡♥tr❡ ♣❡rí♠❡tr♦ ❡ ❛ ár❡❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ r❡t❛♥❣✉❧❛r✳

❯♠ ♦✉tr♦ t✐♣♦ ❞❡ ❡q✉❛çã♦ r❡s♦❧✈✐❞♦ ♣❡❧♦s ❜❛❜✐❧ô♥✐♦s é r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦s s✐st❡✲ ♠❛s ✭q✉❡ ❞ã♦ ♦r✐❣❡♠ ❛ ❡q✉❛çõ❡s ❞❡ ✷♦ _{❣r❛✉✮✿}

x2

+y2

=b, x+y=a,

x2

+y2

=b, x₋y=a ♦✉

x2

+y2

=b, x_·y=s

❆ ♠❛♥❡✐r❛ ❞❡ ♦s ♠❛t❡♠át✐❝♦s ❣r❡❣♦s ❛♣r❡s❡♥t❛r❡♠ s❡✉s r❡s✉❧t❛❞♦s s♦❜r❡ s♦❧✉✲ çõ❡s ❞❡ ❡q✉❛çõ❡s ❞❡ ✷♦ _{❣r❛✉✱ é ❣❡♦♠étr✐❝❛✱ ❝♦♠♦ ♥♦s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✱ ❡s❝r✐t♦s}

❡♠ ✸✵✵ ❛❈✳ ❆ ❢❡rr❛♠❡♥t❛ ❣❡♦♠étr✐❝❛ q✉❡ ♣❡r♠✐t❡ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞❡ ✷♦ _{❣r❛✉ é ❛}

❛♣❧✐❝❛çã♦ ❞❡ ár❡❛s✱ q✉❡ ♦❝✉♣❛ ✉♠ ❧✉❣❛r ✐♠♣♦rt❛♥t❡ ♥❛ ❣❡♦♠❡tr✐❛ ❣r❡❣❛✳

❋✐❣✉r❛ ✶✳✾✿ ❉✐♦❢❛♥t♦

❏á ❉✐♦❢❛♥t♦✱ q✉❡ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❣r❡❣♦ q✉❡ ✈✐✈❡✉ ❡♠ t♦r♥♦ ❞❡ ✷✺✵ ❛❈✳ ❞❡❞✐❝❛♥❞♦✲s❡ ❛♦ ❡st✉❞♦ ❞❛s ❡q✉❛çõ❡s ❡ ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ❢✉❣✐❛ ❞♦s ♠ét♦❞♦s ❣❡♦♠étr✐❝♦s ♥❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞❡ ✷♦ _{❣r❛✉✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❛❝❤❛r ❞♦✐s ♥ú♠❡✲}

r♦s ❞❡ ♠❛♥❡✐r❛ q✉❡ s❡✉ ♣r♦❞✉t♦ ❡ s✉❛ s♦♠❛ s❡❥❛♠ ✐❣✉❛✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ❞♦✐s ♥ú♠❡r♦s ❞❛❞♦s✱ ♦✉ s❡❥❛✱ x+y= b✱ xy =a ❢♦✐ s♦❧✉❝✐♦♥❛❞❛ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♦s ❜❛❜✐❧ô♥✐♦s✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❡ ♠♦str❛ ❝♦♠♦ r❡s♦❧✈❡r s✐st❡♠❛s✱ q✉❡ ❧❡✈❛♠ ❛ ❡q✉❛çõ❡s ❞❡ ✷♦ _{❣r❛✉✿}

x2 −y2

=a, x+y=b,

xy=a, x₋y=b,

x y =p,

x2

+y2

x+y =q.

❆s ❡q✉❛çõ❡s ❞❡ ✷♦ _{❣r❛✉ s✉r❣❡♠ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♥❛ ♠❛t❡♠át✐❝❛ ❍✐♥❞✉ ♥♦s}

s✉❧✈❛s✉tr❛s ✭♣❛rt❡s ❞❡ t❡①t♦s ❞❛s ♠❛✐s ❛♥t✐❣❛s ❡s❝r✐t✉r❛s ❞♦ ❍✐♥❞✉ís♠♦✱ ❡ ❝♦♥s✐❞❡✲ r❛❞❛s ❛s ú♥✐❝❛s ❢♦♥t❡s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛ ✐♥❞✐❛♥❛ ❞❡ss❡ ♣❡rí♦❞♦✮✱ s♦❜ ❛s ❢♦r♠❛s ax2

=c ❡ax2

+bx=c✱ s❡♠ ❛♣r❡s❡♥t❛r s♦❧✉çõ❡s✳ ◆♦ ♠❛♥✉s❝r✐t♦ ❇❛❦s❤❛❧✐

é ❛♣r❡s❡♥t❛❞♦ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ s♦❧✉çã♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ à ❢ór♠✉❧❛

x=

√

b2_{+ 4ac} −b

2a ,

♣❛r❛ ❛ ❡q✉❛çã♦ ax2

+bx₋c= 0✳

❇r❛♠❛❣✉♣t❛ ✭✺✾✽ ✲ ✻✻✺✮ ♠♦str❛ ❝♦♠♦ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ax2

+bx =c ❝♦♠ a✱ b ❡c ♣♦s✐t✐✈♦s✱ ✉t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛

x=

√

4ac+b2 −b

2a ♦✉

x=

s

ac+

b

2

2 − b₂

a

❆❧✲❑❤♦✇❛r✐③♠✐ ✭✼✽✵ ✲ ✽✺✵✮ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ♠❛t❡♠át✐❝♦ ♠✉ç✉❧♠❛♥♦ ❛ ❡s❝r❡✈❡r s♦❜r❡ ❛ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ✉s❛♥❞♦ ❛❧✲❥❛❜r ❡ ❛❧✲♠✉q❛❜❛❧❛✳ ❆❧✲❥❛❜r s✐❣♥✐✜❝❛ ❛❞✐❝✐♦✲ ♥❛r t❡r♠♦s ✐❣✉❛✐s ❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❡ ✉♠❛ ❡q✉❛çã♦✱ ❛✜♠ ❞❡ ❡❧✐♠✐♥❛r ♦s t❡r♠♦s ♥❡❣❛t✐✈♦s✳ ❏á ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ❛❧✲♠✉q❛❜❛❧❛ é ❛ r❡❞✉çã♦ ❞❡ t❡r♠♦s ♣♦s✐t✐✈♦s ♣♦r ♠❡✐♦ ❞❛ s✉❜tr❛çã♦ ❞❡ q✉❛♥t✐❞❛❞❡s ✐❣✉❛✐s ❞❡ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦✳

❋✐❣✉r❛ ✶✳✶✵✿ ❆❧✲❑❤♦✇❛r✐③♠✐

❱❡r❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ❝♦♠♦ ❆❧✲❑❤♦✇❛r✐③♠✐ r❡s♦❧✈✐❛ ✉♠❛ ❡q✉❛çã♦ ❞❡ ✷♦ _{❣r❛✉✿}

◗✉❛❧ ❞❡✈❡ s❡r ♦ q✉❛❞r❛❞♦ q✉❡✱ ❛✉♠❡♥t❛❞♦ ❞❡ ❞❡③ ❞❡ s✉❛s ♣ró♣r✐❛s r❛í③❡s✱ é ❡q✉✐✈❛✲ ❧❡♥t❡ ❛ tr✐♥t❛ ❡ ♥♦✈❡❄

❆ ❡q✉❛çã♦ ❡s❝r✐t❛ ❡♠ ❧✐♥❣✉❛❣❡♠ ❛❧❣é❜r✐❝❛ é ❞❛❞❛ ♣♦r✿ x2

+ 10x= 39✳

❙♦❧✉çã♦✿ ❚♦♠❡ ❛ ♠❡t❛❞❡ ❞♦ ♥ú♠❡r♦ ❞❡ r❛í③❡s✳ ■ss♦ ✈♦❝ê ♠✉❧t✐♣❧✐❝❛ ♣♦r ❡❧❡ ♣ró✲ ♣r✐♦✳ ❆❞✐❝✐♦♥❡ ✐ss♦ ❛ tr✐♥t❛ ❡ ♥♦✈❡✳ ❆❣♦r❛✱ t♦♠❡ ❛ r❛✐③ ❞✐ss♦✱ ❡ s✉❜tr❛✐❛ ❞❡❧❛ ❛ ♠❡t❛❞❡ ❞♦ ♥ú♠❡r♦ ❞❡ r❛í③❡s✳ ❖ r❡s✉❧t❛❞♦ é três✳ ■ss♦ é ❛ r❛✐③ ❞♦ q✉❛❞r❛❞♦ q✉❡ ✈♦❝ê ♣r♦❝✉r❛✈❛✳

■ss♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ ✉s❛r ❛ ❢ór♠✉❧❛

x=

s

b

2

2

+c₋ b

2 =

√

b2 _{+ 4c} −b

2

❇❤❛s❦❛r❛ ❆❦❛r✐❛ ✭✶✶✶✹ ✲ ✶✶✽✺✮✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❇❤❛s❦❛r❛ ■■✱ ♠♦str❛ ❝♦♠♦ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ax2

+bx=c✳ ❊❧❡ ♠✉❧t✐♣❧✐❝❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦

♣♦r a✿

(ax)2

+ (ax)b =ac

❊♠ s❡❣✉✐❞❛✱ é só ❝♦♠♣❧❡t❛r q✉❛❞r❛❞♦s✿

(ax)2

+ (ax)b+

b

2

2

=ac+

b

2

2

❋✐❣✉r❛ ✶✳✶✶✿ ❇❤❛s❦❛r❛

▼✉✐t♦s ♦✉tr♦s ♠❛t❡♠át✐❝♦s t❛♠❜é♠ tr❛❜❛❧❤❛r❛♠ ♥❛ t❡♥t❛t✐✈❛ ❞❡ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ r❡s♦❧✈❡r ✉♠❛ ❡q✉❛çã♦ ❞❡ ✷♦ _{❣r❛✉✳ ❆r✐❛❜❛t❛ ■ ✭✈✐✈❡✉ ❡♠ t♦r♥♦ ❞❡}

✹✼✻ ❞❈✳✮ ❝❤❡❣❛ ❛ ✉♠❛ ❡q✉❛çã♦ ❞❡ ✷♦ _{❣r❛✉ ❛ ♣❛rt✐r ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♣r♦❣r❡ssã♦}

❛r✐t♠ét✐❝❛✳ ❚❛❜✐t ❇❡♥ ◗✉rr❛ ✭✽✸✻ ✲ ✾✵✶✮ ❡s❝r❡✈❡✉ ✉♠ ♣❡q✉❡♥♦ tr❛t❛❞♦ s♦❜r❡ ❛ ✈❡r✐✜❝❛çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ á❧❣❡❜r❛ ♣♦r ❞❡♠♦♥str❛çõ❡s ❣❡♦♠étr✐❝❛s✱ ♠♦str❛♥❞♦ ❝♦♠♦ r❡s♦❧✈❡r ❡q✉❛çõ❡s x2

+mx=n✱ x2

+b=ax ❡x2

=ax+b✳

❊ss❡ ♣❡q✉❡♥♦ r❡s✉♠♦✱ ♥♦s ♠♦str❛ q✉❡✱ ❛♦ ❧♦♥❣♦ ❞♦s sé❝✉❧♦s✱ ♦s ♠ét♦❞♦s ❞❡ r❡s♦❧✈❡r ❛s ❡q✉❛çõ❡s ❞❡ ✷♦ _{❣r❛✉ ♠✉❞❛r❛♠✱ ❛té ❝❤❡❣❛r ♥❛ ❢ór♠✉❧❛ q✉❡ ❝♦♥❤❡❝❡♠♦s}

❤♦❥❡ ❝♦♠♦ ❋ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛✳ ❱❡r ♠❛✐s ❞❡t❛❧❤❡s ❞❛ ❤✐stór✐❛ ❞❛ ❡q✉❛çã♦ ❞❡ ✷♦

❣r❛✉ ❡♠ ❬✻❪✳

✶✳✸✳✷ ❘❛í③❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞♦ ✷

_❣r❛✉

❈♦♠♦ ✈✐♠♦s ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♠ét♦❞♦s ♣❛r❛ s♦❧✉❝✐♦♥❛r ❛s ❡q✉❛çõ❡s ❞♦ ✷♦ _❣r❛✉

r❡♠♦t❛ às ❝✐✈✐❧✐③❛çõ❡s ❞❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❝♦♠♦ ♦s ❜❛❜✐❧ô♥✐♦s ❡ ❡❣í♣❝✐♦s✳ ◆❡st❛ s❡çã♦ ✐r❡♠♦s ❞❡❞✉③✐r ❛ ❢❛♠♦s❛ ❢ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛ ✉t✐❧✐③❛♥❞♦ ✉♠ ♠ét♦❞♦ s✐♠♣❧❡s ❞❡ ❝♦♠♣❧❡t❛r q✉❛❞r❛❞♦s ❡♠ q✉❡ ❝♦♥s✐st❡ ❡♠ ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ♥✉♠❛ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡ q✉❡ ♥♦s ♣❡r♠✐t❛ ❞❡t❡r♠✐♥❛r ❛ s♦❧✉çã♦ ❞✐r❡t❛♠❡♥t❡✳ ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦✿

x2

−6x+ 8 = 0.

❊s❝r❡✈❡♥❞♦ ❛ ♠❡s♠❛ ❡q✉❛çã♦ ❝♦♠♦ s❡♥❞♦✿

x2₋6x=₋8.

❙♦♠❛♥❞♦ ✾ ❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦✱ ♦❜t❡♠♦s✿

x2

−6x+ 9 =₋8 + 9.

(x₋3)2

= 1.

❉❛í t❡♠♦s✿

x₋3 =√1 ou x₋3 = ₋√1.

P♦rt❛♥t♦✱ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ é ❞❛❞❛ ♣❡❧♦s s❡❣✉✐♥t❡s ✈❛❧♦r❡s ❞❡ x✿

x1 = 4 ou x2 = 2.

❉❡✜♥✐çã♦ ✶✳✹ ❆ ❡q✉❛çã♦ ❞♦ ✷♦ _{❣r❛✉ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❛✱❜ ❡ ❝ é ✉♠❛ ❡①♣r❡ssã♦ ❞❛}

❢♦r♠❛✿

ax2

+bx+c= 0,

♦♥❞❡ a₆= 0✱ b✱ c_∈R ❡ x é ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧ ❛ s❡r ❞❡t❡r♠✐♥❛❞❛✳

❱❛♠♦s ❛❣♦r❛✱ ❡♥❝♦♥tr❛r ❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ax2

+bx+c= 0✿

■s♦❧❛♥❞♦ ♦ t❡r♠♦ q✉❡ ♥ã♦ ❝♦♥té♠ ❛ ✈❛r✐á✈❡❧ x❞❛ ❡q✉❛çã♦✱ ♦❜t❡♠♦s✿

ax2

+bx =₋c,

❞✐✈✐❞✐♥❞♦ ❛❣♦r❛✱ ♦s ❞♦✐s ♠❡♠❜r♦s ♣♦r a✱ ♦❜t❡♠♦s✿

x2

+ b

ax=− c a.

❆❞✐❝✐♦♥❛♥❞♦ b2

/4a2 _{❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣❛r❛ ♦❜t❡r ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ❞♦}

❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ✐❣✉❛❧❞❛❞❡✳ ❆ss✐♠✱ ♦❜t❡♠♦s✿

x2

+ b

ax+ b2

4a2 =

b2

4a2 −

c a

(x+ b 2a)

2

= b

2 −4ac

4a2 .

P❛r❛ q✉❡ ❡①✐st❛ ❛❧❣✉♠ ♥ú♠❡r♦ r❡❛❧ s❛t✐s❢❛③❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ❞❡✈❡♠♦s t❡r q✉❡ b2

−4ac_≥0✱ ❥á q✉❡ ♦ t❡r♠♦s ❞❛ ❡sq✉❡r❞❛ ♥❛ ✐❣✉❛❧❞❛❞❡ é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ③❡r♦✳

❊①tr❛✐♥❞♦ ❛ r❛✐③ q✉❛❞r❛❞❛ q✉❛♥❞♦ b2

−4ac_≥0✱ t❡♠♦s ❛s s♦❧✉çõ❡s✿

x+ b 2a =

√

b2_−4ac

2a e x+

b

2a =−

√

b2_−4ac

2a .

❆ss✐♠✱ ♦❜t❡♠♦s ❛s s❡❣✉✐♥t❡s s♦❧✉çõ❡s✿

x= −b±

√

b2 −4ac

2a

❈❤❛♠❛♠♦s ❛ ❡①♣r❡ssã♦ b2

−4ac ❞❡ ❞✐s❝r✐♠✐♥❛♥t❡ ❞❛ ❡q✉❛çã♦ ax2

+bx+c= 0

❡ ❞❡♥♦t❛♠♦s ♣❡❧❛ ❧❡tr❛ ❣r❡❣❛ ∆ ✭▲ê✲s❡ ❞❡❧t❛✮✳ ❆♥❛❧✐s❛♥❞♦ ♦ ❞✐s❝r✐♠✐♥❛♥t❡✿

✐✮ ❙❡ ∆>0✱ ❡①✐st❡♠ ❞✉❛s s♦❧✉çõ❡s r❡❛✐s✳

✐✐✮ ❙❡∆ = 0✱ só ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ r❡❛❧✱ ♦✉ s❡❥❛✱ x1 =x2✳

✐✐✐✮ ❙❡∆<0✱ ♥ã♦ ❡①✐st❡ s♦❧✉çã♦ r❡❛❧✳

✶✳✹ ❉❡✜♥✐çã♦ ❞❛s ❝ô♥✐❝❛s ❝♦♠♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦

❯♠❛ s❡çã♦ ❝ô♥✐❝❛ ♦✉✱ s✐♠♣❧❡s♠❡♥t❡✱ ✉♠❛ ❝ô♥✐❝❛ é ✉♠❛ ❝✉r✈❛ ♦❜t✐❞❛ ❝♦rt❛♥❞♦✲s❡ q✉❛❧q✉❡r ❝♦♥❡ ❞❡ ❞✉❛s ❢♦❧❤❛s ♣♦r ✉♠ ♣❧❛♥♦ q✉❡ ♥ã♦ ♣❛ss❛ ♣❡❧♦ ✈ért✐❝❡✱ ❝❤❛♠❛❞♦ ❞❡ ♣❧❛♥♦ s❡❝❛♥t❡✳

❙❡ ♦ ♣❧❛♥♦ s❡❝❛♥t❡ é ♣❛r❛❧❡❧♦ ❛ ✉♠❛ ❣❡r❛tr✐③ ❞♦ ❝♦♥❡✱ ❛ ❝ô♥✐❝❛ é ✉♠❛ ♣❛rá❜♦❧❛ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✶✷✳

❋✐❣✉r❛ ✶✳✶✷✿ P❛rá❜♦❧❛✳

❙❡ ♦ ♣❧❛♥♦ s❡❝❛♥t❡ ♥ã♦ é ♣❛r❛❧❡❧♦ ❛ ✉♠❛ ❣❡r❛tr✐③ ❡ ❝♦rt❛ só ✉♠❛ ❞❛s ❞✉❛s ❢♦❧❤❛s ❞♦ ❝♦♥❡✱ ❛ ❝ô♥✐❝❛ é ✉♠❛ ❡❧✐♣s❡ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✶✸✳

❋✐❣✉r❛ ✶✳✶✸✿ ❊❧✐♣s❡✳

❙❡ ♦ ♣❧❛♥♦ s❡❝❛♥t❡ ♥ã♦ é ♣❛r❛❧❡❧♦ ❛ ♠❛ ❣❡r❛tr✐③ ❡ ❝♦rt❛ ❛♠❜❛s ❛s ❢♦❧❤❛s ❞♦ ❝♦♥❡✱ ❛ ❝ô♥✐❝❛ é ✉♠❛ ❤✐♣ér❜♦❧❡ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✶✹✳

❋✐❣✉r❛ ✶✳✶✹✿ ❍✐♣ér❜♦❧❡✳

❖❜s❡r✈❛çã♦✿ ◆♦ ❝❛s♦ ❞❡ ✉♠ ♣❧❛♥♦ q✉❡ ♣❛ss❛ ♣❡❧♦ ✈ért✐❝❡ ❞♦ ❝♦♥❡ ♦❜t❡♠♦s ✉♠ ♣♦♥t♦✱ ✉♠❛ r❡t❛ ♦✉ ✉♠ ♣❛r ❞❡ r❡t❛s ❝♦♥❝♦rr❡♥t❡s✳

❙ã♦ ❝♦♥❤❡❝✐❞♦s tr❛t❛❞♦s s♦❜r❡ ❛s s❡çõ❡s ❝ô♥✐❝❛s ❛♥t❡s ❞❛ é♣♦❝❛ ❞❡ ❊✉❝❧✐❞❡s✳ ◆❛ ❤✐stór✐❛ ❞❡ss❛s ❝✉r✈❛s✱ t❡♠♦s ❆♣♦❧ô♥✐♦ q✉❡ ♥❛s❝❡✉ ♥❛ ❝✐❞❛❞❡ ❞❡ P❡r❣❛ ♣♦r ✈♦❧t❛ ❞❡ ✷✻✷ ❛❈✳ s❡♥❞♦ ♦ ♠❛t❡♠át✐❝♦ q✉❡ ♠❛✐s ❡st✉❞♦✉ ❡ ❞❡s❡♥✈♦❧✈❡✉ ❛s s❡çõ❡s ❝ô♥✐❝❛s ❞❛ ❛♥t✐❣✉✐❞❛❞❡✳

❯♠ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❝♦♥s✐st❡ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ ✭♦✉ ❡s♣❛ç♦✮ q✉❡ ❣♦③❛♠ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣r♦♣r✐❡❞❛❞❡✳ ❆ss✐♠✱ ♦s ❧✉❣❛r❡s ❣❡♦♠étr✐❝♦s ♣♦❞❡♠ s❡r ❞❛❞♦s ♣♦r r❡t❛s ❡ ❝✉r✈❛s✳

❱❡r❡♠♦s à s❡❣✉✐r✱ ❛s ❞❡✜♥✐çõ❡s ❞❛s ❝ô♥✐❝❛s ❝♦♠♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✿

❛✮ ❊❧✐♣s❡✿ ➱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s P ❞♦ ♣❧❛♥♦ t❛✐s q✉❡ é ❝♦♥st❛♥t❡ ❛ s♦♠❛ d1 +d2 ❞❛s ❞✐stâ♥❝✐❛s d1 ❡ d2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ P ❛ ❞♦✐s ♣♦♥t♦s ✜①♦sF1 ❡

F2✱ ❝❤❛♠❛❞♦s ❢♦❝♦s ❞❛ ❡❧✐♣s❡ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✶✺✳

❋✐❣✉r❛ ✶✳✶✺✿ ❊❧✐♣s❡ ❝♦♠ s❡✉s ❢♦❝♦s F1 ❡ F2✳

❜✮ ❍✐♣ér❜♦❧❡✿ ➱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦sP ❞♦ ♣❧❛♥♦ t❛✐s q✉❡ é ❝♦♥st❛♥t❡ ♦ ♠ó❞✉❧♦ ❞❛ ❞✐❢❡r❡♥ç❛ |d1−d2|❞❛s ❞✐stâ♥❝✐❛s d1 ❡d2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡P ❛ ❞♦✐s

♣♦♥t♦s ✜①♦s F1 ❡ F2✱ ❝❤❛♠❛❞♦s ❢♦❝♦s ❞❛ ❤✐♣ér❜♦❧❡ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✶✻✳

❋✐❣✉r❛ ✶✳✶✻✿ ❍✐♣ér❜♦❧❡ ❝♦♠ s❡✉s ❢♦❝♦s F1 ❡ F2✳

❝✮ P❛rá❜♦❧❛✿ ➱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s P ❞♦ ♣❧❛♥♦ t❛✐s q✉❡ ❛ ❞✐stâ♥❝✐❛ d1 ❞❡P ❛ ✉♠ ♣♦♥t♦ ✜①♦ F✱ ❝❤❛♠❛❞♦ ❢♦❝♦ ❞❛ ♣❛rá❜♦❧❛✱ é ✐❣✉❛❧ à ❞✐stâ♥❝✐❛ d2 ❞❡P

❛ ✉♠❛ r❡t❛ ✜①❛ D✱ ❝❤❛♠❛❞❛ ❞✐r❡tr✐③ ❞❛ ♣❛rá❜♦❧❛ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✶✼✳

❋✐❣✉r❛ ✶✳✶✼✿ P❛rá❜♦❧❛ ❝♦♠ s❡✉ ❢♦❝♦ F ❡ r❡t❛ ❞✐r❡tr✐③ D✳

❉❡s❧♦❝❛♠❡♥t♦ ♣❛r❛❧❡❧♦

❊st✉❞❛r❡♠♦s ❛❣♦r❛ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s r❡❧❛❝✐♦♥❛♥❞♦ ❛ ❣❡♦♠❡tr✐❛ ❞♦ tr✐â♥❣✉❧♦ ❡ ❞♦s ♣♦❧í❣♦♥♦s ❝♦♥✈❡①♦s ❝♦♠ ❛s ❡q✉❛çõ❡s ❞❡ ✷♦ _{❣r❛✉✳ ❱❡r ❬✶❪✳}

✷✳✶ ❉❡s❧♦❝❛♠❡♥t♦ ♣❛r❛❧❡❧♦ ❛♦ tr✐â♥❣✉❧♦

∆

❉❡✜♥✐çã♦ ✷✳✶ ❉❛❞♦ ✉♠ tr✐â♥❣✉❧♦ ∆ = ∆(a, b, c)❡ ♥ú♠❡r♦ r❡❛❧r >0❝♦♥s✐❞❡r❛♠♦s

♦ ❝♦♥❥✉♥t♦s ❞♦s ♣♦♥t♦s p_∈R2 t❛✐s q✉❡

d(p,∆) =r,

♦♥❞❡ d(p,∆) ❞❡♥♦t❛ ❛ ❞✐stâ♥❝✐❛ ❞❡ p ❛♦ tr✐â♥❣✉❧♦ ∆ ❡ ❛ ♠❡s♠❛ é ❞❡✜♥✐❞❛ ❝♦♠♦

s❡♥❞♦ ❛ ♠❡♥♦r ❞✐stâ♥❝✐❛ ❞❡ p ❛♦s ✈ért✐❝❡s ❡ ❛♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛s s✉♣♦rt❡s ❞♦ tr✐â♥❣✉❧♦✳

❖ tr✐â♥❣✉❧♦ ∆ ❞❡❧✐♠✐t❛ ❞✉❛s r❡❣õ❡s ♥♦ ♣❧❛♥♦ ✭✐♥t❡r♥❛ ❡ ❡①t❡r♥❛✮ ❡ t❡♠♦s q✉❡ ♦

❝♦♥❥✉♥t♦ {p_∈R2

:d(p,∆) =r_} ♣♦ss✉✐ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s✱ ∆+

(r) ❡ ∆−_(r)✱

❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✷✳✶✱ ❛s q✉❛✐s sã♦ ❝❤❛♠❛❞❛s ❞❡ ❞❡s❧♦❝❛♠❡♥t♦s ♣❛r❛❧❡❧♦s ❞❡ ∆✳

❋✐❣✉r❛ ✷✳✶✿ ❈♦♥❥✉♥t♦s∆+

(r)❡ ∆−_{(r)❡q✉✐❞✐st❛♥t❡s ❛♦ tr✐â♥❣✉❧♦ ∆✳}

❆s ❡♥✈♦❧tór✐❛s ✭❡♥✈❡❧♦♣❡s✮✱ ❢♦r❛♠ ✐♥✐❝✐❛❧♠❡♥t❡ ❡st✉❞❛❞❛s ♣♦r ▲❡✐❜♥✐③ ❡ ❇❡r♥♦✉❧❧✐ ♥♦s ❝❤❛♠❛❞♦s ♣r♦❜❧❡♠❛s ❞❡ t❛♥❣ê♥❝✐❛✳ ❆s ❡♥✈♦❧tór✐❛s ❞❡ ❝✉r✈❛s ♣❧❛♥❛s sã♦ ✉t✐❧✐③❛❞❛s ♣❛r❛ ❞❡✜♥✐r ♥♦✈♦s t✐♣♦s ❞❡ ❝✉r✈❛s✳

❋✐❣✉r❛ ✷✳✷✿ ❆ ❢❛♠í❧✐❛ ❞❡ ❝ír❝✉❧♦s ❡ s✉❛ ❡♥✈♦❧tór✐❛ ✭❡♥✈❡❧♦♣❡✮✳

▲❡♠❛ ✷✳✷ ❙❡❥❛ ∆ = ∆(a, b, c) ✉♠ tr✐â♥❣✉❧♦✳ ❊♥tã♦✿

✐✮ P❛r❛ t♦❞♦ r >0 ♦ ❝♦♥❥✉♥t♦ ∆+

(r) é ❢♦r♠❛❞♦ ❞❡ s❡❣♠❡♥t♦ ❞❡ r❡t❛s ❡ ❛r❝♦s ❞❡

❝✐r❝✉♥❢❡rê♥❝✐❛s ✭❝ír❝✉❧♦s✮ ❡ ❡stá ❝♦♥t✐❞♦ ♥❛ r❡❣✐ã♦ ❡①t❡r✐♦r ❛♦ tr✐â♥❣✉❧♦ ∆✳

✐✐✮ P❛r❛ r >0 ♣❡q✉❡♥♦✱ ♦ ❝♦♥❥✉♥t♦ ∆−_(r)é ✉♠ tr✐â♥❣✉❧♦ s❡♠❡❧❤❛♥t❡ ❛ _∆ ❡ ❡stá

❝♦♥t✐❞♦ ♥❛ r❡❣✐ã♦ ✐♥t❡r♥❛ ❛♦ tr✐â♥❣✉❧♦ ∆✳

❉❡♠♦♥str❛çã♦✿ P❛r❛ ❝❛❞❛ ♣♦♥t♦ ❞❡ p _∈ ∆ ❝♦♥s✐❞❡r❛♠♦s ✉♠ ❝ír❝✉❧♦ ❝❡♥tr❛❞♦

❡♠ p ❡ r❛✐♦ r✱ ♦ q✉❛❧ ❞❡♥♦t❛♠♦s ♣♦r Cp(r)✳ ❖s ❝♦♥❥✉♥t♦s ∆+(r) ❡ ∆−(r) sã♦ ❛s

❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ❞♦ ❡♥✈❡❧♦♣❡ ❞❛ ❢❛♠í❧✐❛ ❞❡ ❝ír❝✉❧♦s {Cp(r) :p∈∆}✳

❖ ❝♦♥❥✉♥t♦ ∆+

(r)é ❢♦r♠❛❞♦ ❞❡ s❡❣♠❡♥t♦s ❞❡ r❡t❛s ♣❛r❛❧❡❧❛s ❛♦s ❧❛❞♦s ❞♦ tr✐â♥✲

❣✉❧♦ ∆ ❡ ❞❡ ❛r❝♦s ❞❡ ❝ír❝✉❧♦s q✉❡ ❢❛③ ❛ ❝♦♥❝♦r❞â♥❝✐❛ ❝♦♠ ❡st❡s s❡❣♠❡♥t♦s✳ ❱❡❥❛ ❛

✜❣✉r❛ ✷✳✸✳

❋✐❣✉r❛ ✷✳✸✿ ❊♥✈❡❧♦♣❡ ❞❡ ❝ír❝✉❧♦s q✉❡ ❞❡✜♥❡♠ ♦s ❝♦♥❥✉♥t♦s ∆+

(r) ❡ ∆−_{(r)✱ ❡q✉✐❞✐s✲}

t❛♥t❡s ❛♦ tr✐â♥❣✉❧♦ ∆✳

❏á ♦ ❝♦♥❥✉♥t♦ ∆−_{(r) é ❢♦r♠❛❞♦ s♦♠❡♥t❡ ❞❡ s❡❣♠❡♥t♦s ❞❡ r❡t❛s ❡ ❞❡✜♥❡ ✉♠}

tr✐â♥❣✉❧♦✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✷✳✸✳ ❖❜s❡r✈❛♠♦s q✉❡ ♦s ✈ért✐❝❡s ❞❡∆−_{(r)❡stã♦ ♥❛s}

❜✐ss❡tr✐③❡s ❞♦s â♥❣✉❧♦s ❞❡ ∆ ❡ ❛ ❞✐stâ♥❝✐❛ r ❞♦s ❧❛❞♦s ❞❡ ∆✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛

✷✳✹✳ ▲♦❣♦✱ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ s❡♠❡❧❤❛♥ç❛s ❞❡ tr✐â♥❣✉❧♦s ❝♦♥❝❧✉✐✲s❡ q✉❡ ∆−_{(r) é}

s❡♠❡❧❤❛♥t❡ ❛ ∆✳

❋✐❣✉r❛ ✷✳✹✿ ❈♦♥❥✉♥t♦∆−_{(r)♥❛ ✈✐③✐♥❤❛♥ç❛ ❞♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ ∆✳}

❖❜s❡r✈❛çã♦✿ ❆ ♦♣❡r❛çã♦ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ♣❛r❛❧❡❧♦ ♥ã♦ é s✐♠étr✐❝❛✳ ❖❜s❡r✈❡ q✉❡ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ♣❛r❛❧❡❧♦ ❞❡ ∆−_{(r) ♥ã♦ t❡♠∆ ❝♦♠♦ ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s}

❞♦ ❡♥✈❡❧♦♣❡ ❛ss♦❝✐❛❞♦✳

Pr♦♣♦s✐çã♦ ✷✳✸ ❙❡❥❛A❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣❡❧♦ tr✐â♥❣✉❧♦∆❡s= (a+b+c)/2

♦ s❡✉ s❡♠✐ ♣❡rí♠❡tr♦✳ ❊♥tã♦✿

✐✮ ❆ ár❡❛ ❞❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ∆+

(r) é A+ 2sr+πr2_✳

✐✐✮ ❆ ár❡❛ ❞❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r∆−_{(r) é A}

−2sr+s

2

Ar

2✳

❖❜s❡r✈❛çã♦✿ ❉❡♥♦t❛r❡♠♦s ♣♦r A+

(r)❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r∆+

(r)❡ ♣♦r

A−_{(r) ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ∆}−_(r)✳

❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❛♥❞♦ ❛s ✜❣✉r❛s ✷✳✶ ❡ ✷✳✺✱ ✈❡♠♦s q✉❡ ♦ ❛❝rés❝✐♠♦ ❡♠ ár❡❛ ❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣❡❧♦ tr✐â♥❣✉❧♦ ∆ ❝♦rr❡s♣♦♥❞❡ ❛ ❞❡ três r❡tâ♥❣✉❧♦s ❝✉❥❛s ❜❛s❡s

sã♦ ♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ∆ ❡ ❛❧t✉r❛ r ❡ ❞❡ três ❛r❝♦s ❝✐r❝✉❧❛r❡s ❞❡ r❛✐♦ r ❡ â♥❣✉❧♦s ❝❡♥tr❛✐s ✐❣✉❛✐s ❛ π₋α✱π₋β ❡π₋γ✱ ♦✉ s❡❥❛✱

A+

(r) = A+r_·a+r_·b+r_·c+ 1 2r

2

(π₋α) + 1 2r

2

(π₋β) + 1 2r

2

(π₋γ)

= A+r(a+b+c) + 1 2r

2

(π₋α+π₋β+π₋γ).

❈♦♠♦ α+β +γ = π ❡ a+b+c = 2s✱ ♣♦rt❛♥t♦ ❛ ár❡❛ ❞❡❧✐♠✐t❛❞❛ ♣❡❧❛ r❡❣✐ã♦ ∆+

(r)é ❞❛❞❛ ♣♦r✿

A+

(r) = A+ 2sr+πr2 ✭✷✳✶✮

❋✐❣✉r❛ ✷✳✺✿ ❈♦♥❥✉♥t♦∆+

(r) ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ✈ért✐❝❡ ❞♦ tr✐â♥❣✉❧♦∆✳

❖❜s❡r✈❛♥❞♦ ❛s ✜❣✉r❛s ✷✳✶ ❡ ✷✳✹✱ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ∆−_{(r) é ❞❛❞❛}

♣♦r✿

A−_{(r) = A}

−r(a₋hb −hc)−r(b−ha−hc)−(c−ha−hb)−rha−rhb−rhc

= A₋r(a+b+c) +r(ha+hb+hc),

♦♥❞❡ ❞❡♥♦t❛♠♦s ♣♦r ha✱ hb ❡ hc ♦s ❧❛❞♦s ❞♦s tr✐â♥❣✉❧♦s ❢♦r♠❛❞♦s ♥♦s ✈ért✐❝❡s✳

❖❜s❡r✈❛♠♦s q✉❡ ha = rcot(α2)✱ hb = rcot(

β

2) ❡ hc = rcot(

γ

2)✳ ▲♦❣♦ ❛ ár❡❛ ❞❡

∆−_{(r)é ❞❛❞❛ ♣♦r✿}

A−_{(r) =A}

−(a+b+c)r+r2

[cot(α

2) + cot(

β

2) + cot(

γ

2)].

❉❡ ✶✳✺ t❡♠♦s q✉❡

A−(r) =A₋2sr+r2 ·

"s

s(s₋a) (s₋b)(s₋c) +

s

s(s₋b) (s₋a)(s₋c)+

s

s(s₋c) (s₋a)(s₋b)

#

A−(r) = A₋2sr+r2

· s

2

p

s(s₋a)(s₋b)(s₋c)

P♦rt❛♥t♦✱ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r∆−_{(r)é ❞❛❞❛ ♣♦r✿}

A−_{(r) = A−2sr+} s 2

Ar

2 _{✭✷✳✷✮}

❊①❡♠♣❧♦✿ ❙❡❥❛ ♦ tr✐â♥❣✉❧♦ ∆(a, b, c) ❝♦♠ ♠❡❞✐❞❛sa = 5✱ b = 5 ❡c= 6✳ ❈❛❧❝✉❧❛r

❛s ár❡❛s ❞❛s r❡❣✐õ❡s ❞❡❧✐♠✐t❛❞❛ ♣♦r ∆+

(r) ❡ ∆−_{(r)✱ ♣❛r❛ r= 1/2✳}

❈❛❧❝✉❧❛♥❞♦ ♦ s❡♠✐✲♣❡rí♠❡tr♦s ❝♦♠ ❛s ♠❡❞✐❞❛s ❞❡a✱ b ❡ c✱ t❡♠♦s✿

s= (5 + 5 + 6)/2 = 8

❆❣♦r❛ ❝❛❧❝✉❧❛♥❞♦ ❛ ár❡❛ A ❞♦ tr✐â♥❣✉❧♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ❞❡ ❍❡r♦♥ ✭✶✳✶✵✮✱ t❡♠♦s✿

A=p8(8₋5)(8₋5)(8₋6) =√8_·3_·3_·2 = 12

❙✉❜st✐t✉✐♥❞♦ ❛s ✈❛❧♦r❡s ❞❡ s ❡ A ❡♠ ✭✷✳✶✮ ❡ ✭✷✳✷✮✱ t❡♠♦s ❛s ár❡❛s ❞❛s r❡❣✐õ❡s ❞❡❧✐♠✐t❛❞❛ ♣♦r ∆+

(r) ❡ ∆−_{(r)✱ ❡♠ ❢✉♥çã♦ ❞❡r}

A+

(r) = 12 + 16r+πr2

❡

A−_{(r) = 12}

−16r+ 16 3 r

2

❋❛③❡♥❞♦ ❛ s✉❜st✐t✉✐çã♦ ❞♦ r ❞❛❞♦✱ t❡♠♦s✿

A+

1 2

= 12 + 16_· 1 2 +π

1 2

2

= 20 +π 4 ❡ A− 1 2

= 12₋16_· 1 2+ 16 3 1 2 2

= 4 + 4 3 =

16 3 .

P♦rt❛♥t♦✱ A+

1 2

= 20 + π

4 ❡ A

❈♦r♦❧ár✐♦ ✷✳✸✳✶ ❖ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ✐♥s❝r✐t♦ ❛ ✉♠ tr✐â♥❣✉❧♦ ∆(a, b, c) é r= A

s✳

❉❡♠♦♥str❛çã♦✿ P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✸ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ∆−_{(r) é ✐❣✉❛❧ ❛}

A−(r) =A₋2sr+s

2

Ar

2

= (A−sr)

2

A .

❚❡♠♦s q✉❡ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ∆−_{(r) é ✐❣✉❛❧ ❛ ③❡r♦ q✉❛♥❞♦ ∆}−_(r)

r❡❞✉③✐r ❛ ✉♠ ♣♦♥t♦✱ ❞✐❣❛♠♦s p0✳ P❡❧❛ ❞❡✜♥✐çã♦ p0 é ❡q✉✐❞✐st❛♥t❡ ❛♦s três ❧❛❞♦s ❞❡

∆ ❡ é ♦ ✐♥❝❡♥tr♦ ❛ss♦❝✐❛❞♦✳ ❋❛③❡♥❞♦ A−_{(r) = 0✱ ♦✉ s❡❥❛✱}

(A₋sr)2

A = 0.

❚❡♠♦s✱ ♣♦rt❛♥t♦ A=sr ❝♦♠ r s❡♥❞♦ ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ✐♥s❝r✐t♦✳

❊①❡♠♣❧♦✿ ❈❛❧❝✉❧❛r ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ✐♥s❝r✐t♦ ❛ ✉♠ tr✐â♥❣✉❧♦∆(a, b, c)❝♦♠ ♠❡❞✐❞❛s

a= 5✱ b= 5 ❡ c= 6✳

❱✐♠♦s✱ ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ q✉❡ ❛ ár❡❛ A ❡ ♦ s❡♠✐✲♣❡rí♠❡tr♦ s ❞♦ tr✐â♥❣✉❧♦ ∆

sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ 12❡ 8✳

❙✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❞❡A ❡ s ❡♠ ✭✷✳✷✮✱ t❡♠♦s✿

A−(r) = 12₋2_·8_·r+ 8

2

12r

2

= 12₋16r+16 3 r

2

= (12−8r)

2

12

❆❣♦r❛✱ ❢❛③❡♥❞♦ A−_{(r) = 0✱ ♦✉ s❡❥❛✱}

(12₋8r)2

12 = 0

12₋8r = 0,

❡ ♣♦rt❛♥t♦

r= 3 2.