COORDENADAS BARICÊNTRICAS E APLICAÇÕES

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❆❇❈

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❡str❛❞♦ Pr♦❢✐ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦

❘❡❣✐s ❑❛♣✐t❛♥♦✈❛s

❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ❡ ❆♣❧✐❝❛çõ❡s

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❡str❛❞♦ Pr♦❢✐ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦

❘❡❣✐s ❑❛♣✐t❛♥♦✈❛s

❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ❡ ❆♣❧✐❝❛çõ❡s

❚r❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞♦ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡s✲ tr❡ ❡♠ ▼❛t❡♠át✐❝❛✱ s♦❜ ♦r✐❡♥t❛çã♦ ❞♦ Pr♦❢❡ss♦r ❉♦✉t♦r ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛✳

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣♦r t✉❞♦✳

❆♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✭P❘❖❋▼❆❚✮✱ à ❈❆P❊❙ ♣❡❧♦ ❛✉①í❧✐♦ ❝♦♥❝❡❞✐❞♦✱ à ❯❋❆❇❈ ❡ s❡✉s ♣r♦❢❡ss♦r❡s✱ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛✱ ❡ ❛♦s ❝♦❧❡❣❛s ❞❡ t✉r♠❛ ❞❡ ♠❡str❛❞♦✳

✈✐✐✐

❘❡s✉♠♦

■♥s♣✐r❛❞♦s ♣❡❧♦s tr❛❜❛❧❤♦s ❞❡ ❏✳ ❇♦rt♦❧♦ss✐ ❬✶❪ ❡ ❋✳ ❈❛♣✐tá♥ ❬✷❪✱ ❡st✉❞❛♠♦s ✉♠ ♠ét♦❞♦ ♣❛r❛ ❝❛❧❝✉❧❛r ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ✉♠ ♣♦♥t♦ ❡♠R2✱ ❛ ♣❛rt✐r ❞♦s ✈ért✐❝❡s ❞❡ ✉♠ ❞❛❞♦ tr✐â♥❣✉❧♦✳ ❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛ ♣❛r❛ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✱ ❛♣❧✐❝❛♠♦s ❡st❡ ♠ét♦❞♦ ♣❛r❛ ♣♦♥t♦s ♥♦tá✈❡✐s ❞♦ tr✐â♥❣✉❧♦✳ ❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♥✇❛②✱ ❝❛❧❝✉❧❛♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞♦ P♦♥t♦ ❞❡ ❋❡r♠❛t ❞♦ tr✐â♥❣✉❧♦✳

P❛❧❛✈r❛s✲❈❤❛✈❡

✐①

❆❜str❛❝t

❇❛s❡❞ ♦♥ t❤❡ ♣❛♣❡rs ♦❢ ❏✳ ❇♦rt♦❧♦ss✐ ❬✶❪ ❛♥❞ ❋✳ ❈❛♣✐tá♥ ❬✷❪✱ ✐♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② ❛ ♠❡t❤♦❞ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❜❛r②❝❡♥tr✐❝ ❝♦♦r❞✐♥❛t❡s ♦❢ ❛ ♣♦✐♥t ✐♥ R2 ❢r♦♠ t❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❣✐✈❡♥ tr✐❛♥❣❧❡✳ ❇② ✉s✐♥❣ ❈❡✈❛✬s t❤❡♦r❡♠ ❢♦r ❜❛r②❝❡♥tr✐❝ ❝♦♦r❞✐♥❛t❡s ✇❡ ❛♣♣❧② t❤❡ ♠❡t❤♦❞ t♦ ♥♦t❛❜❧❡ ♣♦✐♥ts ♦❢ t❤❡ tr✐❛♥❣❧❡✳ ❇② ✉s✐♥❣ ❈♦♥✇❛②✬s t❤❡♦r❡♠ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ❜❛r②❝❡♥tr✐❝ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❋❡r♠❛t P♦✐♥t ♦❢ t❤❡ tr✐❛♥❣❧❡✳

❑❡②✇♦r❞s

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ✸

✷✳✶ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ❡ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✷ ❊q✉❛çã♦ ❞❡ ❘❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸ ❈❡✈✐❛♥❛s ❡ ❚r❛ç♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✹ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛ ♣❛r❛ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✸ P♦♥t♦s ◆♦tá✈❡✐s ❞❡ ✉♠ ❚r✐â♥❣✉❧♦ ✷✶

✸✳✶ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ❞♦ ■♥❝❡♥tr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✷ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ❞♦ ❈✐r❝✉♥❝❡♥tr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✸ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ❞♦ ❖rt♦❝❡♥tr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✹ ❚❡♦r❡♠❛ ❞❡ ❈♦♥✇❛② ❡ ♦ P♦♥t♦ ❞❡ ❋❡r♠❛t ✷✾

✹✳✶ ❆ ❋ór♠✉❧❛ ❞❡ ❈♦♥✇❛② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✷ ❖ P♦♥t♦ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✸ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ❞♦ P♦♥t♦ ❞❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✺ ❆t✐✈✐❞❛❞❡s ♣❛r❛ ❛ s❛❧❛ ❞❡ ❛✉❧❛ ✹✸

✺✳✶ ❆t✐✈✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✻ ❈♦♥❝❧✉sã♦ ✺✸

❇✐❜❧✐♦❣r❛✜❛ ✺✺

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ❈♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ✉♠ ♣♦♥t♦ P✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ ➪r❡❛s ❢♦r♠❛❞❛s ♣❡❧♦ ♣♦♥t♦ P ❡ ♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ ABC✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✸ ❙✐♥❛❧ ❞❛ ❝♦♦r❞❡♥❛❞❛ u✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✹ ❙✐♥❛❧ ❞❛ ❝♦♦r❞❡♥❛❞❛ v✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✺ ❙✐♥❛❧ ❞❛ ❝♦♦r❞❡♥❛❞❛ w✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✻ ❙✐♥❛✐s ❞❛s ❝♦♦r❞❡♥❛❞❛s u✱ v ❡w✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✼ ❈♦♥str✉çã♦ ❈❛rt❡s✐❛♥❛ ❞♦ ❊①❡♠♣❧♦ ✷✳✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✽ ❈♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❞❡ ✈ért✐❝❡s✱ ♣♦♥t♦s ♠é❞✐♦s ❡ ❜❛r✐❝❡♥tr♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✾ ❈❡✈✐❛♥❛s ❡ tr❛ç♦s ❞♦ ♣♦♥t♦ P✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✵ ❖ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛ ♣❛r❛ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✶ ■♥❝❡♥tr♦ ✭I✮ ❞❡ ✉♠ tr✐â♥❣✉❧♦ABC✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✷ ❈✐r❝✉♥❝❡♥tr♦ ✭O✮ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ABC✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✸ ❖rt♦❝❡♥tr♦ ✭H✮ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ABC✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✶ ➶♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ABC✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✹✳✷ P ❡①t❡r♥♦ ❛♦ tr✐â♥❣✉❧♦ ABC✱ ❡ s❡✉s â♥❣✉❧♦s ❝♦♠ ♦ ❧❛❞♦ BC✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✸ ❚❡♦r❡♠❛ ❞❡ ❈♦♥✇❛② ♣❛r❛ três ♣♦♥t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✹ ❘♦t❛çã♦ ❞♦ △P AB ❡♠ 60o_{✱ ❝♦♠ ❡✐①♦ ❡♠ B✱ ❡♥❝♦♥tr❛♥❞♦ ♦}

△C′_P′_B✳ ✳ ✳ ✳ ✸✸ ✹✳✺ ❆ s♦❧✉çã♦ ♣❛r❛ ♦ P♦♥t♦ ❞❡ ❋❡r♠❛t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✻ ❯♠❛ ♣♦ssí✈❡❧ ❝♦♥str✉çã♦ ♣❛r❛ ♦ P♦♥t♦ ❞❡ ❋❡r♠❛t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✼ ❖ ♣♦♥t♦ P ✏❡♥①❡r❣❛✑ ♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ❝♦♠ â♥❣✉❧♦sα✱ β ❡ γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✽ ❈♦♥str✉çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♣♦r B✱ P ❡C✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✾ ❉❡t❡r♠✐♥❛♥❞♦ ♦s ♣♦♥t♦s A′✱ _B′ ❡ _C′✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❛❜♦r❞❛r❡♠♦s ♦ t❡♠❛ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s✱ ❝r✐❛❞♦ ♣♦r ❆✉❣✉st✉s ❋❡r❞✐♥❛♥❞ ▼ö❜✐✉s ✭✶✼✾✵✲✶✽✻✽✮✱ ❡ ❞❡s❝r✐t♦ ♥♦ s❡✉ ❧✐✈r♦ ❉❡r ❇❛r②❝❡♥t✐s❝❤❡ ❈❛❧❝✉❧ ❞❡ ✶✽✷✼✳ ❊st❛s ❝♦♦r❞❡♥❛❞❛s s❡rã♦ ❝❛❧❝✉❧❛❞❛s ♣❛r❛ ❛❧❣✉♥s ♣♦♥t♦s ♥♦tá✈❡✐s ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ❝♦♠♦ ✈ért✐❝❡s✱ ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s✱ ❜❛r✐❝❡♥tr♦✱ ✐♥❝❡♥tr♦✱ ♦rt♦❝❡♥tr♦✱ ❝✐r❝✉♥❝❡♥tr♦✱ ❛❧é♠ ❞♦ P♦♥t♦ ❞❡ ❋❡r♠❛t✳ ❱❡r❡♠♦s q✉❡ ♦ ✉s♦ ❞❡ss❛s ❝♦♦r❞❡♥❛❞❛s ❢❛❝✐❧✐t❛♠ ♦s ❝á❧❝✉❧♦s ❡♥✈♦❧✈❡♥❞♦ t❛✐s ♣♦♥t♦s✳

❆ ❞✐ss❡rt❛çã♦ ❡stá ❞✐✈✐❞✐❞❛ ❡♠ ✺ ❝❛♣ít✉❧♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✷✱ ❞❡✜♥✐r❡♠♦s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ✉♠ ♣♦♥t♦ ♥♦ ♣❧❛♥♦ ❛ ♣❛rt✐r ❞♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❞❡ r❡❢❡rê♥❝✐❛✳ ❆♣r❡s❡♥t❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛ ♣❛r❛ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✳

◆♦ ❈❛♣ít✉❧♦ ✸✱ ✉s❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛ ♣❛r❛ ❝❛❧❝✉❧❛r ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥✲ tr✐❝❛s ❞♦s ♣♦♥t♦s ♥♦tá✈❡✐s ❞❡ ✉♠ tr✐â♥❣✉❧♦✿ ❜❛r✐❝❡♥tr♦✱ ✐♥❝❡♥tr♦✱ ❝✐r❝✉♥❝❡♥tr♦ ❡ ♦rt♦❝❡♥tr♦✳

◆♦ ❈❛♣ít✉❧♦ ✹✱ s❡rá ❞❡♠♦♥str❛❞♦ ✉♠ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥✲ tr✐❝❛s✱ q✉❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♥✇❛②✳ ❈♦♠ ❡❧❡✱ ❝❛❧❝✉❧❛r❡♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞♦ P♦♥t♦ ❞❡ ❋❡r♠❛t ❞❡ ✉♠ tr✐â♥❣✉❧♦✳

◆♦ ❈❛♣ít✉❧♦ ✺✱ ❛♣❧✐❝❛r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡①♣❧♦r❛❞♦s ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ❝♦♠♦ ❛t✐✈✐❞❛❞❡s ♣❛r❛ s❛❧❛ ❞❡ ❛✉❧❛✳ P❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❛s ❛t✐✈✐❞❛❞❡s✱ s✉❣❡r✐♠♦s ♦ ✉s♦ ❞❡ ❛❧❣✉♠ s♦❢t✇❛r❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛✱ ❝♦♠♦ ♦ ●❡♦❣❡❜r❛✱ q✉❡ ❢♦✐ ✉t✐❧✐③❛❞♦ ♥❡st❛ ❞✐ss❡rt❛çã♦❀ ❡ t❛♠❜é♠ ✉♠ s♦❢t✇❛r❡ ❞❡ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ♣❛r❛ ♦s ❝á❧❝✉❧♦s q✉❡ ❞❡t❡r♠✐♥❛rã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❡ ❝❛rt❡s✐❛♥❛s✳ ❊st❛ é ✉♠❛ ❡①❝❡❧❡♥t❡ ♦♣♦rt✉♥✐❞❛❞❡ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❛ ❤❛❜✐❧✐❞❛❞❡ ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ✉s❛♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ❡ ❛❧❣é❜r✐❝♦✳

❈❛♣ít✉❧♦ ✷

❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s

◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐r❡♠♦s ❡ ❞✐s❝✉t✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛ ❜❛r✐❝ê♥tr✐❝❛ ♣❛r❛ ✉♠ ♣♦♥t♦ ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ❡♠ ❢✉♥çã♦ ❞♦s ✈ért✐❝❡s ❞❡ ✉♠ ❞❛❞♦ tr✐â♥❣✉❧♦ ❞❡ r❡❢❡rê♥❝✐❛✳ ❆♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❡st❡ t❡♠❛✱ ❝♦♠♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛ ♣❛r❛ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✳

✷✳✶ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ❡ ❊①❡♠♣❧♦s

Pr✐♠❡✐r❛♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛ ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ✉♠ ♣♦♥t♦✳

❉❡✜♥✐çã♦ ✷✳✶ ❙❡❥❛♠ A✱ B ❡ C ♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦ _△ABC ❡ P ✉♠ ♣♦♥t♦ ❞♦ ♣❧❛♥♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s (x, y)✳ ❉✐③❡♠♦s q✉❡ u✱ v ❡ w sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐✲ ❝ê♥tr✐❝❛s ❞❡ P✱ ❡♠ r❡❧❛çã♦ ❛♦ tr✐â♥❣✉❧♦ _△ABC✱ s❡

P = (x, y) = uA+vB+wC

u+v +w .

■st♦ é✱ s❡ ♦ ♣♦♥t♦P ♣♦❞❡ s❡r ♦❜t✐❞♦ ❝♦♠♦ ♠é❞✐❛ ♣♦♥❞❡r❛❞❛ ❞♦s ✈ért✐❝❡sA✱B ❡C ❝♦♠ ♣❡s♦su✱ v ❡w✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦♥❞❡ u✱ v ❡w sã♦ ♥ú♠❡r♦s r❡❛✐s t❛✐s q✉❡ u+v+w₆= 0✳

❉❡st❛ ♠❛♥❡✐r❛✱ ♦ ♣♦♥t♦ P ♣❛ss❛ ❛ s❡r ✐❞❡♥t✐✜❝❛❞♦ ♣♦r ❡ss❡s ♣❡s♦s ❡✱ ♥❡st❡ ❝❛s♦✱ ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦✿

P = (u:v :w).

✹ ❈❆P❮❚❯▲❖ ✷✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙

❋✐❣✉r❛ ✷✳✶✿ ❈♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ✉♠ ♣♦♥t♦ P✳

❯♠❛ q✉❡stã♦ q✉❡ s✉r❣❡ ♥❛t✉r❛❧♠❡♥t❡ ❛ ♣❛rt✐r ❞❛ ❉❡✜♥✐çã♦ ✷✳✶ é s❡♠♣r❡ s❡rá ♣♦ssí✈❡❧ ♦❜t❡r ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ✉♠ ♣♦♥t♦✳ ❙❡❥❛♠ A = (xa, ya)✱ B = (xb, yb) ❡

C = (xc, yc) ❛s ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❞♦s ✈ért✐❝❡s A✱ B ❡ C ❞♦ tr✐â♥❣✉❧♦ △ABC ✱ ❡

P = (xp, yp) ❛s ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❞❡ ✉♠ ♣♦♥t♦ ❞♦ ♣❧❛♥♦✳ P❡❧❛ ❉❡✜♥✐çã♦ ✷✳✶✱ ❛s

❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡P ❞❡✈❡♠ s❛t✐s❢❛③❡r ❛ r❡❧❛çã♦ P = uA+vB+wC

u+v+w =

u(xa, ya) +v(xb, yb) +w(xc, yc)

u+v+w = (xp, yp). ✭✷✳✶✮ ❖❜t❡♠♦s ♦ s❡❣✉✐♥t❡ s✐st❡♠❛✿

(

u(xa−xp) +v(xb−xp) +w(xc−xp) = 0

u(ya−yp) +v(yb−yp) +w(yc −yp) = 0.

✭✷✳✷✮ ✭✷✳✸✮ P❛r❛ ❢❛❝✐❧✐t❛r ❛s ❝♦♥t❛s✱ ❢❛r❡♠♦s ❛s s❡❣✉✐♥t❡s s✉❜st✐t✉✐çõ❡s✿

a = xa−xp, b=xb−xp, c=xc−xp, ✭✷✳✹✮

d = ya−yp, e=yb−yp, f =yc−yp.

❞❡ ♠♦❞♦ q✉❡ ♦ s✐st❡♠❛ ❛♥t❡r✐♦r s❡ r❡❡s❝r❡✈❡ ❝♦♠♦

ua+vb+wc= 0

ud+ve+wf = 0.

✭✷✳✺✮ ✭✷✳✻✮ P❡❧❛ ❉❡✜♥✐çã♦ ✷✳✶✱ ✈❡♠♦s q✉❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞♦s ✈ért✐❝❡s A✱ B ❡ C ❞♦ tr✐â♥❣✉❧♦ △ABC ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❢❛❝✐❧♠❡♥t❡ ❝♦♠♦

✷✳✶✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ❊ ❊❳❊▼P▲❖❙ ✺ ❆❧é♠ ❞✐st♦✱ s❡ P ♣❡rt❡♥❝❡ à r❡t❛ s✉♣♦rt❡ ❞♦ ❧❛❞♦ AB✱ ♦✉ s❡❥❛✱ s❡ P✱ A ❡ B ❡st✐✈❡r❡♠

❛❧✐♥❤❛❞♦s✱ ❡♥tã♦

xp yp 1

xa ya 1

xb yb 1

= 0.

❉❡s❡♥✈♦❧✈❡♥❞♦✲s❡ ❡st❡ ❞❡t❡r♠✐♥❛♥t❡✱ ♦❜t❡♠♦s ae₋bd = 0✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♦ ❛❧✐♥❤❛✲

♠❡♥t♦ ❞❡P✱A ❡C ❡q✉✐✈❛❧❡ à ❝♦♥❞✐çã♦cd₋af = 0✱ ❡ ♦ ❛❧✐♥❤❛♠❡♥t♦ ❞❡P✱B ❡C ❡q✉✐✈❛❧❡ à ❝♦♥❞✐çã♦ bf ₋ce= 0✳

❙✉♣♦♥❤❛ q✉❡ P ♥ã♦ s❡❥❛ ✉♠ ✈ért✐❝❡ ❞♦ tr✐â♥❣✉❧♦ _△ABC ❡ ♥ã♦ ♣❡rt❡♥ç❛ às r❡t❛s s✉♣♦rt❡s ❞♦s ❧❛❞♦s ❞❡st❡ tr✐â♥❣✉❧♦✳

❉❡ ✭✷✳✺✮✱ ♦❜t❡♠♦s

u= −vb−wc

a ✭✷✳✼✮

❡ s✉❜st✐t✉✐♥❞♦✲s❡ ✭✷✳✼✮ ❡♠ ✭✷✳✻✮✱ t❡♠♦s

(ae₋bd).v =w.(cd₋af), ✭✷✳✽✮ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡

v =w.cd−af

ae₋bd. ✭✷✳✾✮

❙✉❜st✐t✉✐♥❞♦✲s❡ ✭✷✳✾✮ ❡♠ ✭✷✳✼✮✱ ♦❜t❡♠♦s

u=₋

w.(cd−af)

(ae₋bd)

.b a −w.

c a =w.

abf ₋bcd₋ace+bcd

(ae₋bd).a

=w.

bf ₋ce ae₋bd

.

P♦r ✭✷✳✽✮✱ t❡♠♦s q✉❡ s❡ P✱ A ❡ B ❡st✐✈❡ss❡♠ ❛❧✐♥❤❛❞♦s✱ t❡rí❛♠♦sw= 0✱ ❞❡ ♠♦❞♦ q✉❡

❛ r❡t❛ ←→AB t❡♠ ❝♦♠♦ ❡q✉❛çã♦ w= 0✱ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ❛

r❡t❛ ←→BC é ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ u= 0 ❡ ❛ r❡t❛ ←→AC é ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ v = 0✳

❉❡ ❛❝♦r❞♦ ❝♦♠ ♦s ❝á❧❝✉❧♦s ❛♥t❡r✐♦r❡s✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r P ❡♠ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐✲ ❝❛s ❡♠ ❢✉♥çã♦ ❞❛ ✈❛r✐á✈❡❧w✳ ❖✉ s❡❥❛✱

P = (u:v :w) =

w

bf ₋ce ae₋bd

:w

cd₋af ae₋bd

:w

. ✭✷✳✶✵✮

❚♦♠❛♥❞♦✲s❡ w= 1✱ ✜❝❛♠♦s ❝♦♠

P =

bf ₋ce ae₋bd :

cd₋af ae₋bd : 1

.

❖❜s❡r✈❡♠♦s q✉❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡P ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ♣❛r❛ ❝❛❞❛w₆= 0✳

✻ ❈❆P❮❚❯▲❖ ✷✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ❉❡✜♥✐çã♦ ✷✳✷ ❈♦♥s✐❞❡r❡ ✉♠ tr✐â♥❣✉❧♦ △ABC ❝♦♠ ✈ért✐❝❡s A✱ B ❡C✳ ❙❡❥❛♠ P1 = (u1 :

v1 : w1) ❡ P2 = (u2 :v2 : w2)✳ ❉✐③❡♠♦s q✉❡ P1 = P2 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦

r❡❛❧ k ₆= 0 t❛❧ q✉❡ u2 =k.u1✱ v2 =k.v1 ❡ w2 =k.w1✳

P❛r❛ ♦♣❡r❛r♠♦s ❝♦♠ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ♣♦♥t♦s✱ ❛♣r❡s❡♥t❛♠♦s ❛s s❡❣✉✐♥✲ t❡s ❞❡✜♥✐çõ❡s✳

❉❡✜♥✐çã♦ ✷✳✸ ❙❡❥❛♠ P1 ❡ P2 ♣♦♥t♦s ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s P1 = (u1 :v1 :w1) ❡

P2 = (u2 :v2 :w2)✳ ❆♦ ♣♦♥t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s

P1+P2 = (u1+u2 :v1+v2 :w1+w2),

❞❡♥♦♠✐♥❛♠♦s s♦♠❛ ❞❡ P1 ❡ P2✳

❉❡✜♥✐çã♦ ✷✳✹ ❙❡❥❛♠ k ✉♠ ♥ú♠❡r♦ r❡❛❧ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ ❡ P1 ✉♠ ♣♦♥t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s

❜❛r✐❝ê♥tr✐❝❛s P1 = (u1 :v1 :w1)✳ ❆♦ ♣♦♥t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s P✱ ❡♠ q✉❡

P =k.P1 = (k.u1 :k.v1 :k.w1),

❞❡♥♦♠✐♥❛♠♦s ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ P1 ♣❡❧♦ ❡s❝❛❧❛r k✳

❉❡✜♥✐çã♦ ✷✳✺ ❙❡❥❛♠ P1 ❡ P2 ♣♦♥t♦s ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s P1 = (u1 :v1 :w1) ❡

P2 = (u2 :v2 :w2)✳ ❆♦ ♣♦♥t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s

P =P1−P2 =P1+ (−1).P2 = (u1−u2 :v1−v2 :w1−w2),

❞❡♥♦♠✐♥❛♠♦s ❞❡ ❞✐❢❡r❡♥ç❛ ❞❡ P1 ❡ P2✳

❊♠ s❡❣✉✐❞❛✱ ❞✐✈✐❞✐♠♦s ❝❛❞❛ ❝♦♦r❞❡♥❛❞❛ ❜❛r✐❝ê♥tr✐❝❛ ♣♦r u+v+w✳ ❆ ❡st❡ ♣r♦❝❡ss♦✱ ❝❤❛♠❛♠♦s ❞❡ ♥♦r♠❛❧✐③❛çã♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✳ ❚❡♠♦s q✉❡

u+v+w= bf −ce

ae₋bd +

cd₋af

ae₋bd + 1 =

bf ₋ce+cd₋af +ae₋bd ae₋bd .

P❡❧❛ ❉❡✜♥✐çã♦ ✷✳✷✱ ♠❡s♠♦ ❞✐✈✐♥❞♦✲s❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡P ❞❛❞❛s ❡♠ ✭✷✳✷✮ ♣♦ru+v+w✱ ❛✐♥❞❛ t❡♠♦s ♦ ♣♦♥t♦ P✳ ❆ss✐♠✱

P = ae−bd bf−ce+cd−af+ae−bd

. bf−ce ae−bd

: ae−bd bf−ce+cd−af+ae−bd

. cd−af ae−bd

: ae−bd bf−ce+cd−af+ae−bd

.

▲♦❣♦✱

P = bf−ce

bf−ce+cd−af+ae−bd :

cd−af

bf−ce+cd−af+ae−bd :

ae−bd bf−ce+cd−af+ae−bd

✷✳✶✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ❊ ❊❳❊▼P▲❖❙ ✼

u= xbyc−xbyp−xpyc−xcyb+xcyp+xpyb

xbyc −xcyb+xcya−xayc +xayb−xbya

.

❖ ♠ó❞✉❧♦ ❞♦ ♥ú♠❡r♦ r❡❛❧xbyc−xbyp−xpyc−xcyb+xcyp+xpyb ❝♦rr❡s♣♦♥❞❡ ❛♦ ❞♦❜r♦ ❞❛

ár❡❛ ❞♦ tr✐â♥❣✉❧♦△P BC✱ ❡♥q✉❛♥t♦ q✉❡ ♦ ♠ó❞✉❧♦ ❞❡xbyc−xcyb+xcya−xayc+xayb−xbya

❝♦rr❡s♣♦♥❞❡ ❛♦ ❞♦❜r♦ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ △ABC✳ ❊♥tã♦✱

u= SP BC

SABC

,

♦♥❞❡ SP BC ❡ SABC ❞❡♥♦t❛♠ ❛s ár❡❛s ❝♦♠ s✐♥❛❧ ❞♦s tr✐â♥❣✉❧♦s △P BC ❡ △ABC✳ P❛r❛

✜①❛r ♥♦t❛çã♦✱ ❝♦♥s✐❞❡r❛♠♦s

SABC =

1 2

xa ya 1

xb yb 1

xc yc 1

✭✷✳✶✶✮

❆ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ SABC s❡rá ❞❡♥♦t❛❞❛ ♣♦r S✳ P♦r ✭✷✳✹✮✱ t❡♠♦s t❛♠❜é♠

v = xcya−xcyp−xpya−xayc +xayp+xpyc

xbyc−xcyb+xcya−xayc+xayb−xbya

.

❆ss✐♠✱

v = SAP C

SABC

,

♦♥❞❡ SAP C ❞❡♥♦t❛ ❛ ár❡❛ ❝♦♠ s✐♥❛❧ ❞♦ tr✐â♥❣✉❧♦△AP C✳ ❊

w = ae−bd

bf ₋ce+cd₋af +ae₋bd =

xayb −xayp −xpyb−xbya+xbyp+xpya

xbyc−xcyb+xcya−xayc+xayb−xbya

= SABP

SABC

,

♦♥❞❡ SABP ❞❡♥♦t❛ ❛ ár❡❛ ❝♦♠ s✐♥❛❧ ❞♦ tr✐â♥❣✉❧♦ △ABP✳

P♦rt❛♥t♦✱ ✈✐♠♦s q✉❡ ❞❛❞♦ ✉♠ ♣♦♥t♦ P = (xp, yp) ❡ ✉♠ tr✐â♥❣✉❧♦ ❞❡ ✈ért✐❝❡s A =

(xa, ya)✱B = (xb, yb)❡C = (xc, yc)✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ✉♠ ♣♦♥t♦P = (xp, yp)

♣♦❞❡♠ s❡r ♦❜t✐❞♦s ❛ ♣❛rt✐r ❞❛s ár❡❛s ❝♦♠ s✐♥❛❧ ❞♦s tr✐â♥❣✉❧♦s △ABP✱_△AP C✱_△P BC ❡

△ABC✳

P = (xp, yp) =

uA+vB +wC

u+v+w = (u:v :w) =

SP BC

SABC

: SAP C

SABC

: SABP

SABC

✽ ❈❆P❮❚❯▲❖ ✷✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙

❋✐❣✉r❛ ✷✳✷✿ ➪r❡❛s ❢♦r♠❛❞❛s ♣❡❧♦ ♣♦♥t♦ P ❡ ♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ ABC✳

❖s s✐♥❛✐s ❞❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s u✱ v ❡ w♣♦❞❡♠ s❡r ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛s ár❡❛s ❝♦♠ s✐♥❛❧ SP BC✱ SAP C ❡ SABP✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚♦♠❛♥❞♦✲s❡ ❛ ♦r✐❡♥t❛çã♦ ❞❛❞❛ ♣❡❧♦s

✈ért✐❝❡s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❝♦♠ s✐♥❛❧✱ ❝♦♠♦ ❡♠ ✭✷✳✶✶✮✱ t❡♠♦s q✉❡✿

✶✳ u > 0 q✉❛♥❞♦ P ❡ A ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❡♠ r❡❧❛çã♦ à r❡t❛ ←→BC ❡ u < 0 q✉❛♥❞♦

❡❧❡s ❡st✐✈❡r❡♠ ❡♠ ❧❛❞♦s ♦♣♦st♦s❀

✷✳ v > 0 q✉❛♥❞♦ P ❡ B ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❡♠ r❡❧❛çã♦ à r❡t❛ ←→AC ❡ v < 0 q✉❛♥❞♦

❡❧❡s ❡st✐✈❡r❡♠ ❡♠ ❧❛❞♦s ♦♣♦st♦s❀

✸✳ w >0 q✉❛♥❞♦ P ❡ C ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❡♠ r❡❧❛çã♦ à r❡t❛ ←→AB ❡ w <0 q✉❛♥❞♦

❡❧❡s ❡st✐✈❡r❡♠ ❡♠ ❧❛❞♦s ♦♣♦st♦s✳

✷✳✶✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ❊ ❊❳❊▼P▲❖❙ ✾

❋✐❣✉r❛ ✷✳✸✿ ❙✐♥❛❧ ❞❛ ❝♦♦r❞❡♥❛❞❛ u✳

✶✵ ❈❆P❮❚❯▲❖ ✷✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙

❋✐❣✉r❛ ✷✳✺✿ ❙✐♥❛❧ ❞❛ ❝♦♦r❞❡♥❛❞❛ w✳

❋✐❣✉r❛ ✷✳✻✿ ❙✐♥❛✐s ❞❛s ❝♦♦r❞❡♥❛❞❛su✱ v ❡ w✳

❆ ✜♠ ❞❡ ❡①♣❧♦r❛r ♦s ❝♦♥❝❡✐t♦s ❛♣r❡s❡♥t❛❞♦s✱ ❛ s❡❣✉✐r ❢❛r❡♠♦s ❛❧❣✉♥s ❝á❧❝✉❧♦s✳

❊①❡♠♣❧♦ ✷✳✶ ❈♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ △ABC ❞❡ ✈ért✐❝❡s ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s A= (1,1)✱B = (3,4)❡C = (6,2)❡ s❡❥❛P ♦ ♣♦♥t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛sP = (3,2)✳

✷✳✶✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ❊ ❊❳❊▼P▲❖❙ ✶✶

❋✐❣✉r❛ ✷✳✼✿ ❈♦♥str✉çã♦ ❈❛rt❡s✐❛♥❛ ❞♦ ❊①❡♠♣❧♦ ✷✳✶✳

❖❜s❡r✈❛♠♦s q✉❡ P ♥ã♦ ♣❡rt❡♥❝❡ às r❡t❛s s✉♣♦rt❡s ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ ❞❛❞♦✳ P♦r ✭✷✳✹✮ ❡ ✭✷✳✶✵✮✱ t❡♠♦s

a = ₋2, b= 0, c= 3, d = ₋1, e= 2, f = 0,

❞♦♥❞❡ s❡❣✉❡ q✉❡

P = (3 2w:

3

4w:w).

◆♦t❛♠♦s q✉❡ u+v+w= 13₄w₆= 0✱ ♣♦✐s w₆= 0✳

❚♦♠❛♥❞♦✲s❡ w= 4✱ ❡♥tã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ P sã♦P = (6 : 3 : 4)

❆s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ sã♦ ❞❛❞❛s ♣♦r✿

A= 1(1,1) + 0(3,4) + 0(6,2)

1 + 0 + 0 = (1,1), ❞♦♥❞❡ A= (1 : 0 : 0).

B = 0(1,1) + 1(3,4) + 0(6,2)

0 + 1 + 0 = (3,4), ❞♦♥❞❡ B = (0 : 1 : 0).

C = 0(1,1) + 0(3,4) + 1(6,2)

0 + 0 + 1 = (6,2), ❞♦♥❞❡ C = (0 : 0 : 1).

◆♦t❛♠♦s q✉❡ u+v +w = 1✳ ❊st❡ é ✉♠ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s✱

✶✷ ❈❆P❮❚❯▲❖ ✷✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ❉❡✜♥✐çã♦ ✷✳✻ ❙❡❥❛♠ A✱ B ❡ C ♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❞❡ r❡❢❡rê♥❝✐❛ _△ABC✳ ❙❡ P é ✉♠ ♣♦♥t♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s P = (u : v : w)✱ s✉❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s

s❡rã♦ ❞❡♥♦♠✐♥❛❞❛s ❤♦♠♦❣ê♥❡❛s q✉❛♥❞♦ u+v+w= 1✳

◆♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ tí♥❤❛♠♦su+v+w= 13₄ w✳ ❉❡st❛ ❢♦r♠❛✱ ♣❛r❛ q✉❡ ❛s ❝♦♦r❞❡♥❛❞❛s

❞❡ P s❡❥❛♠ ❤♦♠♦❣ê♥❡❛s✱ ❞❡✈❡♠♦s t❡r w = ₁₃4✳ ❆ss✐♠✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s

❤♦♠♦❣ê♥❡❛s ❞❡ P sã♦P = 6 13, 3 13, 4 13 ✳

❆s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ P t❛♠❜é♠ ♣♦❞❡r✐❛♠ s❡r ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞❛s ár❡❛s ❝♦♠ s✐♥❛❧ SP BC✱SAP C ✱ SABP ❡SABC✱ ❝♦♠♦ ❡♠ ✭✷✳✶✷✮✿

SABC =

1 2.

xa ya 1

xb yb 1

xc yc 1

= 1 2.

1 1 1 3 4 1 6 2 1

=− 13 2 .

SP BC =

1 2.

xp yp 1

xb yb 1

xc yc 1

= 1 2.

3 2 1 3 4 1 6 2 1

=−3.

SAP C =

1 2.

xa ya 1

xp yp 1

xc yc 1

= 1 2.

1 1 1 3 2 1 6 2 1

=− 3 2.

SABP =

1 2.

xa ya 1

xb yb 1

xp yp 1

= 1 2.

1 1 1 3 4 1 3 2 1

=−2.

❆ss✐♠✱ P = (u:v :w) =SP BC

SABC :

SAP C

SABC :

SABP SABC = 6 13 : 3 13 : 4 13 ✳

◆♦ ❝❛s♦ ❞❡ ❝❛❧❝✉❧❛r♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❛ ♣❛rt✐r ❞❛s ár❡❛s ❝♦♠ s✐♥❛❧✱ ❡❧❛s sã♦ ❤♦♠♦❣ê♥❡❛s✳ ■st♦ ♥ã♦ é ✉♠❛ ♠❡r❛ ❝♦✐♥❝✐❞ê♥❝✐❛✱ ✉♠❛ ✈❡③ q✉❡

u+v+w= SP BC

SABC

+ SAP C

SABC

+SABP

SABC

✷✳✶✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ❊ ❊❳❊▼P▲❖❙ ✶✸

❋✐❣✉r❛ ✷✳✽✿ ❈♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❞❡ ✈ért✐❝❡s✱ ♣♦♥t♦s ♠é❞✐♦s ❡ ❜❛r✐❝❡♥tr♦✳

❆❧é♠ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ✈ért✐❝❡s✱ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ❛s ❝♦✲ ♦r❞❡♥❛❞❛s ❞♦s ♣♦♥t♦s ♠é❞✐♦s D✱ E ❡ F ❞♦s ❧❛❞♦s AB✱ BC ❡ AC✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♠ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✱ t❡♠♦s✿

D= A+B

2 = (2, 5

2)✱ E =

B+C

2 = ( 9

2,3) ❡F =

A+C

2 = ( 7 2,

3 2)✳

❊ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❤♦♠♦❣ê♥❡❛s✱ t❡♠♦s✿

D= 1 2A+

1

2B+ 0C 1

2 + 1 2 + 0

= A+B

2 , ❡ ❛ss✐♠D = ( 1 2 :

1 2 : 0).

E = 0A+ 1 2B+

1 2C 0 + 1₂ +1₂ =

B+C

2 , ❡ ❛ss✐♠ E = (0 : 1 2 :

1 2).

F = 1

2A+ 0B+ 1 2C 1

2 + 0 + 1 2

= A+C

2 , ❡ ❛ss✐♠F = ( 1 2 : 0 :

1 2).

❖ ❜❛r✐❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ♦ ♣♦♥t♦ ❞❡ ❡♥❝♦♥tr♦ ❞❛s ♠❡❞✐❛♥❛s ❞♦ tr✐â♥❣✉❧♦✳ ❉❛❞♦ ♦ tr✐â♥❣✉❧♦△ABC✱ ❡❧❡ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❝♦♠♦ s❡♥❞♦ ♦ ♣♦♥t♦G= A+B+C

3 ✳ ❊♠ ❝♦♦r❞❡♥❛❞❛s

❜❛r✐❝ê♥tr✐❝❛s ❤♦♠♦❣ê♥❡❛s✱ ❡❧❡ é ❞❛❞♦ ❝♦♠♦

G= 1 3A+

1 3B+

1 3C 1 3 + 1 3 + 1 3 =

A+B+C

3

1 =

A+B +C

3 , ❡ ❛ss✐♠, G=

1 3 : 1 3 : 1 3 .

✶✹ ❈❆P❮❚❯▲❖ ✷✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙

✷✳✷ ❊q✉❛çã♦ ❞❡ ❘❡t❛

◆❡st❛ s❡çã♦✱ ❞❡t❡r♠✐♥❛♠♦s ❛ ❢♦r♠❛ ❣❡r❛❧ ❞❛ ❡q✉❛çã♦ ❞❡ ✉♠❛ r❡t❛ ✉t✐❧✐③❛♥❞♦ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✳ ▼❛✐s ❛❞✐❛♥t❡✱ ✉t✐❧✐③❛r❡♠♦s ❢♦rt❡♠❡♥t❡ ❡st❡ ❝♦♥❝❡✐t♦✳

❆ ❡q✉❛çã♦ ❞❡ ✉♠❛ r❡t❛ ♥♦ ♣❧❛♥♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s éa.x+b.y+c= 0✱ ♦♥❞❡

a, b❡csã♦ ♥ú♠❡r♦s r❡❛✐s✳ ❙❡u, v ❡w❞❡♥♦t❛♠ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ✉♠ ♣♦♥t♦ P✱ ♦✉ s❡❥❛✱ s❡ P = (u : v : w)✱ ❡♠ r❡❧❛çã♦ ❛ ✉♠ tr✐â♥❣✉❧♦ ❞❡ r❡❢❡rê♥❝✐❛ _△ABC✱ ♥♦ss♦

♦❜❥❡t✐✈♦ é ❞❡t❡r♠✐♥❛r ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❡st❛ r❡t❛ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✳

◆♦✈❛♠❡♥t❡✱ s❡❥❛♠A= (xa, ya)✱B = (xb, yb)❡C = (xc, yc)❛s ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s

❞♦s ✈ért✐❝❡sA✱ B ❡ C ❞♦ tr✐â♥❣✉❧♦ _△ABC ✱ ❡ P = (xp, yp)✳ P♦r ✭✷✳✶✮✱ ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛

r❡t❛ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s é

r.u+s.v+t.w = 0,

♦♥❞❡ r=a.xa+b.ya+c✱s =a.xb +b.yb+c❡ t=a.xc+b.yc+c✳

P♦r ❡①❡♠♣❧♦✱ ❞❡t❡r♠✐♥❡♠♦s ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r A = (1 : 0 : 0) ❡

B = (0 : 1 : 0)✳ ❈♦♠♦ A ❡ B ♣❡rt❡♥❝❡♠ ❛ ❡st❛ r❡t❛ ❡♥tã♦ r = 0 ❡ s = 0✳ ❆ss✐♠✱

tw = 0✳ ❈♦♠♦ t ₆= 0✱ ♣♦✐s ♥❡st❡ ❝❛s♦✱ C t❛♠❜é♠ ♣❡rt❡♥❝❡r✐❛ ❛ ❡st❛ r❡t❛✱ ❞❡ ♠♦❞♦ q✉❡ A, B ❡C s❡r✐❛♠ ❝♦❧✐♥❡❛r❡s✱ ❡♥tã♦ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛←→AB✱ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✱ é w = 0✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❛s ❡q✉❛çõ❡s ❞❡ ←→BC ❡ ❞❡ ←→AC ✱ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✱ sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ u= 0 ❡v = 0✳

✷✳✸ ❈❡✈✐❛♥❛s ❡ ❚r❛ç♦s

❆ s❡❣✉✐r✱ ❞❡✜♥✐♠♦s ❝❡✈✐❛♥❛s ❡ tr❛ç♦s ❞❡ ♣♦♥t♦s✳

✷✳✸✳ ❈❊❱■❆◆❆❙ ❊ ❚❘❆➬❖❙ ✶✺

❋✐❣✉r❛ ✷✳✾✿ ❈❡✈✐❛♥❛s ❡ tr❛ç♦s ❞♦ ♣♦♥t♦ P✳

❙❡❥❛♠ X✱ Y ❡ Z ♦s tr❛ç♦s ❞❛s ❝❡✈✐❛♥❛s ♣♦r A✱ B ❡ C✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♠♦ P =SP BC

SABC :

SAP C

SABC :

SABP

SABC

= (SP BC :SAP C :SABP)✱ ❞❡✈❡♠♦s ❡♥❝♦♥tr❛r ❛s ❝♦✲

♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞♦s tr❛ç♦s X✱ Y ❡ Z ♣♦r ♠❡✐♦ ❞❛ ✐♥t❡rs❡❝çã♦ ❡♥tr❡ r❡t❛s✳ P♦r ❡①❡♠♣❧♦✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ X sã♦ ♦❜t✐❞❛s ♣❡❧❛ ✐♥t❡rs❡❝çã♦ ❡♥tr❡ ❛s r❡t❛s

←→

BC ❡ ←→AP✳ ❏á ✈✐♠♦s q✉❡ ❛ ❡q✉❛çã♦ ❞❡ ←→BC é u = 0✳ ❊♠ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✱ ❛

❡q✉❛çã♦ ❞❛ r❡t❛←→AP ér.u+s.v+t.w= 0✳ ❈♦♠♦A= (1 : 0 : 0)♣❡rt❡♥❝❡ ❛ ❡st❛ r❡t❛✱ t❡♠♦s

r = 0✳ ❆ss✐♠✱ ✉s❛♥❞♦ t❛♠❜é♠ q✉❡ P ♣❡rt❡♥❝❡ ❛ ❡st❛ r❡t❛✱ t❡♠♦s s.SAP C +t.SABP = 0✳

❈♦♠♦ SAP C 6= 0✱ t❡♠♦ss =−t.S_SABP

AP C✳ ❉❡st❛ ❢♦r♠❛✱

t.

w₋ SABP SAP C

.v

= 0.

❈♦♠♦ t₆= 0✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦ t❡rí❛♠♦sr=s=t= 0✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ❡q✉❛çã♦ ❞❛

r❡t❛ ←→AP é

w= SABP

SAP C

.v.

P♦rt❛♥t♦✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ X sã♦

0 :v : SABP

SAP C

.v

= (0 : SAP C.v :SABP.v) = (0 :SAP C :SABP).

❆♥❛❧♦❣❛♠❡♥t❡✱

Y = (SP BC :SAP C :SABP) = (SP BC : 0 : SABP).

✶✻ ❈❆P❮❚❯▲❖ ✷✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙

Z = (SP BC :SAP C :SABP) = (SP BC :SAP C : 0).

◆♦ss♦ ♣ró①✐♠♦ ♦❜❥❡t✐✈♦ é ❡♥❝♦♥tr❛r ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞♦s tr❛ç♦s X✱Y ❡Z ❡♠ ❢✉♥çã♦ ❞❡ ♣r♦♣♦rçõ❡s ❞❡ s❡❣♠❡♥t♦s✳ ❚❡♠♦s ❞♦✐s ❝❛s♦s ♣❛r❛ ❡st✉❞❛r✿

❈❛s♦ ✶✿ P ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✐♦r ❞♦ tr✐â♥❣✉❧♦_△ABC ✳

◆❡st❡ ❝❛s♦✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛su✱v❡w❞♦s tr❛ç♦s s❡rã♦ ♥ã♦✲♥❡❣❛t✐✈❛s✳ ❚❡♠♦s q✉❡ SABP

SAP C =

BX XC✱ ♣♦✐s

SABP

SAP C

= SXAB −SXP B

SXAC−SXP C

= BX.hA 2 − BX.hP 2 XC.hA 2 − XC.hP 2 =

BX.(hA−hP)

2

XC.(hA−hP)

2

= BX

XC,

♦♥❞❡ hA ❡ hP sã♦ ❛s ❛❧t✉r❛s ❛ ♣❛rt✐r ❞♦s ♣♦♥t♦s A ❡ P✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♠ r❡❧❛çã♦ ❛♦

❧❛❞♦ BC✳

❆♥❛❧♦❣❛♠❡♥t❡✱ SAP C

SP BC =

AZ ZB ❡

SABP

SP BC =

AY Y C✳

❉❡st❛ ❢♦r♠❛✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞♦s tr❛ç♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ♣❡❧❛ ♣r♦♣♦rçã♦ ❞❡ s❡❣♠❡♥t♦s✿

X = (0 :SAP C :SABP) =

0 : 1 : SABP

SAP C

=

0 : 1 : BX

XC

= (0 :XC:BX)

Y = (SP BC : 0 :SABP) =

1 : 0 : SABP

SP BC

=

1 : 0 : AY

Y C

= (Y C : 0 :AY) ✭✷✳✶✸✮

Z = (SP BC :SAP C : 0) =

1 : SAP C

SP BC

: 0

=

1 : AZ

ZB : 0

= (ZB :AZ : 0).

❈❛s♦ ✷✿ P ♣❡rt❡♥❝❡ ❛♦ ❡①t❡r✐♦r ❞♦ tr✐â♥❣✉❧♦_△ABC ✳

◆❡st❡ ❝❛s♦✱ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ♥ã♦ ✈❛❧❡✿ X ❡B ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛ ←→AC✱ X ❡ C ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛←→AB✱ Y ❡ A ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛ ←→BC✱ Y ❡ C ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛ ←→AB✱ Z ❡ A ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛ ←→BC✱ Z ❡B ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛ ←→AC✳ ❉❡♣❡♥❞❡♥❞♦ ❞❡ q✉❛❧ ❝♦♥❞✐çã♦ ♥ã♦ ❢♦r ✈á❧✐❞❛✱ ♦ s✐♥❛❧ ❞❡ u✱ v ♦✉ w ♣♦❞❡rá s❡r ♥❡❣❛t✐✈♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❛♥á❧✐s❡ ❢❡✐t❛ ♥❛ ❙❡çã♦ ✷✳✶✳

❆ ♠❡♥♦s ❞❡ s✐♥❛❧✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ❞♦s tr❛ç♦s X✱ Y ❡ Z sã♦ ❞❛❞❛s ♣♦r ✭✷✳✶✸✮✳ P♦r ❡①❡♠♣❧♦✱ ♥♦ ❝❛s♦ ❡♠ X ❡ B ❡stã♦ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛ ←→AC✱ ♠❛s X ❡ C ❡stã♦ ❡♠ ❧❛❞♦s ♦♣♦st♦s ❞❛ r❡t❛ ←→AB✱ t❡r❡♠♦s

X = (0 :SAP C :SABP) =

0 : 1 : SABP

SAP C

=

0 : 1 : −BX

XC

✷✳✹✳ ❚❊❖❘❊▼❆ ❉❊ ❈❊❱❆ P❆❘❆ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ✶✼ ❊st❡ é ♦ ❝❛s♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ q✉❡ P é ♦ ♦rt♦❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ _△ABC✱ ♦❜t✉s♦ ♥♦ ✈ért✐❝❡ B✳

✷✳✹ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛ ♣❛r❛ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s

◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ✈❡rsã♦ ❞♦ ❝♦♥❤❡❝✐❞♦ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛ ❞❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✳ ❖ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛ ❞á ✉♠❛ ❝♦♥❞✐çã♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r s❡ ❛s ❝❡✈✐❛♥❛s ❞❡ ✉♠ tr✐â♥❣✉❧♦ s❡ ✐♥t❡r❝❡♣t❛♠✳ ❆ s❛❜❡r✱

❚❡♦r❡♠❛ ✷✳✶ ❊♠ ✉♠ tr✐â♥❣✉❧♦ △ABC✱ ❛s ❝❡✈✐❛♥❛sAX✱ BY ❡ CZ s❡ ✐♥t❡r❝❡♣t❛rã♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

BX XC.

CY Y A.

AZ ZB = 1.

◆♦ ❡♥✉♥❝✐❛❞♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛✱ ❛s ❝❡✈✐❛♥❛s ❢♦r❛♠ t♦♠❛❞❛s ❝♦♠♦ s❡❣♠❡♥t♦s q✉❡ ✉♥❡♠ ♦ ✈ért✐❝❡ ❞♦ tr✐â♥❣✉❧♦ ❛ ✉♠ tr❛ç♦ ❞❡ P✳

❆ s❡❣✉✐r✱ s❡rá ❡♥✉♥❝✐❛❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛ ♣❛r❛ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✳ ❊❧❡ ❞á ✉♠❛ ❝♦♥❞✐çã♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r s❡ ✉♠ ♣♦♥t♦ é ✉♠ tr❛ç♦ ❞❡ P✱ ❡ s❡rá ✉t✐❧✐③❛❞♦ ♥♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡ ♥♦ ❝á❧❝✉❧♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ ❛❧❣✉♥s ♣♦♥t♦s ♥♦tá✈❡✐s✳

❚❡♦r❡♠❛ ✷✳✷ ❚rês ♣♦♥t♦s X✱ Y ❡ Z sã♦ ♦s tr❛ç♦s ❞♦ ♣♦♥t♦ P = (u:v :w)s❡ ❡ s♦♠❡♥t❡

s❡✱ X✱ Y ❡ Z sã♦ ❞❛ ❢♦r♠❛

    

X = (0 :v :w)

Y = (u: 0 : w)

Z = (u:v : 0)

♣❛r❛ ❛❧❣✉♠ u✱ v ❡ w✳

❈♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡✱ t♦♠❛♠♦s △ABC ❝♦♠♦ tr✐â♥❣✉❧♦ ❞❡ r❡❢❡rê♥❝✐❛✱ ❡♠ q✉❡ A = (1 : 0 : 0)✱B = (0 : 1 : 0)✱C= (0 : 0 : 1)✳ ◆❡st❡ ❝❛s♦✱ ✈✐♠♦s q✉❡P = (SP BC :SAP C :SABP)

❡ _

 

 

X = (0 :SAP C :SABP)

Y = (SP BC : 0 :SABP)

Z = (SP BC :SAP C : 0)

.

Pr♦✈❛✿ P❡❧❛ ❝♦♥str✉çã♦ ❢❡✐t❛ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✱ s❡ X✱ Y ❡ Z ❢♦r❡♠ ♦s tr❛ç♦s ❞❡ ✉♠ ♣♦♥t♦ P = (u : v : w)✱ ❡♥tã♦ X = (0 :SAP C :SABP)✱ Y = (SP BC : 0 :SABP) ❡

Z = (SP BC :SAP C : 0)✳ ❖✉ s❡❥❛✱ ♥♦ ❝❛s♦ ❞❡X✱ ❜❛st❛ t♦♠❛ru= 0✱v =SAP C ❡w=SABP✱

❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳

✶✽ ❈❆P❮❚❯▲❖ ✷✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙

♠♦str❡♠♦s q✉❡ ❡❧❡s sã♦ ♦s tr❛ç♦s ❞❡P✳ P❛r❛ ✐st♦✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡P = (SP BC :SAP C :SABP)✳

■st♦ s❡rá ❢❡✐t♦ ♦❜t❡♥❞♦✲s❡ ❛ ✐♥t❡rs❡❝çã♦ ❞❛s ❝❡✈✐❛♥❛sAX✱BY ❡ CZ✳

❙❡♥❞♦ r.u+s.v+t.w = 0 ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r A = (1 : 0 : 0) ❡ X = (0 :SAP C :SABP) ❡♥tã♦

(

r.1 +s.0 +t.0 = 0_⇒r = 0

r.0 +s.SAP C +t.SABP = 0 ⇒s=−S_SABP

AP C.t

▲♦❣♦✱

0.u+

−S_SABP

AP C

.t

.v+t.w= 0 =_⇒←→AX :w= SABP

SAP C

.v

❆♥❛❧♦❣❛♠❡♥t❡✱ ❝♦♠♦ Y = (SP BC : 0 :SABP) ❡ Z = (SP BC :SAP C : 0)✱ ❛s r❡t❛s ←→BY ❡

←→

CZ t❡rã♦ ❡q✉❛çõ❡s u = SP BC

SABP.w ❡ v =

SAP C

SP BC.u✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ ✐♥t❡rs❡❝çã♦ ❞❛s r❡t❛s

←→

AX✱ ←→BY ❡ ←→CZ ♥♦s ❞á ♦ ♣♦♥t♦ P✿

    

u= SP BC

SABP.w

v = SAP C

SP BC.u

w= SABP

SAP C.v

❆ss✐♠✱

P = (u:v :w) =

u: SAP C

SP BC

.u: SABP

SAP C

.v

=

u: SAP C

SP BC

.u: SABP

SAP C

.SAP C SP BC

.u

=

=

1 : SAP C

SP BC

: SABP

SP BC

✷✳✹✳ ❚❊❖❘❊▼❆ ❉❊ ❈❊❱❆ P❆❘❆ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ✶✾

❈❛♣ít✉❧♦ ✸

P♦♥t♦s ◆♦tá✈❡✐s ❞❡ ✉♠ ❚r✐â♥❣✉❧♦

◆❡st❡ ❝❛♣ít✉❧♦✱ ❝❛❧❝✉❧❛♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ❛❧❣✉♥s ♣♦♥t♦s ♥♦tá✈❡✐s ❞❡ ✉♠ tr✐â♥❣✉❧♦✳ ❖ ❜❛r✐❝❡♥tr♦ G ❥á ❢♦✐ ❡st✉❞❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ❡ s✉❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐✲ ❝ê♥tr✐❝❛s sã♦ ❞❛❞❛s ♣♦rG= 1

3 : 1 3 :

1 3

✳ ❋❛r❡♠♦s ♦ ♠❡s♠♦ ♣❛r❛ ♦ ✐♥❝❡♥tr♦✱ ♦ ❝✐r❝✉♥❝❡♥tr♦ ❡ ♦ ♦rt♦❝❡♥tr♦✳

✸✳✶ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ❞♦ ■♥❝❡♥tr♦

▲❡♠❜r❛♠♦s q✉❡ ♦ ✐♥❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ❛ ❡st❡ tr✐â♥❣✉❧♦✳ ❊❧❡ ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞♦ ❝♦♠♦ s❡♥❞♦ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❡♥tr❡ ❛s ❜✐ss❡tr✐③❡s ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ tr✐â♥❣✉❧♦✳

❙❡❥❛♠A✱B ❡C♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦_△ABC ❡I ♦ s❡✉ ✐♥❝❡♥tr♦✳ ❙❡❥❛♠a=BC✱ b = AC ❡ c =AB ❡ r ♦ r❛✐♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ❛♦ tr✐â♥❣✉❧♦ _△ABC✳ P♦r ✭✷✳✶✷✮✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ I sã♦

I = (u:v :w) =

SIBC

SABC

: SAIC

SABC

: SABI

SABC

.

✷✷ ❈❆P❮❚❯▲❖ ✸✳ P❖◆❚❖❙ ◆❖❚➪❱❊■❙ ❉❊ ❯▼ ❚❘■➶◆●❯▲❖

❋✐❣✉r❛ ✸✳✶✿ ■♥❝❡♥tr♦ ✭I✮ ❞❡ ✉♠ tr✐â♥❣✉❧♦ABC✳

❈♦♠♦ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ❛♦ tr✐â♥❣✉❧♦△ABC t❛♥❣❡♥❝✐❛ s❡✉s ❧❛❞♦s✱ ♦ r❛✐♦r❞❡st❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ♠❡❞✐❞❛s ❞❡ ❛❧t✉r❛s ❞♦s tr✐â♥❣✉❧♦s SIBC✱SAIC ❡ SABI✳

❆ss✐♠✱

I = (u:v :w) = 1

SABC

(SIBC :SAIC :SABI) = (SIBC :SAIC :SABI) =

= (a.r 2 :

b.r

2 :

c.r

2 ) =

_r

2.a:

r

2.b :

r

2.c

= (a :b :c).

P♦rt❛♥t♦✱ ❞❛❞♦ ✉♠ tr✐â♥❣✉❧♦△ABC✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ s❡✉ ✐♥❝❡♥tr♦ sã♦

I = (BC :AC :AB). ✭✸✳✶✮

✸✳✷ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ❞♦ ❈✐r❝✉♥❝❡♥tr♦

▲❡♠❜r❛♠♦s q✉❡ ♦ ❝✐r❝✉♥❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝✐r❝✉♥s❝r✐t❛ ❛ ❡st❡ tr✐â♥❣✉❧♦✳ ❊❧❡ ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞♦ ❝♦♠♦ s❡♥❞♦ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❡♥tr❡ ❛s ♠❡❞✐❛tr✐③❡s ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦✳

❙❡❥❛♠ A✱ B ❡ C ♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦ _△ABC ❡ O ♦ s❡✉ ❝✐r❝✉♥❝❡♥tr♦✳ ❙❡❥❛♠ a=BC✱b =AC ❡c=AB ❡ R♦ r❛✐♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝✐r❝✉♥s❝r✐t❛ ❛♦ tr✐â♥❣✉❧♦_△ABC✳ P♦r ✭✷✳✶✷✮✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ O sã♦

O = (u:v :w) =

SOBC

SABC

: SAOC

SABC

: SABO

SABC

✸✳✷✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ❉❖ ❈■❘❈❯◆❈❊◆❚❘❖ ✷✸

❋✐❣✉r❛ ✸✳✷✿ ❈✐r❝✉♥❝❡♥tr♦ ✭O✮ ❞❡ ✉♠ tr✐â♥❣✉❧♦ABC✳

❉❡♥♦t❡♠♦s ♣♦r _Aˆ_{✱ B}ˆ _{❡ C}ˆ _{❛s ♠❡❞✐❞❛s ❞♦s â♥❣✉❧♦s ∠BAC✱ ∠ABC ❡ ∠ACB✱ r❡s♣❡❝✲}

t✐✈❛♠❡♥t❡✳ P❡❧♦ t❡♦r❡♠❛ ❞♦ â♥❣✉❧♦ ✐♥s❝r✐t♦✱ t❡♠♦s q✉❡ ❛ ♠❡❞✐❞❛ ❞♦ ❛r❝♦ BC✱ q✉❡ ♥ã♦ ❝♦♥té♠ A✱ é ✐❣✉❛❧ ❛♦ ❞♦❜r♦ ❞❡ _{A✳ ❆ss✐♠✱ ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ❝❡♥tr❛❧}ˆ _{∠BOC é ✐❣✉❛❧ ❛} 2 ˆA✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

SOBC =

R2_{.sen (2 ˆA)}

2 =R

2_{.s❡♥A.❝♦sˆ A.ˆ}

❆♥❛❧♦❣❛♠❡♥t❡✱

SAOC =

R2_{.sen (2 ˆB)}

2 =R

2_{.s❡♥B.ˆ ❝♦sB.ˆ}

SABO =

R2_{.sen (2 ˆC)}

2 =R

2_{.s❡♥C.❝♦sˆ C.ˆ}

❆ss✐♠✱

O = R2.s❡♥A.❝♦sˆ Aˆ:R2.s❡♥B.❝♦sˆ Bˆ :R2.s❡♥C.❝♦sˆ Cˆ=

= 2.R.s❡♥A.❝♦sˆ Aˆ: 2.R.s❡♥B.❝♦sˆ Bˆ : 2.R.s❡♥C.❝♦sˆ Cˆ.

P❡❧❛ ▲❡✐ ❞♦s ❙❡♥♦s✱ t❡♠♦s q✉❡ a

✷✹ ❈❆P❮❚❯▲❖ ✸✳ P❖◆❚❖❙ ◆❖❚➪❱❊■❙ ❉❊ ❯▼ ❚❘■➶◆●❯▲❖ P❡❧❛ ▲❡✐ ❞♦s ❈♦ss❡♥♦s✱ t❡♠♦s q✉❡

a2 =b2+c2₋2.b.c.❝♦sAˆ_⇒❝♦sAˆ= −a

2_+b2_+c2

2.b.c .

❆♥❛❧♦❣❛♠❡♥t❡✱

2.R.s❡♥Bˆ =b ❡ ❝♦sBˆ = a

2_−b2_+c2 2.a.c . ❡

2.R.s❡♥Cˆ =c ❡ ❝♦sCˆ = a

2_+b2 _−c2 2.a.b .

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

O =

a.

−a2_+b2_+c2 2.b.c

:b.

a2_−b2_+c2 2.a.c

:c.

a2_+b2_−c2 2.a.b

=

a2.

−a2_+b2_+c2 2.a.b.c

:b2.

a2_−b2 _+c2 2.a.b.c

:c2.

a2_+b2_−c2 2.a.b.c

= a2. ₋a2+b2+c2:b2. a2₋b2+c2 :c2. a2+b2₋c2.

P♦rt❛♥t♦✱ ❞❛❞♦ ✉♠ tr✐â♥❣✉❧♦△ABC✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ s❡✉ ❝✐r❝✉♥❝❡♥tr♦

sã♦

O = a2. ₋a2+b2+c2:b2. a2₋b2+c2 :c2. a2 +b2₋c2. ✭✸✳✷✮

✸✳✸ ❈♦♦r❞❡♥❛❞❛s ❇❛r✐❝ê♥tr✐❝❛s ❞♦ ❖rt♦❝❡♥tr♦

▲❡♠❜r❛♠♦s q✉❡ ♦ ♦rt♦❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❡♥tr❡ ❛s ❛❧t✉r❛s ❞♦ tr✐â♥❣✉❧♦✱ tr❛ç❛❞❛s ❛ ♣❛rt✐r ❞❡ s❡✉s ✈ért✐❝❡s✳ ❙❡❥❛♠A✱B ❡ C ♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦

△ABC✱ ❝♦♠a=BC✱b =AC ✱c=AB✱ ❡H ♦ s❡✉ ♦rt♦❝❡♥tr♦✳ ❚♦♠❛♠♦s ♦s tr❛ç♦sX✱Y ❡Z ❞❡H ❝♦♠♦ s❡♥❞♦ ♦s ♣és ❞❛s ❛❧t✉r❛s r❡❧❛t✐✈❛s ❛♦s ✈ért✐❝❡sA✱B ❡C✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚❡♠♦s ❛❧❣✉♥s ❝❛s♦s ♣❛r❛ ❛♥❛❧✐s❛r✳

✸✳✸✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ❉❖ ❖❘❚❖❈❊◆❚❘❖ ✷✺

❋✐❣✉r❛ ✸✳✸✿ ❖rt♦❝❡♥tr♦ ✭H✮ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ABC✳

❆♣❧✐❝❛♥❞♦✲s❡ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❛♦s tr✐â♥❣✉❧♦s △ABX ❡ _△AXC✱ t❡♠♦s✿

(

BX2+XA2 =AB2 XC2+XA2 =AC2

❖✉ s❡❥❛✱ ₍

BX2_+XA2 _=c2 (a₋BX)2_+XA2 _=b2

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

BX = +a

2_−b2_+c2 2.a ❡

XC =a₋XB = +a

2 _+b2_−c2 2.a .

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ❛♣❧✐❝❛♥❞♦✲s❡ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❛♦s tr✐â♥❣✉❧♦s △BAY ❡

△BCY✱ t❡r❡♠♦s✿

AY = −a

2_+b2_+c2

2.b ❡

Y C = +a

2_+b2_−c2 2.b ,

❊ ❛♣❧✐❝❛♥❞♦✲s❡ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❛♦s tr✐â♥❣✉❧♦s △CAZ ❡ _△CZB✱ t❡r❡♠♦s✿ AZ = −a

2_+b2_+c2

✷✻ ❈❆P❮❚❯▲❖ ✸✳ P❖◆❚❖❙ ◆❖❚➪❱❊■❙ ❉❊ ❯▼ ❚❘■➶◆●❯▲❖ ❡

ZB = +a

2_−b2_+c2 2.c .

P♦r ✭✷✳✶✸✮         

X = (0 :XC :BX) = 0 : +a2

+b2 −c2

2.a :

+a2 −b2

+c2

2.a

Y = (Y C : 0 :AY) = +a2

+b2 −c2

2.b : 0 : − a2

+b2

+c2

2.b

Z = (ZB :AZ : 0) =+a2−b2

+c2

2.c :

−a2

+b2

+c2

2.c : 0

❆ss✐♠✱ _        

X =0 : +a2+b2−c2

(+a2_+b2

−c2_).(+a2

−b2_+c2₎ :

+a2 −b2

+c2

(+a2_+b2

−c2_).(+a2 −b2_+c2₎

Y = +a2

+b2 −c2

(+a2

+b2 −c2

).(−a2

+b2

+c2

) : 0 : −

a2

+b2

+c2

(+a2

+b2 −c2

).(−a2

+b2

+c2

)

Z = +a2 −b2

+c2

(+a2 −b2

+c2

).(−a2

+b2

+c2

) : −

a2

+b2

+c2

(+a2 −b2

+c2

).(−a2

+b2

+c2

) : 0

❖✉ s❡❥❛✱

X =

0 : 1

+a2_−b2_+c2 :

1 +a2_+b2_−c2

Y =

1

−a2_+b2_+c2 : 0 :

1 +a2_+b2_−c2

✭✸✳✸✮

Z =

1

−a2_+b2_+c2 :

1

+a2_−b2_+c2 : 0

P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❈❡✈❛ ♣❛r❛ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s✱ ❝♦♥❝❧✉í♠♦s q✉❡

H =

1

−a2_+b2_+c2 :

1

+a2_−b2_+c2 :

1 +a2_+b2_−c2

. ✭✸✳✹✮

❈❛s♦ ✷✿ ♦ tr✐â♥❣✉❧♦ △ABC é ♦❜t✉sâ♥❣✉❧♦✳ ◆❡st❡ ❝❛s♦✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞♦ ♦rt♦❝❡♥tr♦ t❛♠❜é♠ sã♦ ❞❛❞❛s ♣♦r ✭✸✳✹✮✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ♣r♦✈❛ ❞❡st❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❡st❡ ❝❛s♦ ❡①✐❣❡ ❛♥á❧✐s❡ ❞❡ s✐♥❛✐s✳ P♦r ❡①❡♠♣❧♦✱ s✉♣♦♥❤❛♠♦s q✉❡ ♦ â♥❣✉❧♦ ♦❜t✉s♦ ♦❝♦rr❛ ♥♦ ✈ért✐❝❡ B✳ ❆ss✐♠✱

        

X = (0 :XC :₋BX) = 0 : +a2

+b2 −c2

2.a :

+a2 −b2

+c2

2.a

Y = (Y C : 0 :AY) = +a2

+b2 −c2

2.b : 0 : − a2

+b2

+c2

2.b

Z = (₋ZB:AZ : 0) =+a2 −b2

+c2

2.c :

−a2

+b2

+c2

2.c : 0

✸✳✸✳ ❈❖❖❘❉❊◆❆❉❆❙ ❇❆❘■❈✃◆❚❘■❈❆❙ ❉❖ ❖❘❚❖❈❊◆❚❘❖ ✷✼ ❉❡st❛ ❢♦r♠❛✱

H =

1

−a2_+b2_+c2 :

1

+a2_−b2_+c2 :

1 +a2_+b2_−c2

.

❈❛s♦ ✸✿ ♦ tr✐â♥❣✉❧♦ △ABC é r❡tâ♥❣✉❧♦✳ ◆❡st❡ ❝❛s♦✱ é ✐♠♣♦rt❛♥t❡ ❞❡st❛❝❛r q✉❛❧ é ♦ â♥❣✉❧♦ r❡t♦ ❞♦ tr✐â♥❣✉❧♦✳

✶✳ ♦ â♥❣✉❧♦ r❡t♦ ♦❝♦rr❡ ♥♦ ✈ért✐❝❡ A✳ ◆❡st❡ ❝❛s♦✱ H =A= (1 : 0 : 0).

✷✳ ♦ â♥❣✉❧♦ r❡t♦ ♥♦ ✈ért✐❝❡ B✳ ◆❡st❡ ❝❛s♦✱

H =B = (0 : 1 : 0).

❈❛♣ít✉❧♦ ✹

❚❡♦r❡♠❛ ❞❡ ❈♦♥✇❛② ❡ ♦ P♦♥t♦ ❞❡

❋❡r♠❛t

◆❡st❡ ❝❛♣ít✉❧♦✱ ❞❡t❡r♠✐♥❛r❡♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ✉♠ ✐♠♣♦rt❛♥t❡ ♣♦♥t♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ♦ ❝❤❛♠❛❞♦ P♦♥t♦ ❞❡ ❋❡r♠❛t ❞♦ tr✐â♥❣✉❧♦✳ ❊❧❛s s❡rã♦ ❝❛❧❝✉❧❛❞❛s ❞❡ ❞✉❛s ♠❛♥❡✐r❛s✿ ❡♠ ❢✉♥çã♦ ❞❡ ár❡❛s ❡ ❡♠ ❢✉♥çã♦ ❞❡ â♥❣✉❧♦s✳

✹✳✶ ❆ ❋ór♠✉❧❛ ❞❡ ❈♦♥✇❛②

◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♥✇❛② ✭❬✺❪ ❡ ❬✻❪✮✱ ❝♦♠ ❛ q✉❛❧ ♦❜t❡r❡♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ✉♠ ♣♦♥t♦ ❡♠ ❢✉♥çã♦ ❞❡ ❝❡rt♦s â♥❣✉❧♦s✳ ❈♦♠❡ç❛♠♦s ✐♥tr♦✲ ❞✉③✐♥❞♦ ❛❧❣✉♠❛s ♥♦t❛çõ❡s✳

❈♦♥s✐❞❡r❡ ✉♠ tr✐â♥❣✉❧♦ ❞❡ ✈ért✐❝❡s A✱ B ❡ C✳ ❉❡♥♦t❛♠♦s ♣♦r Se ♦ ❞♦❜r♦ ❞❛ ár❡❛ ❞❡st❡ tr✐â♥❣✉❧♦✳ ❙❡ θ ❢♦r ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ â♥❣✉❧♦✱ ♦ sí♠❜♦❧♦Sθ s❡rá ✉s❛❞♦ ♣❛r❛ ❞❡♥♦t❛r ♦

♥ú♠❡r♦ r❡❛❧ S.e❝♦t❣θ✳

✸✵ ❈❆P❮❚❯▲❖ ✹✳ ❚❊❖❘❊▼❆ ❉❊ ❈❖◆❲❆❨ ❊ ❖ P❖◆❚❖ ❉❊ ❋❊❘▼❆❚

❋✐❣✉r❛ ✹✳✶✿ ➶♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ABC✳

❉❡♥♦t❛♥❞♦ ♣♦r _A✱ˆ _Bˆ _{❡ C}ˆ _{❛s ♠❡❞✐❞❛s ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ tr✐â♥❣✉❧♦ △ABC✱ H ♦}

♣é ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❜❛✐①❛❞❛ ♣♦r C✱ a = BC✱ b = AC ❡ c = AB ✭❝♦♠♦ ♥❛ ✜❣✉r❛ ✹✳✶✮✱ t❡♠♦s

SA =b.c.s❡♥A.ˆ

❝♦s_Aˆ

s❡♥_Aˆ =b.c.❝♦sA.ˆ

P❡❧❛ ▲❡✐ ❞♦s ❈♦ss❡♥♦s✱ ❝♦♠♦ a2 _=b2_+c2_{−2.b.c.❝♦sA✱ t❡♠♦s✿}ˆ

SA= −

a2_+b2_+c2

2 .

❆♥❛❧♦❣❛♠❡♥t❡✱

SB =a.c.❝♦sBˆ =

+a2_−b2_+c2

2 ❡ SC =a.b.❝♦sCˆ =

+a2_+b2_−c2

2 .

❆❧é♠ ❞✐ss♦✱

SB+SC =

+a2 _−b2_+c2

2 +

+a2 _+b2_−c2

2 =a

2_.

❆ s❡❣✉✐r✱ ❡♥✉♥❝✐❛♠♦s ❡ ❞❡♠♦♥str❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♥✇❛② ✭✈✐❞❡ ❬✷❪✮✱ q✉❡ ♥♦s ❞á ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ ✉♠ ♣♦♥t♦ ❡♠ ❢✉♥çã♦ ❞❡ â♥❣✉❧♦s q✉❡ ❡st❡ ♣♦♥t♦ ❢♦r♠❛ ❝♦♠ ✉♠ ❧❛❞♦ ❞♦ tr✐â♥❣✉❧♦ ❞❡ r❡❢❡rê♥❝✐❛✳

❚❡♦r❡♠❛ ✹✳✶ ✭❚❡♦r❡♠❛ ❞❡ ❈♦♥✇❛②✮ ❉❛❞♦s ✉♠ tr✐â♥❣✉❧♦ △ABC ❡ ✉♠ ♣♦♥t♦ P t❛❧ q✉❡ P ❡ A ❡st❡❥❛♠ ❡♠ ❧❛❞♦s ♦♣♦st♦s ❞❛ r❡t❛ ←→BC ✱ s❡❥❛♠ a=BC ✱ θ =∠CBP ❡ ϕ =∠BCP ❛s ♠❡❞✐❞❛s ❞♦s â♥❣✉❧♦s q✉❡ P ❢♦r♠❛ ❝♦♠ ♦ ❧❛❞♦BC✳ ❊♥tã♦✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❡ P sã♦

P = ₋a2 :SC+Sϕ :SB+Sθ

✹✳✶✳ ❆ ❋Ó❘▼❯▲❆ ❉❊ ❈❖◆❲❆❨ ✸✶

❋✐❣✉r❛ ✹✳✷✿ P ❡①t❡r♥♦ ❛♦ tr✐â♥❣✉❧♦ ABC✱ ❡ s❡✉s â♥❣✉❧♦s ❝♦♠ ♦ ❧❛❞♦ BC✳

Pr♦✈❛✿ P♦r ✭✷✳✶✷✮✱ t❡♠♦s q✉❡ P = (SP BC : SAP C : SABP)✳ ❊♥tã♦✱ ❛♣❧✐❝❛♥❞♦✲s❡ ❛ ▲❡✐

❞♦s ❙❡♥♦s ❛♦ tr✐â♥❣✉❧♦ △P CB✱ t❡♠♦s✿

s❡♥ϕ BP =

s❡♥θ CP =

s❡♥_Pˆ

BC . ❆ss✐♠✱

BP = BC.s❡♥ϕ

s❡♥_Pˆ =

a.s❡♥ϕ

s❡♥(π₋(θ+ϕ)) =

a.s❡♥ϕ s❡♥(θ+ϕ)

❡

CP = BC.s❡♥θ

s❡♥_Pˆ =

a.s❡♥θ

s❡♥(π₋(θ+ϕ)) =

a.s❡♥θ s❡♥(θ+ϕ).

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ △P BC é ❞❛❞❛ ♣♦r

1 2.a.

a.s❡♥ϕ

s❡♥(θ+ϕ).s❡♥θ=

a2_{s❡♥ϕ.s❡♥θ} 2.s❡♥(θ+ϕ)

❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ △P CAé ❞❛❞❛ ♣♦r

1 2.

a.s❡♥θ

s❡♥(θ+ϕ).b.s❡♥(π−( ˆC+ϕ)) =

✸✷ ❈❆P❮❚❯▲❖ ✹✳ ❚❊❖❘❊▼❆ ❉❊ ❈❖◆❲❆❨ ❊ ❖ P❖◆❚❖ ❉❊ ❋❊❘▼❆❚ ❚❛♠❜é♠ t❡♠♦s q✉❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ △P BA é ❞❛❞❛ ♣♦r

1 2.

a.s❡♥ϕ

s❡♥(θ+ϕ).c.s❡♥(π−( ˆB +θ)) =

a.c.s❡♥ϕ.s❡♥( ˆB+θ) 2.s❡♥(θ+ϕ) .

P♦r ✜♠✱ ❝♦♠♦ P = (SP BC :SAP C :SABP)✱ ❡♥tã♦✱

P = ₋a

2_{s❡♥ϕ.s❡♥θ} 2.s❡♥(θ+ϕ) :

a.b.s❡♥θ.s❡♥( ˆC+ϕ) 2.s❡♥(θ+ϕ) :

a.c.s❡♥ϕ.s❡♥( ˆB+θ) 2.s❡♥(θ+ϕ)

!

=

= ₋a2s❡♥ϕ.s❡♥θ:a.b.s❡♥θ.s❡♥( ˆC+ϕ) :a.c.s❡♥ϕ.s❡♥( ˆB+θ)=

= ₋a

2_{s❡♥ϕ.s❡♥θ}

s❡♥ϕ.s❡♥θ :

a.b.s❡♥θ.s❡♥( ˆC+ϕ)

s❡♥ϕ.s❡♥θ :

a.c.s❡♥ϕ.s❡♥( ˆB+θ)

s❡♥ϕ.s❡♥θ

!

=

= ₋a2 : a.b.s❡♥( ˆC+ϕ)

s❡♥ϕ :

a.c.s❡♥( ˆB+θ)

s❡♥θ

!

=

= ₋a2 :a.b.s❡♥C.❝♦t❣ϕˆ +a.b.❝♦sCˆ :a.c.s❡♥B.❝♦t❣θˆ +a.c.❝♦sBˆ=

= ₋a2 :a.b.s❡♥C.❝♦t❣ϕˆ +SC :a.c.s❡♥B.❝♦t❣θˆ +SB

= ₋a2 :Sϕ+SC :Sθ+SB

.

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ❞❛❞♦s ✉♠ tr✐â♥❣✉❧♦ △ABC✱ X′✱ _Y′ ❡ _Z′ ♣♦♥t♦s t❛✐s q✉❡ _X′ ❡ _A ❡stã♦ ❡♠ ❧❛❞♦s ♦♣♦st♦s ❞❛ r❡t❛ ←→BC✱ Y′ ❡ _B ❡stã♦ ❡♠ ❧❛❞♦s ♦♣♦st♦s ❞❛ r❡t❛ ←→AC✱ _Z′ ❡ C ❡stã♦ ❡♠ ❧❛❞♦s ♦♣♦st♦s ❞❛ r❡t❛ ←→AB✱ s❡ ♦s â♥❣✉❧♦s X′✱ _Y′ ❡ _Z′ ❢♦r♠❛♠ ❝♦♠ ♦s s❡✉s r❡s♣❡❝t✐✈♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ â♥❣✉❧♦s ❞❡ ♠❡❞✐❞❛s θi ❡ ϕi✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ ✹✳✸✱ ❡♥tã♦ X′✱

Y′ ❡ _Z′ tê♠ ❝♦♦r❞❡♥❛❞❛s ❜❛r✐❝ê♥tr✐❝❛s ❞❛❞❛s ♣♦r