❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈
❈❡♥tr♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❈♦♠♣✉t❛çã♦ ❡ ❈♦❣♥✐çã♦
Pr♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦
❖❡rt❡s ❆❧✈❡s ❙♦✉③❛
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❥✉♥t♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧✐③❛♥t❡ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
Pr♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦
❊st❡ ❡①❡♠♣❧❛r ❝♦rr❡s♣♦♥❞❡ à r❡❞❛çã♦ ✜♥❛❧ ❞❛ ❞✐ss❡rt❛çã♦ ❞❡✈✐❞❛♠❡♥t❡ ❝♦r✲ r✐❣✐❞❛ ❡ ❞❡❢❡♥❞✐❞❛ ♣♦r ❖❡rt❡s ❆❧✈❡s ❙♦✉③❛ ❡ ❛♣r♦✈❛❞❛ ♣❡❧❛ ❝♦♠✐ssã♦ ❥✉❧❣❛✲ ❞♦r❛✳
❙❛♥t♦ ❆♥❞ré✱ ✷✷ ❞❡ ♦✉t✉❜r♦ ❞❡ ✷✵✶✹✳
Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛ ❖r✐❡♥t❛❞♦r
❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿
✶✳ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛ ✭❖r✐❡♥t❛❞♦r✮ ✲ ❯❋❆❇❈ ✷✳ Pr♦❢✳ ❉r✳ ❆♥❞ré ❘✐❝❛r❞♦ ❖❧✐✈❡✐r❛ ❞❛ ❋♦♥s❡❝❛ ✲ ❯❋❆❇❈ ✸✳ Pr♦❢✳ ❉r❛✳ ❘♦s❛ ▼❛r✐❛ ❞♦s ❙❛♥t♦s ❇❛rr❡✐r♦ ❈❤❛✈❡s ✲ ❯❙P
❊st❡ ❡①❡♠♣❧❛r ❢♦✐ r❡✈✐s❛❞♦ ❡ ❛❧t❡r❛❞♦ ❡♠ r❡❧❛çã♦ à ✈❡rsã♦ ♦r✐❣✐♥❛❧✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ♦❜s❡r✈❛çõ❡s ❧❡✈❛♥t❛❞❛s ♣❡❧❛ ❜❛♥❝❛ ♥♦ ❞✐❛ ❞❛ ❞❡❢❡s❛✱ s♦❜ ❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ú♥✐❝❛ ❞♦ ❛✉t♦r ❡ ❝♦♠ ❛ ❛♥✉ê♥❝✐❛ ❞♦ s❡✉ ♦r✐❡♥t❛❞♦r✳
❙❛♥t♦ ❆♥❞ré✱ ✶✽ ❞❡ ❞❡③❡♠❜r♦ ❞❡ ✷✵✶✹✳
❖❡rt❡s ❆❧✈❡s ❙♦✉③❛ ❆✉t♦r
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠❛t✐❝❛ ✭P❘❖❋▼❆❚✮✱ ❛♦s Pr♦❢❡ss♦r❡s ❞❛ ❯❋❆❇❈ q✉❡ s❡ ❡♥❣❛❥❛r❛♠ ♥❡st❡ ♣r♦❣r❛♠❛✱ ❛♦ ♠❡✉ ❛♠✐❣♦ ❈í❝❡r♦ ❚❤✐❛❣♦ ▼❛❣❛❧❤ã❡s✱ ♣❡❧❛ s✉❣❡stã♦ ❞♦ t❡♠❛✱ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛✱ q✉❡ ❞❡ ♣r♦♥t♦ ❛❝❡✐t♦✉ ♠❡✉ ♣❡❞✐❞♦ ❞❡ ♦r✐❡♥t❛çã♦✱ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ▼❡str❛❞♦ ❘✐❝❛r❞♦ ❆✳ ▼♦r✐✱ ▲❡❛♥❞r♦ ❆✳ ▼♦s❝❛ ❡ ▼❛r❝❡❧♦ ❆✳ ❙♦✉③❛ ♣❡❧❛s ✐♥ú♠❡r❛s ✈❡③❡s ❡♠ q✉❡ ❡st✉❞❛♠♦s ❥✉♥t♦s ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ à ♠✐♥❤❛ ❢❛♠í❧✐❛✱ q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ♥❛ ♠✐♥❤❛ ❝❛rr❡✐r❛✱ ❞❡s❞❡ ♦ ✐♥í❝✐♦✳
❘❡s✉♠♦
■♥s♣✐r❛❞♦ ❡♠ ✉♠ ❝❛♣ít✉❧♦ ❞♦ ▲✐✈r♦ ❞❡ ■✳ ▼✳ ❨❛❣❧♦♠ ❬✼❪✱ ♥❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛r❡♠♦s ❛ ❣❡✲ ♦♠❡tr✐❛ ✐♥✈❡rs✐✈❛ ❛ ✜♠ ❞❡ r❡s♦❧✈❡r ❛❧❣✉♥s ❞♦s ❛♥t✐❣♦s ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦ ❞❡ Pér❣❛♠♦✱ ❛♣❡♥❛s ❝♦♠ ♦ ✉s♦ ❞❡ ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ♦✉ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ✉♠ s♦❢t✇❛r❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛✳
❆❜str❛❝t
❇❛s❡❞ ♦♥ t❤❡ ✇♦r❦ ♦❢ ■✳ ▼✳ ❨❛❣❧♦♠ ❬✼❪✱ ✐♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② t❤❡ ✐♥✈❡rs✐✈❡ ❣❡♦♠❡tr② t♦ s♦❧✈❡ s♦♠❡ ♦❧❞ ♣r♦❜❧❡♠s ♦❢ ❆♣♦❧❧♦♥✐✉s ♦❢ P❡r❣❛♠✉♠ ✱ ❥✉st ✉s✐♥❣ ❛ r✉❧❡r ❛♥❞ ❝♦♠♣❛ss ♦r ✇✐t❤ t❤❡ ❛✐❞ ♦❢ ❛ ❞②♥❛♠✐❝ ❣❡♦♠❡tr② s♦❢t✇❛r❡✳
P❛❧❛✈r❛s✲❝❤❛✈❡
❆♣♦❧ô♥✐♦✱ Pr♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦✱ ●❡♦♠❡tr✐❛ ■♥✈❡rs✐✈❛✳
❑❡②✇♦r❞s
❙✉♠ár✐♦
▲✐st❛ ❞❡ ❋✐❣✉r❛s ✈✐✐✐
■♥tr♦❞✉çã♦ ✶
✶ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✸
✶✳✶ ❈❛s♦s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❞❡ tr✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❈❛s♦s ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✿ AA✱ LAL ❡ LLL ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✸ ❘❡❧❛çõ❡s ♠étr✐❝❛s ❡♠ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ❚❡♦r❡♠❛ ❞♦ ➶♥❣✉❧♦ ■♥s❝r✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✺ ❘❡❧❛çõ❡s ♠étr✐❝❛s ♥✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✷ ❈✐r❝✉♥❢❡rê♥❝✐❛s ❖rt♦❣♦♥❛✐s ✶✸
✷✳✶ ➶♥❣✉❧♦ ❡♥tr❡ r❡t❛ ❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ➶♥❣✉❧♦ ❡♥tr❡ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✸ ❆ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✶✼
✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ ■♥✈❡rsã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✷ ❆ ✐♥✈❡rsã♦ ♣r❡s❡r✈❛ â♥❣✉❧♦ ❡♥tr❡ ❞✉❛s ❝✉r✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✹ ❖s ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦ ✸✶
✹✳✶ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ♣❛r❛ três ❝✐r❝✉♥❢❡rê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✺ ❊①❡r❝í❝✐♦s ✹✸
✺✳✶ ❘❡s♦❧✉çõ❡s ❞♦s ❡①❡r❝í❝✐♦s ♣r♦♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✺✳✷ P❧❛♥♦ ❞❡ ❛✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✈✐✐✐ ❙❯▼➪❘■❖ ✺✳✷✳✶ ❆✉❧❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✺✳✷✳✷ ❆✉❧❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✺✳✸ ❚✉t♦r✐❛❧ ❞♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❝❛s♦ ▲❆▲✿ ∆ABC ∼= ∆DEF ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✷ ❝❛s♦ ❆▲❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❝❛s♦ ▲▲▲ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✹ ❝❛s♦ ❆❆ ❞❡ s❡♠❡❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✺ ❝❛s♦ ▲❆▲ ❞❡ s❡♠❡❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✻ ❝❛s♦ ▲▲▲ ❞❡ s❡♠❡❧❤❛♥ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✼ ❘❡❧❛çõ❡s ♠étr✐❝❛s ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✽ ❚❡♦r❡♠❛ ❞❛s ❈♦r❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✾ ❚❡♦r❡♠❛ ❞❛s ❙❡❝❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✵ ❚❡♦r❡♠❛ ❞❛ ❙❡❝❛♥t❡ ❡ ❚❛♥❣❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✶ ❖ â♥❣✉❧♦ ❡♥tr❡ ❛ r❡t❛ ❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ é ♥✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ❖ â♥❣✉❧♦ ❡♥tr❡ ❛ r❡t❛ ❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♠❡❞❡ α ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷✳✸ ❖ â♥❣✉❧♦ ❡♥tr❡ ❛ r❡t❛ ❡ ❛ ❝✐r❝✉♥❢❡rê❝✐❛ ♠❡❞❡ 90♦ ✭❛ r❡t❛ ❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛
sã♦ ♦rt♦❣♦♥❛✐s✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✹ ❖s â♥❣✉❧♦s θ1 ❡θ2 sã♦ ❝♦♥❣r✉❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✺ ♦ â♥❣✉❧♦ ❡♥tr❡ Γ1 ❡ Γ2 éθ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✸✳✶ ❆ ❡ ❆✬ s❛♦ s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ à r❡t❛ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸✳✷ Γ1 ❡Γ2 sã♦ ♦rt♦❣♦♥❛✐s ❛ Γ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✸✳✸ ✐♥✈❡rsã♦ ❞❡ ✉♠ ♣♦♥t♦ ✐♥t❡r♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✹ ◆❛ ✐♥✈❡rsã♦ ❝♦♠ r❡❧❛çã♦ ❛ Γ✱ ♦ â♥❣✉❧♦ OABˆ é tr❛♥s❢♦r♠❛❞♦ ♥♦ â♥❣✉❧♦
OBˆ′A′ ❡ ♦ â♥❣✉❧♦ OBAˆ é tr❛♥s❢♦r♠❛❞♦ ♥♦ â♥❣✉❧♦ OAˆ′B′✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✸✳✺ ✐♥✈❡rsã♦ ❞❡ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r O ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✸✳✻ s′ é ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❞✐â♠❡tr♦ OP′✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✸✳✼ Γ′
1 é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦rP′ ❡ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ OP′ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸✳✽ ✐♥✈❡rsã♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝♦♠ O ♥♦ ✐♥t❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
① ▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✸✳✾ ✐♥✈❡rsã♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝♦♠ O ♥♦ ❡①t❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✶✵ Ωé ✉♠❛ r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✶✶ Ωé ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝♦♠ O ∈Ω ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✶✷ Ωé ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝♦♠ O /∈Ω ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✶✸ ✐♥✈❡rsã♦ ♣r❡s❡r✈❛ â♥❣✉❧♦ ❡♥tr❡ r❡t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✹✳✶ ❈❛s♦ ❞♦✐s ♣♦♥t♦s ❡ ✉♠❛ r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✷ Γ1 ❡Γ2 sã♦ ❛s s♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✹✳✸ Σé ♦rt♦❣♦♥❛❧ ❛ Γ1 ❡ Γ2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✹✳✹ Γ′
1 ❡Γ′2 sã♦ ❝♦♥❝ê♥tr✐❝❛s ❞❡ ❝❡♥tr♦ {C}=s∩Σ′ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✹✳✺ Ω′
1 ❡ Ω′2 sã♦ t❛♥❣❡♥t❡s ❡①t❡r♥❛♠❡♥t❡ ❛Γ′2 ❡ ❛ Γ′3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✹✳✻ Ω′
3 ❡ Ω′4 sã♦ t❛♥❣❡♥t❡s ❡①t❡r♥❛♠❡♥t❡ ❛Γ′2 ❡ ✐♥t❡r♥❛♠❡♥t❡ ❛ Γ′3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✹✳✼ Ω′
5 ❡ Ω′6 sã♦ t❛♥❣❡♥t❡s ❡①t❡r♥❛♠❡♥t❡ ❛Γ′3 ❡ ✐♥t❡r♥❛♠❡♥t❡ ❛ Γ′2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✹✳✽ Ω′
7 ❡ Ω′8 sã♦ t❛♥❣❡♥t❡s ✐♥t❡r♥❛♠❡♥t❡ ❛Γ′2 ❛ Γ′3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✹✳✾ Ω′i, i= 1,2,3, ...,8✱ sã♦ t❛♥❣❡♥t❡s ❛Γ′
1✱Γ′2 ❡ Γ′3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✹✳✶✵ Ωi, i= 1,2,3, ...,8✱ sã♦ t❛♥❣❡♥t❡s ❛Γ1✱Γ2 ❡ Γ3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✺✳✶ ❙♦❧✉çõ❡s Ω′
1✱ Ω′2✱ Ω′3 ❡ Ω′4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✺✳✷ ❙♦❧✉çõ❡s Ω1✱ Ω2✱ Ω3 ❡ Ω4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✺✳✸ ❙♦❧✉çõ❡s Ω′
1 ❡ Ω′2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✺✳✹ ❙♦❧✉çõ❡s Ω1 ❡ Ω2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✺✳✺ ❋❡rr❛♠❡♥t❛ ✐♥✈❡rsã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✺✳✻ ■♥✈❡rt❡♥❞♦ ♦s ♣♦♥t♦s A✱ B ❡C✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
✺✳✼ Σé ♦rt♦❣♦♥❛❧ ❛ Γ1 ❡ ❛ Γ2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✺✳✽ Γ′
■♥tr♦❞✉çã♦
❆♣♦❧ô♥✐♦ ❞❡ Pér❣❛♠♦ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❡ ❛strô♥♦♠♦ ❣r❡❣♦ q✉❡ t❡r✐❛ ✈✐✈✐❞♦ ❡♥tr❡ ♦s ❛♥♦s ✷✻✷ ❛ ✶✾✵ ❛✳❝✳ P♦✉❝♦ s❡ s❛❜❡ s♦❜r❡ s✉❛ ✈✐❞❛ ❡ s✉❛s ♦❜r❛s✱ ♣♦✐s ❡st❛s ❞❡s❛♣❛r❡❝❡r❛♠ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✳ ❊♠ ✉♠ ❞♦s s❡✉s tr❛❜❛❧❤♦s✱ ❞❡♥♦♠✐♥❛❞♦ ❚❛♥❣ê♥❝✐❛s ✱ ❛♣❛r❡❝❡ ♦ ❝❤❛♠❛❞♦ Pr♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦✱ q✉❡ ♣♦❞❡ s❡r ❛ss✐♠ ❡♥✉♥❝✐❛❞♦✿ ❞❛❞♦s três ♦❜❥❡t♦s ❞✐st✐♥t♦s✱ q✉❡ ♣♦❞❡♠ s❡r ✉♠ ♣♦♥t♦✱ ✉♠❛ r❡t❛ ♦✉ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ tr❛ç❛r ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ s❡❥❛ t❛♥❣❡♥t❡ ❛ ❡st❡s três ♦❜❥❡t♦s ✭♦♥❞❡ t❛♥❣ê♥❝✐❛ ❛ ✉♠ ♣♦♥t♦ s✐❣♥✐✜❝❛ q✉❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♣❛ss❛ ♣♦r ❛q✉❡❧❡ ♣♦♥t♦✮✳
◗✉❛♥t♦s ❛♦s ♦❜❥❡t♦s✱ t❡♠♦s ✶✵ s✐t✉❛çõ❡s ♣♦ssí✈❡✐s✿ ✶✮ três ♣♦♥t♦s❀
✷✮ ❞♦✐s ♣♦♥t♦s ❡ ✉♠❛ r❡t❛❀
✸✮ ❞♦✐s ♣♦♥t♦s ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛❀ ✹✮ ✉♠ ♣♦♥t♦ ❡ ❞✉❛s r❡t❛s❀
✺✮ ✉♠ ♣♦♥t♦ ❡ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s❀
✻✮ ✉♠ ♣♦♥t♦✱ ✉♠❛ r❡t❛ ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛❀ ✼✮ ✉♠❛ r❡t❛ ❡ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s❀
✽✮ ❞✉❛s r❡t❛s ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛❀ ✾✮ três r❡t❛s❀
✶✵✮ três ❝✐r❝✉♥❢❡rê♥❝✐❛s✳
❊✈✐❞❡♥t❡♠❡♥t❡✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ❞✐s♣♦s✐çã♦ ❞♦s três ♦❜❥❡t♦s✱ ♦ ♣r♦❜❧❡♠❛ ♣♦❞❡ ♥ã♦ t❡r s♦❧✉çã♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ q✉❛♥❞♦ ♥♦ ❝❛s♦ ✶ ♦s três ♣♦♥t♦s ❢♦r❡♠ ❝♦❧✐♥❡❛r❡s ♦✉ ♥♦ ❝❛s♦ ✷✱ q✉❛♥❞♦ ♦s ♣♦♥t♦s ♣❡rt❡♥❝❡r❡♠ ❛ s❡♠✐♣❧❛♥♦s ♦♣♦st♦s ❡♠ r❡❧❛çã♦ à r❡t❛✳ ❆❧é♠ ❞✐ss♦✱ ❡♠ ❝❛❞❛ ❝❛s♦ ♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ♣♦❞❡ ✈❛r✐❛r✳ ❆♣❡♥❛s ♣❛r❛ ❡①❡♠♣❧✐✜❝❛r✱ ♥♦ ❝❛s♦ ✶✱ s❡ ♦s três ♣♦♥t♦s ♥ã♦ ❢♦r❡♠ ❝♦❧✐♥❡❛r❡s✱ ♦ ♣r♦❜❧❡♠❛ ❛❞♠✐t❡ ❡①❛t❛♠❡♥t❡ ✉♠❛ s♦❧✉çã♦✱ q✉❡ é
✷ ■♥tr♦❞✉çã♦ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝✐r❝✉♥s❝r✐t❛ ❛♦ tr✐â♥❣✉❧♦ ❝✉❥♦s ✈ért✐❝❡s sã♦ ❡st❡s ♣♦♥t♦s✳ ◆♦ ❝❛s♦ ✾✱ s❡ ❛s r❡t❛s ❢♦r❡♠ ❞✉❛s ❛ ❞✉❛s ♣❛r❛❧❡❧❛s ❡♥tr❡ s✐✱ ♦ ♣r♦❜❧❡♠❛ ♥ã♦ ❛❞♠✐t❡ s♦❧✉çã♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ❛s três r❡t❛s ❢♦r❡♠ ❝♦♥❝♦rr❡♥t❡s ❞✉❛s ❛ ❞✉❛s✱ ♦ ♣r♦❜❧❡♠❛ ❛❞♠✐t❡ ✹ s♦❧✉çõ❡s ✭❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ❛♦ tr✐â♥❣✉❧♦ ❝✉❥♦s ✈ért✐❝❡s sã♦ ♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❡♥tr❡ ❛s r❡t❛s ❡ ❛s três ❝❤❛♠❛❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡①✲✐♥s❝r✐t❛s ❛ ❡st❡ tr✐â♥❣✉❧♦✮✳
❉❡st❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ♥♦t❛r q✉❡ ❤á ✉♠❛ ❝❡rt❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ tr❛t❛r♠♦s t♦❞♦s ♦s ❝❛s♦s✳ P❛rt✐❝✉❧❛r♠❡♥t❡✱ ♥❡st❡ tr❛❜❛❧❤♦ ✈❛♠♦s r❡s♦❧✈❡r ❛❧❣✉♥s ❞❡st❡s ❝❛s♦s✱ ✐♥❝❧✉✐♥❞♦ ♦ ❝❛s♦ ✶✵✱ s❡♥❞♦ ❛s três ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡①t❡r♥❛s ❞✉❛s ❛ ❞✉❛s✳ ▼♦str❛r❡♠♦s q✉❡✱ ♥❡st❡ ❝❛s♦✱ ❤á ❡①❛t❛♠❡♥t❡ ✽ s♦❧✉çõ❡s✳ P❛r❛ ✐ss♦✱ ❡①♣❧♦r❛♠♦s ❡ ❞❡♠♦♥str❛♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ●❡♦♠❡tr✐❛ ■♥✈❡rs✐✈❛✳ ❖s ♣ré✲r❡q✉✐s✐t♦s ♣❛r❛ ✐ss♦ sã♦ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ P❧❛♥❛ ❡♥s✐♥❛❞♦s ❞❡s❞❡ ♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✱ ❝♦♠♦ ❝♦♥❣r✉ê♥❝✐❛ ❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s ❡ r❡❧❛çõ❡s ♠étr✐❝❛s ❡♠ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❛ ❝♦♥❝❡✐t♦s ✉♠ ♣♦✉❝♦ ♠❛✐s ❡s♣❡❝í✜❝♦s✱ ❝♦♠♦ r❡❧❛çõ❡s ♠étr✐❝❛s ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♣♦tê♥❝✐❛ ❞❡ ♣♦♥t♦ ❡ t❡♦r❡♠❛ ❞♦ â♥✲ ❣✉❧♦ ✐♥s❝r✐t♦✱ ❡♥s✐♥❛❞♦s ❣❡r❛❧♠❡♥t❡ ♥♦ s❡❣✉♥❞♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❙❡r✐❛ ✐♥t❡r❡ss❛♥t❡✱ t❛♠❜é♠✱ q✉❡ ♦ ❧❡✐t♦r ✜③❡ss❡ ✉s♦ ❞❡ ❛❧❣✉♠ s♦❢t✇❛r❡ ❞❡ ●❡♦♠❡tr✐❛ ❉✐♥â♠✐❝❛✳
❊st❡ tr❛❜❛❧❤♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ ✺ ❝❛♣ít✉❧♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✶ sã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ P❧❛♥❛ q✉❡ sã♦ ❛❜♦r❞❛❞♦s ♥♦s ♣r♦❣r❛♠❛s ❞♦s ❊♥s✐♥♦s ❋✉♥❞❛♠❡♥t❛❧ ❡ ▼é❞✐♦✳ ◆♦ ✐♥í❝✐♦ ❞♦ ❝❛♣ít✉❧♦ ✉t✐❧✐③❛♠♦s ✉♠ ❢❛t♦ r❡❝♦♥❤❡❝✐❞♦✱ ♣♦ré♠ ♥ã♦ ❞❡♠♦♥str❛❞♦✱ ♦ q✉❛❧ ❝❤❛♠❛♠♦s ❞❡ ♣♦st✉❧❛❞♦✳
◆♦ ❝❛♣ít✉❧♦ ✷✱ ❞❡✜♥✐♠♦s ♦ â♥❣✉❧♦ ❡♥tr❡ ✉♠❛ r❡t❛ ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❜❡♠ ❝♦♠♦ ♦ â♥❣✉❧♦ ❡♥tr❡ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s✱ ❡♥❢❛t✐③❛♥❞♦ ♦ ❝❛s♦ ❡♠ q✉❡ ❡st❡s ♦❜❥❡t♦s sã♦ ♦rt♦❣♦♥❛✐s✳
◆♦ ❝❛♣ít✉❧♦ ✸✱ ❛❜♦r❞❛♠♦s ❛ ❣❡♦♠❡tr✐❛ ✐♥✈❡rs✐✈❛✳ ❆❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s sã♦ ❞❡♠♦♥str❛❞❛s✱ ❝♦♠♦ ♦s ✐♥✈❡rs♦s ❞❡ ✉♠ ♣♦♥t♦✱ ❞❡ ✉♠❛ r❡t❛ ♦✉ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❯♠ r❡s✉❧t❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡♠♦♥str❛❞♦ ♥❡st❡ ❝❛♣ít✉❧♦ é q✉❡ ❛ ✐♥✈❡rsã♦ ♣r❡s❡r✈❛ ♦ â♥❣✉❧♦ ❡♥tr❡ ♦s ♦❜❥❡t♦s tr❛t❛❞♦s ♥♦ ❡st✉❞♦ ❞♦s ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦✳
◆♦ ❝❛♣ít✉❧♦ ✹✱ ❡♥✉♥❝✐❛♠♦s✱ ♥♦ ❝❛s♦ ❣❡r❛❧✱ ♦s ♣r♦❜❧❡♠❛s ❞❡ ❆♣♦❧ô♥✐♦✳ ❆❧❣✉♥s ❝❛s♦s sã♦ r❡s♦❧✈✐❞♦s✱ ✐♥❝❧✉✐♥❞♦ ♦ ❝❛s♦ ♦♥❞❡ ♦s três ♦❜❥❡t♦s sã♦ ❝✐r❝✉♥❢❡rê♥❝✐❛s ❡①t❡r♥❛s ❞✉❛s ❛ ❞✉❛s✳
❈❛♣ít✉❧♦ ✶
❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s
◆♦t❛çõ❡s
❘❡♣r❡s❡♥t❛r❡♠♦s ♦ s❡❣♠❡♥t♦ ❞❡ ❡①tr❡♠♦s A ❡B ♣♦rAB✳ ❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s A ❡B✱ ✐st♦ é✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ AB ✐♥❞✐❝❛r❡♠♦s ♣♦r m(AB)✳ P❛r❛ s✐♠♣❧✐✜❝❛r
❛ ♥♦t❛çã♦✱ ✉s❛r❡♠♦s m(AB) = AB✳ ❘❡♣r❡s❡♥t❛r❡♠♦s ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ABCˆ ♣♦r m(ABCˆ )✳ ❉❛❞♦s três ♣♦♥t♦s ❞✐st✐♥t♦s ❡ ❝♦❧✐♥❡❛r❡s A, B ❡C✱ ❞✐③❡♠♦s q✉❡ B ❡stá ❡♥tr❡ A
❡ C✱ ❡ ❡s❝r❡✈❡♠♦s✱ A−B−C✱ q✉❛♥❞♦AB +BC =AC✳
❉❡✜♥✐çã♦ ✶✳✶✳ ❉♦✐s s❡❣♠❡♥t♦s AB ❡ CD sã♦ ❝♦♥❣r✉❡♥t❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ AB =CD✱
✐st♦ é✱ AB ∼=CD ⇐⇒AB =CD✳
❉❡✜♥✐çã♦ ✶✳✷✳ ❉♦✐s â♥❣✉❧♦s ABCˆ ❡ DEFˆ sã♦ ❝♦♥❣r✉❡♥t❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❡s tê♠ ❛
♠❡s♠❛ ♠❡❞✐❞❛✱ ✐st♦ é✱ ABCˆ ∼=DEFˆ ⇐⇒m(ABCˆ ) = m(DEFˆ ).
❉❡✜♥✐çã♦ ✶✳✸✳ ❉♦✐s tr✐â♥❣✉❧♦s ∆ABC ❡ ∆DEF sã♦ ❝♦♥❣r✉❡♥t❡s s❡ ❤♦✉✈❡r ✉♠❛ ❝♦rr❡s✲
♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ s❡✉s ✈ért✐❝❡s✱ A ←→ D✱ B ←→ E ❡ C ←→ F✱ ❞❡ t❛❧ ♠♦❞♦
q✉❡ AB ∼= DE✱ AC ∼= DF ❡ BC ∼= EF ❡✱ ❛❧é♠ ❞✐ss♦✱ BACˆ ∼= EDFˆ ✱ ABCˆ ∼=DEFˆ ❡ ACBˆ ∼=DF Eˆ ✳
◗✉❛♥❞♦ ❡s❝r❡✈❡r♠♦s∆ABC ∼= ∆DEF✱ s❡rá ❛❞♠✐t✐❞❛ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛A ←→D✱ B ←→E ❡ C←→F ❡♥tr❡ s❡✉s ✈ért✐❝❡s✳
❖s ♣ré✲r❡q✉✐s✐t♦s ♣❛r❛ ❛ ❧❡✐t✉r❛ ❞❡st❡ t❡①t♦ sã♦ ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❡ ●❡✲ ♦♠❡tr✐❛ P❧❛♥❛✱ q✉❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ q✉❛❧q✉❡r ❧✐✈r♦ ❞♦s ❊♥s✐♥♦s ❋✉♥❞❛♠❡♥t❛❧ ❡✴♦✉ ▼é❞✐♦✳ ❆ s❡❣✉✐r✱ ♣❛ss❛♠♦s ❛ ❧✐st❛r ♦s r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ♦ ❞❡s❡♥✲ ✈♦❧✈✐♠❡♥t♦ ❞♦ t❡①t♦✳ ❆❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s s❡rã♦ ♦♠✐t✐❞❛s✱ ♠❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✸❪ ♦✉ ❡♠ ❬✹❪✳
✹ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙
✶✳✶ ❈❛s♦s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❞❡ tr✐â♥❣✉❧♦s
❆❞♠✐t✐r❡♠♦s ❝♦♠♦ ♣♦st✉❧❛❞♦✱ ♦ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡ ❝❛s♦LAL✱ ♦ s❡❣✉✐♥t❡ ❢❛t♦✿ ❙❡ ❞♦✐s
tr✐â♥❣✉❧♦s ∆ABC ❡ ∆DEF sã♦ t❛✐s q✉❡ AB =DE✱BC =EF ❡ m(ABCˆ ) =m(DEFˆ )✱
❡♥tã♦ ∆ABC ∼= ∆DEF✳
❋✐❣✉r❛ ✶✳✶✿ ❝❛s♦ ▲❆▲✿∆ABC ∼= ∆DEF
❆ ♣❛rt✐r ❞❡st❡ ♣♦st✉❧❛❞♦✱ ♣♦❞❡♠♦s ♣r♦✈❛r ♦✉tr♦s ❞♦✐s ❝❛s♦s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❞❡ tr✐â♥✲ ❣✉❧♦s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳
❚❡♦r❡♠❛ ✶✳✶✳ ✭❈❛s♦ ❆▲❆✮ ❉❛❞♦s ❞♦✐s tr✐â♥❣✉❧♦s ∆ABC ❡ ∆DEF ❡ ✉♠❛ ❝♦rr❡s♣♦♥✲
❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ s❡✉s ✈ért✐❝❡s A←→D✱ B ←→E ❡ C ←→F✱ s❡
ABCˆ ∼=DEFˆ
BC =EF
ACBˆ ∼=DF Eˆ
❡♥tã♦ ∆ABC ∼= ∆DEF✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ D′ ✉♠ ♣♦♥t♦ ❞❛ s❡♠✐rr❡t❛ −−→ED t❛❧ q✉❡ AB =D′E✱ ❝♦♠♦ ♥❛ ❋✐❣✉r❛
✶✳✷✳ P❡❧♦ ❝❛s♦ LAL✱ t❡♠♦s q✉❡ ∆ABC ∼= ∆D′EF✱ ❞❡ ♠♦❞♦ q✉❡ m(ACBˆ ) =m(D′F Eˆ )✳
▼❛s m(ACBˆ ) = m(DF Eˆ ) ❡✱ ♣♦rt❛♥t♦✱ m(DF Eˆ ) = m(D′F Eˆ ) ❡ ❝♦♠♦ D′ ∈ −−→ED✱ s❡❣✉❡
✶✳✶✳ ❈❆❙❖❙ ❉❊ ❈❖◆●❘❯✃◆❈■❆ ❉❊ ❚❘■➶◆●❯▲❖❙ ✺
❋✐❣✉r❛ ✶✳✷✿ ❝❛s♦ ❆▲❆
❚❡♦r❡♠❛ ✶✳✷✳ ✭❈❛s♦ ▲▲▲✮ ❉❛❞♦s ❞♦✐s tr✐â♥❣✉❧♦s∆ABC ❡∆DEF ❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥✲
❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ s❡✉s ✈ért✐❝❡s A ←→D✱ B ←→E ❡ C ←→F✱ s❡
AB =DE
AC =DF
BC =EF
❡♥tã♦ ∆ABC ∼= ∆DEF.
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛D′ ✉♠ ♣♦♥t♦ ❞♦ s❡♠✐♣❧❛♥♦ ♦♣♦st♦ ❛♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦rA ❡ ♣❡❧❛ r❡t❛
←→
BC t❛❧ q✉❡D′B =DE ❡ D′BCˆ ∼=DEFˆ ✳ P❡❧♦ ❝❛s♦LAL✱ t❡♠♦s q✉❡ ∆DEF ∼= ∆D′BC✳
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ D′C = DF✳ ❈♦♠♦ D′ ❡ A ❡stã♦ ❡♠ ❧❛❞♦s ♦♣♦st♦s ❞❡ ←→BC ❡♥tã♦ D′A∩←→BC 6=φ✳ ❙❡❥❛ {X}=D′A∩←→BC✳
❉❡✈❡♠ ♦❝♦rr❡rX−B−C✱ X=B✱B−X−C✱X =C♦✉B−C−X✳ ❈♦♥t✐♥✉❛♠♦s ❛
♣r♦✈❛ ♣❛r❛ ♦ ❝❛s♦B−X−C✱ ❝♦♠♦ ✐❧✉str❛❞♦ ♥❛ ❋✐❣✉r❛ ✶✳✸✳ ❖s ♦✉tr♦s ❝❛s♦s sã♦ ❛♥á❧♦❣♦s✳
❖s tr✐â♥❣✉❧♦s∆ABD′❡∆ACD′sã♦ ✐sós❝❡❧❡s✱ ❝♦♠ ❜❛s❡ ❝♦♠✉♠AD′✳ ❊♥tã♦✱m(BADˆ ′) = m(BDˆ′A) ❡ m(CADˆ ′) = m(CDˆ′A)✳ ❙❡❣✉❡ q✉❡ m(BACˆ ) = m(BDˆ′C) ❡ ♣❡❧♦ ❝❛s♦ LAL
✻ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙
❋✐❣✉r❛ ✶✳✸✿ ❝❛s♦ ▲▲▲ ❞❡ ❝♦♥❣r✉ê♥❝✐❛
✶✳✷ ❈❛s♦s ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✿
AA
✱
LAL
❡
LLL
❉❡✜♥✐çã♦ ✶✳✹✳ ❉✐③❡♠♦s q✉❡ ❞♦✐s tr✐â♥❣✉❧♦s∆ABC ❡ ∆DEF sã♦ s❡♠❡❧❤❛♥t❡s ✭❡ s✐♠❜♦✲
❧✐③❛r❡♠♦s ♣♦r ∆ABC ∼∆DEF✮ s❡ ❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ABC ←→DEF t❛❧ q✉❡ BACˆ ∼=EDFˆ ✱ ABCˆ ∼=DEFˆ ✱ BCAˆ ∼=EF Dˆ ❡ AB
DE =
AC
DF =
BC EF.
◗✉❛♥❞♦ ❡s❝r❡✈❡r♠♦s ∆ABC ∼ ∆DEF✱ s❡rá ❛❞♠✐t✐❞❛ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ A ←→
D✱ B ←→E ❡ C ←→F ❡♥tr❡ s❡✉s ✈ért✐❝❡s✳
❚❡♦r❡♠❛ ✶✳✸✳ ✭❈❛s♦ ❆❆✮ ❙❡ ❞♦✐s tr✐â♥❣✉❧♦s∆ABC ❡∆DEF sã♦ t❛✐s q✉❡BACˆ ∼=EDFˆ
❡ ABCˆ ∼=DEFˆ ✱ ❡♥tã♦ ∆ABC ∼∆DEF✳
❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ♥♦t❡ q✉❡ m(ACBˆ ) = 180♦ −m(BACˆ ) − m(ABCˆ ) = 180♦−m(EDFˆ )−m(DEFˆ ) =m(DF Eˆ ) ❡✱ ♣♦rt❛♥t♦✱ACBˆ ∼=DF Eˆ ✳ ❆ s❡❣✉✐r✱ s❡❥❛♠E′
❡ F′ ♣♦♥t♦s ❞❛s s❡♠✐rr❡t❛s −→AB ❡ −→AC✱ t❛✐s q✉❡ AE′ =DE ❡ AF′ =DF✱ ❝♦♠♦ ♥❛ ❋✐❣✉r❛
✶✳✹✳ ❆ss✐♠✱ ♦s tr✐â♥❣✉❧♦s ∆AE′F′ ❡ ∆DEF sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ♣♦st✉❧❛❞♦ LAL✱ ❞❡
♦♥❞❡ s❡❣✉❡ q✉❡ m(ABCˆ ) =m(DEFˆ ) =m(AEˆ′F′) ❡ m(ACBˆ ) =m(DF Eˆ ) =m(AFˆ′E′)
❡✱ ♣♦rt❛♥t♦✱←−→E′F′ k←→BC✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❚❛❧❡s t❡♠♦s q✉❡ AE ′
AB =
AF′ AC.▲♦❣♦
AB
DE =
AC DF✳
❉❡ ♠❛♥❡✐r❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ❛♥á❧♦❣❛ ♣♦❞❡♠♦s ♠♦str❛r q✉❡ AB
DE =
BC
EF. P♦rt❛♥t♦✱∆ABC
✶✳✷✳ ❈❆❙❖❙ ❉❊ ❙❊▼❊▲❍❆◆➬❆ ❉❊ ❚❘■➶◆●❯▲❖❙✿AA✱LAL ❊ LLL ✼
❋✐❣✉r❛ ✶✳✹✿ ❝❛s♦ ❆❆ ❞❡ s❡♠❡❧❤❛♥ç❛
❖s ♦✉tr♦s ❞♦✐s ❝❛s♦s ❞❡ s❡♠❡❧❤❛♥ç❛ s❡rã♦ ❡♥✉♥❝✐❛❞♦s ❛ s❡❣✉✐r✱ ♠❛s ❛s ♣r♦✈❛s s❡rã♦ ♦♠✐t✐❞❛s ❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✸❪ ❡ ❬✹❪✳
❚❡♦r❡♠❛ ✶✳✹✳ ✭❈❛s♦ LAL ❞❡ s❡♠❡❧❤❛♥ç❛✮ ❉❛❞♦s ❞♦✐s tr✐â♥❣✉❧♦s ∆ABC ❡ ∆DEF ❡
✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ A ←→D✱ B ←→E ❡ C ←→F ❡♥tr❡ s❡✉s ✈❡rt✐❝❡s✱ s❡
BC
EF =
AC DF
BCAˆ ∼=EF Dˆ
❡♥tâ♦ ∆ABC ∼∆DEF ✭❝♦♠♦ ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ✶✳✺✮✳
✽ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ❚❡♦r❡♠❛ ✶✳✺✳ ✭❈❛s♦ LLL ❞❡ s❡♠❡❧❤❛♥ç❛✮ ❉❛❞♦s ❞♦✐s tr✐â♥❣✉❧♦s ∆ABC ❡ ∆DEF ❡
✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ A ←→D✱ B ←→E ❡ C ←→F ❡♥tr❡ s❡✉s ✈❡rt✐❝❡s✱ s❡
➜
BC
EF =
AC
DF =
AB DE
❡♥tã♦ ∆ABC ∼∆DEF ✭❝♦♠♦ ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ✶✳✻✮
❋✐❣✉r❛ ✶✳✻✿ ❝❛s♦ ▲▲▲ ❞❡ s❡♠❡❧❤❛♥ç❛
✶✳✸ ❘❡❧❛çõ❡s ♠étr✐❝❛s ❡♠ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦
❈♦♥s✐❞❡r❡♠♦s ✉♠ tr✐â♥❣✉❧♦ ABC✱ r❡tâ♥❣✉❧♦ ❡♠ A✱ ❞❡ ❝❛t❡t♦s AB = c✱ AC = b ❡ ❤✐♣♦✲
t❡♥✉s❛ BC =a✳ ❙❡❥❛ D ♦ ♣é ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❜❛✐①❛❞❛ ❞❡ A s♦❜r❡ ♦ ❧❛❞♦ BC✳ ❈♦♠♦ ❛
✶✳✹✳ ❚❊❖❘❊▼❆ ❉❖ ➶◆●❯▲❖ ■◆❙❈❘■❚❖ ✾
❋✐❣✉r❛ ✶✳✼✿ ❘❡❧❛çõ❡s ♠étr✐❝❛s ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦
❚❡♠♦sm(DCAˆ ) = 90♦−m(DACˆ )✳ ▼❛sm(DACˆ ) = 90♦−m(DABˆ )✳ ▲♦❣♦m(DCAˆ ) =
m(DABˆ ) ❡ ♣❡❧♦ ❝❛s♦ AA✱ ✈❛❧❡ q✉❡ ∆DCA ∼ ∆DAB✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ DA
DC =
DB DA✱
✐st♦ é✱ h
m =
n
h✳ ▲♦❣♦✱ h
2
=m·n q✉❡ é ❛ r❡❧❛çã♦ ❞❛❞❛ ❡♠(I)✳
P❡❧♦ ❝❛s♦ AA✱ ✈❛❧❡ q✉❡ ∆DCA∼∆ACB✳ ❆ss✐♠✱ DA
AB =
AC
BC✱ ✐st♦ é✱ h
c = b
a✳ ▲♦❣♦ a·h=b·c✱ ♦❜t❡♥❞♦ ❛ r❡❧❛çã♦ (II)✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s DC
AC =
AC
BC✱ ✐st♦ é✱ m
b =
b a✳
▲♦❣♦b2
=a·m✱ q✉❡ é ❛ r❡❧❛çã♦(III)✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♣❡❧♦ ❝❛s♦AA✱∆DBA ∼∆ABC✳
❆ss✐♠✱ DB
AB =
AB
BC✱ ✐st♦ é✱ n c =
c
a✳ ▲♦❣♦✱ c
2
=a·n✱ q✉❡ é ❛ r❡❧❛çã♦ (IV)✳
P❛r❛ ♦❜t❡r♠♦s ❛ r❡❧❛çã♦ (V)✱ ❜❛st❛ s♦♠❛r♠♦s ❛s ❡q✉❛çõ❡s ❞❛❞❛s ❡♠ (III) ❡(IV)✳
✶✳✹ ❚❡♦r❡♠❛ ❞♦ ➶♥❣✉❧♦ ■♥s❝r✐t♦
❉❡✜♥✐çã♦ ✶✳✺✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠ ❛r❝♦ BCø ♥✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ✉♠ ♣♦♥t♦ ❆ ❞❡st❛
❝✐r❝✉♥❢❡rê♥❝✐❛ ♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛♦ ❛r❝♦ BCø✳ ❉✐③❡♠♦s q✉❡ ♦ â♥❣✉❧♦ BACˆ é ✉♠ â♥❣✉❧♦
✶✵ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ❚❡♦r❡♠❛ ✶✳✻✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O✳ ❙❡❥❛♠ A✱ B ❡ C três
♣♦♥t♦s ❞✐st✐♥t♦s ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ t❛✐s q✉❡ A /∈BCø✳ ❊♥tã♦ m(BOCˆ ) = 2·m(BACˆ ).
❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s s✉♣♦r q✉❡ O ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦ BACˆ ✳ ◆❡st❡ ❝❛s♦✱
tr❛❝❡♠♦s ♦ ❞✐â♠❡tr♦AA′✳ ◆♦t❡ q✉❡ ♦s tr✐â♥❣✉❧♦s∆ABO ❡ ∆ACO sã♦ ✐sós❝❡❧❡s ❞❡ ❜❛s❡s AB ❡ AC✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ▲♦❣♦✱ m(OABˆ ) = m(OBAˆ ) = θ1 ❡ m(OACˆ ) = m(OCAˆ ) =
θ2 ❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ➶♥❣✉❧♦ ❊①t❡r♥♦✱ t❡♠♦sm(BOAˆ ′) = 2θ1 ❡m(A′OCˆ ) = 2θ2✱ ❞❡ ♦♥❞❡
s❡❣✉❡ q✉❡ m(BOCˆ ) =m(BOAˆ ′) +m(A′OCˆ ) = 2·θ
1+ 2·θ2 = 2·(θ1+θ2) = 2·m(BACˆ )✳
❖s ❝❛s♦s ❡♠ q✉❡O♣❡rt❡♥❝❡ ❛ ✉♠ ❞♦s ❧❛❞♦s ♦✉ ❛♦ ❡①t❡r✐♦r ❞♦ â♥❣✉❧♦BACˆ sã♦ ❛♥á❧♦❣♦s✳
❈♦r♦❧ár✐♦ ✶✳✶✳ ❙❡ BCø é ✉♠ ✉♠ ❛r❝♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ A1 ❡ A2 sã♦ ♣♦♥t♦s ❞❡st❛
❝✐r❝✉♥❢❡rê♥❝✐❛ ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛♦ ❛r❝♦ BCø✱ ❡♥tã♦ m(BAˆ1C) =m(BAˆ2C)✱ ✐st♦ é✱ â♥❣✉✲
❧♦s ✐♥s❝r✐t♦s q✉❡ s✉❜t❡♥❞❡♠ ♦ ♠❡s♠♦ ❛r❝♦ sã♦ ❝♦♥❣r✉❡♥t❡s✳
✶✳✺ ❘❡❧❛çõ❡s ♠étr✐❝❛s ♥✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛
◆❡st❛ s❡çã♦ ♦❜t❡r❡♠♦s ❛❧❣✉♠❛s r❡❧❛çõ❡s ♠étr✐❝❛s ❡♥✈♦❧✈❡♥❞♦ ✉♠ ♣♦♥t♦ P ❡ s❡❣♠❡♥t♦s t❛♥❣❡♥t❡s ❡✴♦✉ s❡❝❛♥t❡s ❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♣❛ss❛♥❞♦ ♣♦r P✳
❚❡♦r❡♠❛ ✶✳✼✳ ❚❡♦r❡♠❛ ❞❛s ❈♦r❞❛s
❙❡ AB ❡ CD sã♦ ❞✉❛s ❝♦r❞❛s ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ s❡ ✐♥t❡r❝❡♣t❛♠ ♥✉♠ ♣♦♥t♦ P
✭✐♥t❡r♥♦ à ❝✐r❝✉♥❢❡rê♥❝✐❛✮✱ ❡♥tã♦
P A·P B =P C ·P D ✭✶✳✶✮
❉❡♠♦♥str❛çã♦✳ P❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✶✱ t❡♠♦s q✉❡ BCDˆ ❡ BADˆ sã♦ â♥❣✉❧♦s ✐♥s❝r✐t♦s q✉❡
✐♥t❡r❝❡♣t❛♠ ♦ ♠❡s♠♦ ❛r❝♦ arcDB✱ ♣♦rt❛♥t♦✱ sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♦s â♥❣✉❧♦s ABCˆ ❡ ADCˆ t❛♠❜é♠ ♦s sã♦✱ ❝♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✶✳✽✳ ❆ss✐♠✱ ♦s tr✐â♥❣✉❧♦s ∆P AD ❡
∆P CB sã♦ s❡♠❡❧❤❛♥t❡s ✭❝❛s♦ ❆❆✮ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ P A P C =
P D
✶✳✺✳ ❘❊▲❆➬Õ❊❙ ▼➱❚❘■❈❆❙ ◆❯▼❆ ❈■❘❈❯◆❋❊❘✃◆❈■❆ ✶✶
❋✐❣✉r❛ ✶✳✽✿ ❚❡♦r❡♠❛ ❞❛s ❈♦r❞❛s
❖❜s❡r✈❛çã♦ ✶✳✶✳ ❆ r❡❝í♣r♦❝❛ ❞♦ ❚❡♦r❡♠❛ ❞❛s ❈♦r❞❛s t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✱ ✐st♦ é✱ s❡
AB ❡ CD sã♦ ❞♦✐s s❡❣♠❡♥t♦s ❞✐st✐♥t♦s ❡ s❡ ✐♥t❡r❝❡♣t❛♠ ❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦ P t❛❧ q✉❡ P A·P B =P C·P D✱ ❡♥tã♦ ♦s ♣♦♥t♦sA✱B✱C ❡D♣❡rt❡♥❝❡♠ ❛ ✉♠❛ ♠❡s♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱
✐st♦ é✱ sã♦ ❝♦♥❝í❝❧✐❝♦s✳
❉❡ ❢❛t♦✱ t❡♠♦s q✉❡ AP Dˆ ∼= CP Bˆ ✭♣♦✐s sã♦ ♦♣♦st♦s ♣❡❧♦ ✈ért✐❝❡✮✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ ♣♦r
❤✐♣ót❡s❡ t❡♠♦s P A
P D =
P C
P B✱ s❡❣✉❡ q✉❡ ♦s tr✐â♥❣✉❧♦s∆AP D ❡∆CP B sã♦ s❡♠❡❧❤❛♥t❡s ♣❡❧♦
❝❛s♦ LAL✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ m(PADˆ ) =m(BADˆ ) =m(PCBˆ ) =m(DCBˆ )✱ ✐st♦ é✱ ♦s
♣♦♥t♦s A ❡ C ♣❡rt❡♥❝❡♠ ❛♦ ♠❡s♠♦ ❛r❝♦ ❝❛♣❛③ ❞♦ s❡❣♠❡♥t♦ BD ❡ â♥❣✉❧♦ DABˆ ∼=DCBˆ ✱
♦ q✉❡ ♠♦str❛ q✉❡ A✱ B✱ C ❡ D sã♦ ❝♦♥❝í❝❧✐❝♦s✳
❚❡♦r❡♠❛ ✶✳✽✳ ❚❡♦r❡♠❛ ❞❛s ❙❡❝❛♥t❡s
❙❡ ♣♦r ✉♠ ♣♦♥t♦P ♥♦ ❡①t❡r✐♦r ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛Γ tr❛ç❛r♠♦s ✉♠❛ s❡♠✐rr❡t❛r q✉❡ ❛
✐♥t❡r❝❡♣t❛ ♥♦s ♣♦♥t♦s A ❡ B ❡ ✉♠❛ s❡♠✐rr❡t❛ s q✉❡ ❛ ✐♥t❡r❝❡♣t❛ ♥♦s ♣♦♥t♦s C ❡ D ✭❝♦♠♦
♥❛ ❋✐❣✉r❛ ✶✳✾✮✱ ❡♥tã♦
P A·P B =P C ·P D ✭✶✳✷✮
❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡ ♦s â♥❣✉❧♦s ABCˆ ❡ ADCˆ sã♦ ✐♥s❝r✐t♦s ♥♦ ♠❡s♠♦ ❛r❝♦ ACø✳
❉❡st❛ ❢♦r♠❛✱ ♦s tr✐â♥❣✉❧♦s ∆P DA ❡ ∆P BC sã♦ s❡♠❡❧❤❛♥t❡s ✭❝❛s♦ ❆❆✮ ❡✱ ♣♦rt❛♥t♦✱ P A
P C = P D
✶✷ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙
❋✐❣✉r❛ ✶✳✾✿ ❚❡♦r❡♠❛ ❞❛s ❙❡❝❛♥t❡s
❚❡♦r❡♠❛ ✶✳✾✳ ❚❡♦r❡♠❛ ❞❛ ❙❡❝❛♥t❡ ❡ ❚❛♥❣❡♥t❡
❙❡ ♣♦r ✉♠ ♣♦♥t♦ P ♥♦ ❡①t❡r✐♦r ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ tr❛ç❛r♠♦s ✉♠❛ s❡♠✐rr❡t❛ r q✉❡ ❛
t❛♥❣❡♥❝✐❛ ♥♦ ♣♦♥t♦ A ❡ ✉♠❛ s❡♠✐rr❡t❛ s q✉❡ ❛ ✐♥t❡r❝❡♣t❛ ♥♦s ♣♦♥t♦s C ❡ D ✭❝♦♠♦ ♥❛
❋✐❣✉r❛ ✶✳✶✵✮ ✱ ❡♥tã♦
P A2
=P C ·P D ✭✶✳✸✮
❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡ ♦s â♥❣✉❧♦s PACˆ ❡PDAˆ ❡stã♦ ✐♥s❝r✐t♦s ♥♦ ♠❡s♠♦ ❛r❝♦ACø ❡✱
♣♦rt❛♥t♦✱ sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❆ss✐♠✱ ♦s tr✐â♥❣✉❧♦s ∆P AC ❡ ∆P DA sã♦ s❡♠❡❧❤❛♥t❡s ✭❝❛s♦
❆❆✮ ❡✱ ❡♥tã♦✱ P A
P D =
P C
P A✱ ♦✉ s❡❥❛✱ P A
2
=P C ·P D✳
❈❛♣ít✉❧♦ ✷
❈✐r❝✉♥❢❡rê♥❝✐❛s ❖rt♦❣♦♥❛✐s
◆❡st❡ ❝❛♣ít✉❧♦✱ ❞✐s❝✉t✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ â♥❣✉❧♦s ❡♥tr❡ ✉♠❛ r❡t❛ ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ❡♥tr❡ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s✳ P❛rt✐❝✉❧❛r♠❡♥t❡✱ ♦ ❝❛s♦ ❡♠ q✉❡ ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s sã♦ ♦rt♦✲ ❣♦♥❛✐s✱ ✐st♦ é✱ ♦ â♥❣✉❧♦ ❡♥tr❡ ❡❧❛s é r❡t♦✳ ❊st❛ ❞✐s❝✉ssã♦ s❡rá ✉t✐❧✐③❛❞❛ ♣❛r❛ ♠♦t✐✈❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ✐♥✈❡rsã♦✱ q✉❡ s❡rá tr❛❜❛❧❤❛❞♦ ♥♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✳
✷✳✶ ➶♥❣✉❧♦ ❡♥tr❡ r❡t❛ ❡ ❝✐r❝✉♥❢❡rê♥❝✐❛
❉❡✜♥✐çã♦ ✷✳✶✳ ❉❛❞❛s ✉♠❛ r❡t❛ ❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ s❡ ✐♥t❡r❝❡♣t❛♠ ✭♦✉ s❡❥❛✱ ❛ r❡t❛ ♣♦❞❡ s❡r t❛♥❣❡♥t❡ ♦✉ s❡❝❛♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛✮✱ ❞❡✜♥✐♠♦s ♦ â♥❣✉❧♦ ❡♥tr❡ ❛s ♠❡s♠❛s ❝♦♠♦ s❡♥❞♦ ♦ ♠❡♥♦r â♥❣✉❧♦ ❡♥tr❡ ❛ r❡t❛ ❞❛❞❛ ❡ ❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ ♥♦ ✭s✮ s❡✉✭s✮ ♣♦♥t♦✭s✮ ❞❡ ✐♥t❡rs❡❝çã♦✳
◆♦t❡ q✉❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦✱ s❡ ❛ r❡t❛ r ❢♦r t❛♥❣❡♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❛ r❡t❛
t❛♥❣❡♥t❡ ♥♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ é ❛ ♣ró♣r✐❛ r❡t❛r✱ ♦✉ s❡❥❛✱ ♦ â♥❣✉❧♦ ❡♥tr❡ ❡❧❛s é ♥✉❧♦✳
❙❡ ❛ r❡t❛ ❢♦r s❡❝❛♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ♥ã♦ ♣❛ss❛r ♣❡❧♦ s❡✉ ❝❡♥tr♦✱ s❡♥❞♦ A ❡ B ♦s
♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦✱ ❛s r❡t❛s t❛♥❣❡♥t❡s à ❝✐r❝✉♥❢❡rê♥❝✐❛ ♣♦r ❡st❡s ♣♦♥t♦s s❡ ✐♥t❡r❝❡♣t❛♠ ♥✉♠ ♣♦♥t♦ P✳ ❈♦♠♦ ♦s tr✐â♥❣✉❧♦s P AO ❡ P BO sã♦ r❡tâ♥❣✉❧♦s ❞❡ ❝❛t❡t♦s OA = OB ❡
❤✐♣♦t❡♥✉s❛ ❝♦♠✉♠ ❞❡ ♠❡❞✐❞❛ OP✱ s❡❣✉❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ q✉❡ P A= P B ❡✱
♣♦rt❛♥t♦✱ ♦s tr✐â♥❣✉❧♦s ∆P AO ❡ ∆P BO sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦ ▲▲▲✳ ❈♦♥s❡q✉❡♥t❡✲
♠❡♥t❡✱ ♦ â♥❣✉❧♦ ❡♥tr❡ ❛ r❡t❛r ❡ q✉❛❧q✉❡r ✉♠❛ ❞❡ss❛s ❞✉❛s r❡t❛s t❛♥❣❡♥t❡s é ♦ ♠❡s♠♦✳
❙❡ ❛ r❡t❛r ♣❛ss❛r ♣❡❧♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ s❡♥❞♦A❡B ♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦✱
❛s r❡t❛s t❛♥❣❡♥t❡s à ❝✐r❝✉♥❢❡rê♥❝✐❛ ♣♦r ❡st❡s ♣♦♥t♦s s❡rã♦ ♣❛r❛❧❡❧❛s✳ ❆ss✐♠✱ ♦ â♥❣✉❧♦ ❡♥tr❡ ❛ r❡t❛ r ❡ q✉❛❧q✉❡r ✉♠❛ ❞❡ss❛s ❞✉❛s r❡t❛s t❛♥❣❡♥t❡s ♠❡❞❡ ✾✵♦ ❡✱ ♥❡ss❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡
✶✹ ❈❆P❮❚❯▲❖ ✷✳ ❈■❘❈❯◆❋❊❘✃◆❈■❆❙ ❖❘❚❖●❖◆❆■❙ ❛ r❡t❛ ❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ sã♦ ♦rt♦❣♦♥❛✐s✳
❋✐❣✉r❛ ✷✳✶✿ ❖ â♥❣✉❧♦ ❡♥tr❡ ❛ r❡t❛ ❡ ❛
❝✐r❝✉♥❢❡rê♥❝✐❛ é ♥✉❧♦ ❋✐❣✉r❛ ✷✳✷✿ ❖ â♥❣✉❧♦ ❡♥tr❡ ❛ r❡t❛ ❡ ❛❝✐r❝✉♥❢❡rê♥❝✐❛ ♠❡❞❡ α
❋✐❣✉r❛ ✷✳✸✿ ❖ â♥❣✉❧♦ ❡♥tr❡ ❛ r❡t❛ ❡ ❛ ❝✐r❝✉♥❢❡rê❝✐❛ ♠❡❞❡ 90♦ ✭❛ r❡t❛ ❡ ❛ ❝✐r✲
❝✉♥❢❡rê♥❝✐❛ sã♦ ♦rt♦❣♦♥❛✐s✮
✷✳✷ ➶♥❣✉❧♦ ❡♥tr❡ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s
❆♥t❡s ❞❡ ❞❡✜♥✐r ♦ â♥❣✉❧♦ ❡♥tr❡ ❞✉❛s ❝✐r❝✉♥❢ê♥❝✐❛s✱ ❝♦♠❡❝❡♠♦s ❝♦♠ ♦ s❡❣✉✐♥t❡
❚❡♦r❡♠❛ ✷✳✶✳ ❙❡❥❛♠Γ1 ❡Γ2 ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s s❡❝❛♥t❡s✱ q✉❡ s❡ ✐♥t❡r❝❡♣t❛♠ ♥♦s ♣♦♥t♦s
A ❡ B✱ ❞✐st✐♥t♦s✳ ❙❡❥❛♠ ❛✐♥❞❛ s1 ❡ s2 ❛s r❡t❛s q✉❡ ♣❛ss❛♠ ♣♦r A ❡ sã♦ t❛♥❣❡♥t❡s ❛ Γ1 ❡
Γ2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❜❡♠ ❝♦♠♦ t1 ❡ t2 ❛s r❡t❛s q✉❡ ♣❛ss❛♠ ♣♦r B ❡ sã♦ t❛♥❣❡♥t❡s ❛ Γ1 ❡
Γ2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡♥❞♦ θ1 ❛ ♠❡❞✐❞❛ ❞♦ ♠❡♥♦r â♥❣✉❧♦ ❡♥tr❡ s1 ❡ s2 ❡ θ2 ❛ ♠❡❞✐❞❛ ❞♦
♠❡♥♦r â♥❣✉❧♦ ❡♥tr❡ t1 ❡ t2✱ t❡♠♦s q✉❡ θ1 =θ2.
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ O1 ❡ O2 ♦s ❝❡♥tr♦s ❞❡ Γ1 ❡ Γ2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ P ❡ ◗ ♣♦♥t♦s
✷✳✷✳ ➶◆●❯▲❖ ❊◆❚❘❊ ❉❯❆❙ ❈■❘❈❯◆❋❊❘✃◆❈■❆❙ ✶✺
❋✐❣✉r❛ ✷✳✹✿ ❖s â♥❣✉❧♦s θ1 ❡θ2 sã♦ ❝♦♥❣r✉❡♥t❡s
❏á q✉❡ s1 é t❛♥❣❡♥t❡ ❛ Γ1 ❡ s2 é t❛♥❣❡♥t❡ ❛ Γ2✱ t❡♠♦s q✉❡ ♦ â♥❣✉❧♦ O1APˆ ♠❡❞❡
90♦−θ1 ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ â♥❣✉❧♦ O1AOˆ 2 ♠❡❞❡ 90♦−(90♦ −θ1) = θ1✳ ❉❡ ♠❛♥❡✐r❛
❝♦♠♣❧❡t❛♠❡♥t❡ ❛♥á❧♦❣❛✱ s❡❣✉❡ q✉❡ ♦ â♥❣✉❧♦ O1BOˆ 2 ♠❡❞❡ 90♦−(90♦−θ2) =θ2✳ ▼❛s ♦s
tr✐â♥❣✉❧♦s ∆O1AO2 ❡∆O1BO2 sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦ LLL ❡✱ ♣♦rt❛♥t♦✱ θ1 =θ2 =θ✳
◆♦t❡ q✉❡ ♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ♠♦str❛r é q✉❡ ♦ â♥❣✉❧♦ ❡♥tr❡ ❛s r❡t❛s t❛♥❣❡♥t❡s ❡♠ ❝❛❞❛ ♣♦♥t♦ é ❝♦♥❣r✉❡♥t❡ ❛♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣❡❧♦s r❛✐♦s ❞❛s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ♥❛q✉❡❧❡ ♣♦♥t♦✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ q✉❛♥❞♦ ♦s r❛✐♦s ❢♦r❡♠ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ♥♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦✱ ❞✐r❡♠♦s q✉❡ ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s sã♦ ♦rt♦❣♦♥❛✐s✳
❉❡✜♥✐çã♦ ✷✳✷✳ ❈♦♥s✐❞❡r❡♠♦s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s Γ1 ❡ Γ2 s❡❝❛♥t❡s✱ s❡♥❞♦ A ❡ B ♦s
♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡ ❛♠❜❛s✳ P❡❧♦ ♣♦♥t♦ A tr❛ç❛♠♦s ❛s r❡t❛ss1 ❡s2✱ t❛♥❣❡♥t❡s ❛Γ1 ❡
Γ2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♣❡❧♦ ♣♦♥t♦B tr❛ç❛♠♦s ❛s r❡t❛st1 ❡ t2✳ ❉❡✜♥✐♠♦s
♦ â♥❣✉❧♦ ❡♥tr❡ Γ1 ❡ Γ2 ❝♦♠♦ s❡♥❞♦ ♦ ♠❡♥♦r â♥❣✉❧♦ ❡♥tr❡ s1 ❡ s2 ♦✉ ❡♥tr❡ t1 ❡ t2 ✭❝♦♠♦
✶✻ ❈❆P❮❚❯▲❖ ✷✳ ❈■❘❈❯◆❋❊❘✃◆❈■❆❙ ❖❘❚❖●❖◆❆■❙
❋✐❣✉r❛ ✷✳✺✿ ♦ â♥❣✉❧♦ ❡♥tr❡ Γ1 ❡Γ2 é θ
❖❜s❡r✈❛çã♦ ✷✳✶✳ P♦r ❞❡✜♥✐çã♦✱ s❡ Γ1 ❡ Γ2 ❢♦r❡♠ t❛♥❣❡♥t❡s ✭✐♥t❡r♥❛ ♦✉ ❡①t❡r♥❛♠❡♥t❡✮✱
❈❛♣ít✉❧♦ ✸
❆ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠❛
❝✐r❝✉♥❢❡rê♥❝✐❛
◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ❞❡✜♥✐çõ❡s ❞❡ r❡✢❡①ã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ r❡t❛ ❡ ❞❡ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❜❡♠ ❝♦♠♦ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳
❉❡✜♥✐çã♦ ✸✳✶✳ ❉❛❞❛ ✉♠❛ r❡t❛ r✱ ❛ r❡✢❡①ã♦ ❡♠ r❡❧❛çã♦ à r❡t❛ r é ❛ ❛♣❧✐❝❛çã♦ q✉❡ ✜①❛
♦s ♣♦♥t♦s ❞❡ r ❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ A /∈ r ♦ ♣♦♥t♦ A′ t❛❧ q✉❡ r é ❛ ♠❡❞✐❛tr✐③ ❞❡ AA′✳
◆❡ss❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ A ❡ A′ sã♦ s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ à r❡t❛ r✳
❋✐❣✉r❛ ✸✳✶✿ ❆ ❡ ❆✬ s❛♦ s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ à r❡t❛ r
❖ r❡s✉❧t❛❞♦ s❡❣✉✐♥t❡ ♥♦s ❞á ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ✈❡r✐✜❝❛r♠♦s s❡ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s sã♦ s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ r❡t❛✳
✶✽ ❈❆P❮❚❯▲❖ ✸✳ ❆ ■◆❱❊❘❙➹❖ ❊▼ ❘❊▲❆➬➹❖ ❆ ❯▼❆ ❈■❘❈❯◆❋❊❘✃◆❈■❆ ❚❡♦r❡♠❛ ✸✳✶✳ ❉❛❞❛ ✉♠❛ r❡t❛ r✱ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s A ❡ A′ sã♦ s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦
❛ r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t♦❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ Γ ❝♦♠ ❝❡♥tr♦ ❡♠ ❛❧❣✉♠ ♣♦♥t♦ O ∈r q✉❡ ♣❛ss❛r
♣❡❧♦ ♣♦♥t♦ A t❛♠❜é♠ ♣❛ss❛r ♣♦r A′✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ r ❛ ♠❡❞✐❛tr✐③ ❞❡ AA′ ❡ O ∈ r✳ ❆ss✐♠✱ OA = OA′ ❡✱ ♣♦rt❛♥t♦✱ ❛
❝✐r❝✉♥❢❡rê♥❝✐❛ Γ ❞❡ ❝❡♥tr♦ O ❡ q✉❡ ♣❛ss❛ ♣♦r A ✭❝✉❥♦ r❛✐♦ é OA✮ t❛♠❜é♠ ♣❛ss❛ ♣♦r A′✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ❝♦♥s✐❞❡r❡♠♦s ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ Γ❞❡ ❝❡♥tr♦ O ∈r❡ q✉❡ ♣❛ss❛ ♣♦r A ❡ A′✳ ❉❡ss❛ ❢♦r♠❛✱ OA=OA′ ❡ ❝♦♠♦O é ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❛ r❡t❛r ❡ é ❡q✉✐❞✐st❛♥t❡ ❞❡ A ❡ A′✱ s❡❣✉❡r é ❛ ♠❡❞✐❛tr✐③ ❞❡AA′✱ ❞❡ ♠♦❞♦ q✉❡ A ❡A′ sã♦ s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ ❛ r✳
❉❡✜♥✐çã♦ ✸✳✷✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦O ❡ r❛✐♦r❝♦♥t✐❞❛ ♥✉♠ ♣❧❛♥♦ α✳ ❆ ✐♥✈❡rsã♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ A
❞♦ ♣❧❛♥♦ α✱ ❞✐st✐♥t♦ ❞❡ O✱ ✉♠ ♣♦♥t♦ A′ ♥❛ s❡♠✐rr❡t❛ OA−→ t❛❧ q✉❡ OA·OA′ = r2✳ ◆❡st❡
❝❛s♦✱ ❞✐③❡♠♦s q✉❡ A′ é ♦ ✐♥✈❡rs♦ ❞❡ A ❡♠ r❡❧❛çã♦ ❛ Γ ✭♦✉ q✉❡ A ❡ A′ sã♦ s✐♠étr✐❝♦s ❡♠
r❡❧❛çã♦ ❛ Γ✮✳
▼♦t✐✈❛❞♦s ♣❡❧❛ ❞✐s❝✉ssã♦ ❛♥t❡r✐♦r✱ ✈❡r❡♠♦s q✉❡ ❛ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ ❝✐r❝✉♥❢❡✲ rê♥❝✐❛ ♣♦❞❡ s❡r ❞❛❞❛ ❛ ♣❛rt✐r ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s ♦rt♦❣♦♥❛✐s✳ ❆♥t❡s ❞✐st♦✱ ♣r❡❝✐s❛♠♦s ❞♦s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s ❛✉①✐❧✐❛r❡s✳
▲❡♠❛ ✸✳✶✳ ❙❡ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s Γ1 ❡ Γ2✱ ❞❡ ❝❡♥tr♦s O1 ❡ O2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ sã♦
♦rt♦❣♦♥❛✐s ❡♥tã♦ O1 ❡stá ♥♦ ❡①t❡r✐♦r ❞❡ Γ2 ❡ O2 ❡stá ♥♦ ❡①t❡r✐♦r ❞❡ Γ1✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ P ❡ Q ♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s✱ ♦s q✉❛✐s
❡①✐st❡♠ ♣♦✐s s✉♣♦♠♦s q✉❡ Γ1 é ♦rt♦❣♦♥❛❧ ❛ Γ2✳ ❆s r❡t❛s s✉♣♦rt❡s ❞♦s r❛✐♦s O1P ❡ O2P
sã♦ t❛♥❣❡♥t❡s ❛Γ2 ❡Γ1✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ O1 ❡stá ♥♦ ❡①t❡r✐♦r ❞❡Γ2 ❡
O2 ❡stá ♥♦ ❡①t❡r✐♦r ❞❡ Γ1✳
▲❡♠❛ ✸✳✷✳ ❙❡ ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s Γ1 ❡ Γ2 s❡ ✐♥t❡r❝❡♣t❛♠ ♥♦s ♣♦♥t♦s A ❡ B ❡ sã♦
♦rt♦❣♦♥❛✐s ❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ Γ ❞❡ ❝❡♥tr♦ O✱ ❡♥tã♦ O✱ A ❡ B sã♦ ❝♦❧✐♥❡❛r❡s✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ P ✉♠ ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❡♥tr❡Γ❡Γ1 ✱Q ✉♠ ❞♦s ♣♦♥t♦s ❞❡
✐♥t❡rs❡❝çã♦ ❡♥tr❡ Γ❡Γ2✱{C}=
−→
OA∩Γ1, C 6=A ❡{D}=
−→
OA∩Γ2, D 6=A✳ P❡❧♦ ❚❡♦r❡♠❛
✶✳✾✱ t❡♠♦s OP2
= OA·OC ❡ OQ2
= OA·OD✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ OC = OD✱ ♣♦✐s OP
✶✾ ❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ é ✉♠❛ ❛♥❛❧♦❣✐❛ ❛♦ ❚❡♦r❡♠❛ ✸✳✶✳ ❊❧❡ ♥♦s ❞á ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ✈❡r✐✜❝❛r♠♦s s❡ ❞♦✐s ♣♦♥t♦s sã♦ s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ ❞❛❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳
❚❡♦r❡♠❛ ✸✳✷✳ ❉❛❞♦s ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ Γ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r ❡ ✉♠ ♣♦♥t♦ A6=O✱ t❡✲
♠♦s q✉❡ A′ é ♦ ✐♥✈❡rs♦ ❞❡A ❡♠ r❡❧❛çã♦ ❛ Γs❡ ❡ s♦♠❡♥t❡ s❡ t♦❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♦rt♦❣♦♥❛❧
❛ Γ q✉❡ ♣❛ss❛r ♣♦r A t❛♠❜é♠ ♣❛ss❛r ♣♦r A′✳
❋✐❣✉r❛ ✸✳✷✿ Γ1 ❡Γ2 sã♦ ♦rt♦❣♦♥❛✐s ❛ Γ
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ A ❡ A′ ❞♦✐s ♣♦♥t♦s q✉❛✐sq✉❡r✱ s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ ❛ Γ✱ Γ
1 ✉♠❛
❝✐r❝✉♥❢❡rê♥❝✐❛ ♦rt♦❣♦♥❛❧ ❛ Γ ♣❛ss❛♥❞♦ ♣♦r A ❡ B ♦ ♣♦♥t♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r −→OA ∩Γ1✱
B 6= A✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ ✸✳✷✳ ❆ss✐♠✱ ♣♦r ❞❡✜♥✐çã♦ ❞❡ ✐♥✈❡rsã♦✱ t❡♠♦s OA ·OA′ = r2
❡ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✽✱ s❡❣✉❡ q✉❡ OA·OB = r2✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱
OA′ = OB ❡✱ ♣❡❧❛
❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✸✳✷✱B ♣❡rt❡♥❝❡ à s❡♠✐rr❡t❛−−→OA′ =−→OA✱ ❞❡ ♠♦❞♦ q✉❡ B =A′✱ ✐st♦
é✱ Γ1 t❛♠❜é♠ ♣❛ss❛ ♣♦r A′✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛♠A❡A′❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s t❛✐s q✉❡ t♦❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♦rt♦❣♦✲
♥❛❧ ❛Γq✉❡ ♣❛ss❛ ♣♦rA t❛♠❜é♠ ♣❛ss❛ ♣♦rA′✳ ❙❡❥❛♠Γ
1 ❡Γ2 ❞✉❛s ❞❡st❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s✱
❞❡ ♠♦❞♦ q✉❡Γ1∩Γ2 ={A, A′}✳ P❡❧♦ ▲❡♠❛ ✸✳✶✱O ❡stá ♥♦ ❡①t❡r✐♦r ❞❡Γ1 ❡Γ2 ❡ ♣❡❧♦ ▲❡♠❛
✸✳✷✱ t❡♠♦s q✉❡ O✱ A ❡ A′ ❡stã♦ ❛❧✐♥❤❛❞♦s✳ ❆ss✐♠✱ A′ ∈ −→OA✳ ❙❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ✶✳✽ q✉❡ r2
✷✵ ❈❆P❮❚❯▲❖ ✸✳ ❆ ■◆❱❊❘❙➹❖ ❊▼ ❘❊▲❆➬➹❖ ❆ ❯▼❆ ❈■❘❈❯◆❋❊❘✃◆❈■❆ ❉♦r❛✈❛♥t❡✱ ❞❡s✐❣♥❛r❡♠♦s ♣♦r Γ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝♦♠ ❝❡♥tr♦ ♥✉♠ ♣♦♥t♦ O ❡ ❝♦♠
✉♠ r❛✐♦ r > 0✱ ❡ A′ ❞❡♥♦t❛rá ♦ ✐♥✈❡rs♦ ❞❡ ✉♠ ♣♦♥t♦ A ❡♠ r❡❧❛çã♦ ❛ Γ✳ ❯♠❛ ♦❜s❡r✈❛çã♦
✐♠♣♦rt❛♥t❡ q✉❡ ❞❡✈❡ s❡r ❢❡✐t❛ é q✉❡✱ ♣♦r ❞❡✜♥✐çã♦✱ (A′)′ = A✱ ♣❛r❛ q✉❛❧q✉❡r ♣♦♥t♦ A✱
❞✐st✐♥t♦ ❞❡ O✱ ♦✉ s❡❥❛✱ ♦ ✐♥✈❡rs♦ ❞♦ ✐♥✈❡rs♦ ❞❡ ✉♠ ♣♦♥t♦ é ♦ ♣ró♣r✐♦ ♣♦♥t♦✳ ❉✐③❡♠♦s
t❛♠❜é♠ q✉❡ ❛ ✐♥✈❡rsã♦ é ✉♠❛ ✐♥✈♦❧✉çã♦✱ ✐st♦ é✱ ✉♠❛ ❛♣❧✐❝❛çã♦ I : E2
−→ E2✱ ❝♦♠
I(I(x)) =x✱ ♣❛r❛ t♦❞♦ x∈E2✳
❆ s❡❣✉✐r✱ ✈❛♠♦s ❡♥✉♥❝✐❛r ❡ ❞❡♠♦♥str❛r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ✐♥✈❡rsã♦ q✉❡ s❡rã♦ ❡①tr❡♠❛♠❡♥t❡ r❡❧❡✈❛♥t❡s ♣❛r❛ ♦ ♥♦ss♦ tr❛❜❛❧❤♦✳
Pr♦♣♦s✐çã♦ ✸✳✶✳ ❙❡❥❛ A ✉♠ ♣♦♥t♦ ❞♦ ♣❧❛♥♦ ❞❡✜♥✐❞♦ ♣♦r Γ✱ ❞✐st✐♥t♦ ❞❡ O✳
✭❛✮ s❡ A ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✐♦r ❞❡ Γ ❡♥tã♦ A′ ♣❡rt❡♥❝❡ ❛♦ s❡✉ ❡①t❡r✐♦r❀
✭❜✮ s❡ A ♣❡rt❡♥❝❡ ❛♦ ❡①t❡r✐♦r ❞❡ Γ ❡♥tã♦ A′ ♣❡rt❡♥❝❡ ❛♦ s❡✉ ✐♥t❡r✐♦r❀
✭❝✮ s❡ A ♣❡rt❡♥❝❡ ❛ Γ ❡♥tã♦ A′ =A✳
❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ A ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✐♦r ❞❡ Γ✱ ✐st♦ é✱ OA < r✳
▼✉t✐♣❧✐❝❛♥❞♦✲s❡ ❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣♦r r✱ t❡♠♦sr·OA < r2
=OA·OA′✳ ▲♦❣♦✱ OA′ > r✳
❖ ❝❛s♦ ❡♠ q✉❡ A ♣❡rt❡♥❝❡ ❛♦ ❡①t❡r✐♦r ❞❡ Γ é ❝♦♠♣❧❡t❛♠❡♥t❡ ❛♥á❧♦❣♦✳ P♦r ✜♠✱ s❡ A
♣❡rt❡♥❝❡ ❛ Γ✱ ❡♥tã♦ OA = r ❡ r2
= OA·OA′✱ ❞♦♥❞❡ OA′ = r✳ ❈♦♠♦ A′ ♣❡rt❡♥❝❡ à
s❡♠✐rr❡t❛ −→OA✱ s❡❣✉❡ q✉❡ A=A′✳
◆♦t❡ q✉❡ ❛ ❞❡♠♦♥str❛çã♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❢❛③❡r é ❛❧❣é❜r✐❝❛✳ ◆♦ ❡♥t❛♥t♦✱ ♣♦❞❡♠♦s ❞❛r ✉♠❛ ❞❡♠♦♥str❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ss❛ ♣r♦♣♦s✐çã♦✳ ❆❧é♠ ❞✐ss♦✱ ♠♦str❛r❡♠♦s ❝♦♠♦ ❝♦♥s✲ tr✉✐r✱ ❝♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦✱ ♦ ✐♥✈❡rs♦ ❞❡ ✉♠ ♣♦♥t♦ A ✭❞✐st✐♥t♦ ❞❡ O✮ ❡♠ r❡❧❛çã♦ ❛ Γ✳
❙❡A❡stá ♥♦ ✐♥t❡r✐♦r ❞❡Γ✱ tr❛❝❡ ♣♦rA❛ r❡t❛r✱ ♣❡r♣❡♥❞✐❝✉❧❛r ❛−→OA✱ q✉❡ ✐♥t❡r❝❡♣t❛ráΓ
❡♠ ❞♦✐s ♣♦♥t♦s✳ ❙❡❥❛ P ✉♠ ❞❡st❡s ♣♦♥t♦s✳ P❡❧♦ ♣♦♥t♦P tr❛ç❛♠♦s ❛ r❡t❛ s✱ ♣❡r♣❡♥❞✐❝✉❧❛r
❛ OP ✭♦❜s❡r✈❡ q✉❡ s é ❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ Γ ❡♠ P✮✳ ❙❡♥❞♦ X ❛ ✐♥t❡rs❡❝✲
✸✳✶✳ P❘❖P❘■❊❉❆❉❊❙ ❉❆ ■◆❱❊❘❙➹❖ ✷✶
❋✐❣✉r❛ ✸✳✸✿ ✐♥✈❡rsã♦ ❞❡ ✉♠ ♣♦♥t♦ ✐♥t❡r♥♦✳
❆ ❞❡♠♦♥str❛çã♦ ❡stá ♥♦ ❢❛t♦ ❞❡ q✉❡ ♦s tr✐â♥❣✉❧♦s △OAP ❡ △OP X sã♦ s❡♠❡❧❤❛♥t❡s
♣❡❧♦ ❝❛s♦ AA✳ ❆ss✐♠✱ OA
OP =
OP
OX✳ ❖✉ s❡❥❛✱ OA·OX = OP
2
= r2✳ ❈♦♠♦
X ∈ −→OA✱
s❡❣✉❡ q✉❡ X =A′✳ ❆❧é♠ ❞✐st♦✱ ❝♦♠♦ A′ ∈ s✱ q✉❡ é t❛♥❣❡♥t❡ ❛ Γ ❡♠ P✱ ❡♥tã♦ A′ ❡stá ♥♦
❡①t❡r✐♦r ❞❡ Γ✳
❙❡A ❡stá ♥♦ ❡①t❡r✐♦r ❞❡Γ✱ tr❛❝❡ ♣♦rA♣♦r ✉♠❛ ❞❛s r❡t❛s t❛♥❣❡♥t❡s ❛ Γ✱ ❞✐❣❛♠♦ss✱ ❡
s❡❥❛ P ♦ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛✳ P❡❧♦ ♣♦♥t♦P tr❛❝❡ ❛ r❡t❛ t✱ ♣❡r♣❡♥❞✐❝✉❧❛r à s❡♠✐rr❡t❛ −→OA✳
❙❡♥❞♦ X ❛ ✐♥t❡rs❡❝çã♦ ❞❡ −→OA ❡ t✱ t❡♠♦s q✉❡ O−X−A ❡ X ❡stá ♥♦ ✐♥t❡r✐♦r ❞❡ Γ✱ ♣♦✐s
OA é ❛ ❤✐♣♦t❡♥✉s❛ ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ △OP A✳ ❆ss✐♠✱ X = A′✳ ❆ ❞❡♠♦♥str❛çã♦ é
❛♥á❧♦❣❛ ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✳
◆♦ ❝❛s♦ ❡♠ q✉❡ A ♣❡rt❡♥❝❡ ❛ Γ✱ ♥ã♦ ❤á ♦ q✉❡ ❢❛③❡r✱ ♣♦✐s ❥á q✉❡ OA = OA′ = r ❡ A′ ∈−→OA✱ ❡♥tã♦ A′ =A✳
✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ ■♥✈❡rsã♦
■♥✐❝✐❛❧♠❡♥t❡✱ ❡st✉❞❛r❡♠♦s ❛ ✐♥✈❡rsã♦ ❞❡ r❡t❛s ❡♠ r❡❧❛çã♦ ❛ Γ ❡✱ ❡♠ s❡❣✉✐❞❛✱ ❞❡ ❝✐r❝✉♥❢❡✲
rê♥❝✐❛s✳ ❯s❛r❡♠♦s ♦s s❡❣✉✐♥t❡s ❧❡♠❛s✿
▲❡♠❛ ✸✳✸✳ ❙❡❥❛♠ A ❡ B ♣♦♥t♦s ❞✐st✐♥t♦s ❞❡ ♠♦❞♦ q✉❡ ❖✱ ❆ ❡ ❇ s❡❥❛♠ ♥ã♦ ❝♦❧✐♥❡❛r❡s✳
✷✷ ❈❆P❮❚❯▲❖ ✸✳ ❆ ■◆❱❊❘❙➹❖ ❊▼ ❘❊▲❆➬➹❖ ❆ ❯▼❆ ❈■❘❈❯◆❋❊❘✃◆❈■❆ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ A′ ❡ B′ ♦s ✐♥✈❡rs♦s ❞❡ A ❡ B ❝♦♠ r❡❧❛çã♦ ❛ Γ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❚❡♠♦s✱ ♣♦r ❞❡✜♥✐çã♦✱ q✉❡ OA·OA′ = OB ·OB′ = r2✱ ✐st♦ é✱ OA
OB =
OB′
OA′✳ ❆❧é♠ ❞✐ss♦✱ AOBˆ ∼= B′OAˆ ′ ✭q✉❡ é ♦ â♥❣✉❧♦ ❡♥tr❡ ❛s s❡♠✐rr❡t❛s −→OA ❡ −−→OB✮✳ P❡❧♦ ❝❛s♦ LAL ❞❡ s❡✲
♠❡❧❤❛♥ç❛❛ ❞❡ tr✐â♥❣✉❧♦s✱ s❡❣✉❡ q✉❡✱ ∆OAB ∼ ∆OB′A′ ❡✱ ❡♠ r❡❧❛çã♦ à ❝♦rr❡s♣♦♥❞ê♥❝✐❛ OAB ↔OB′A′✱ t❡♠♦sOABˆ ∼=OBˆ′A′✳
❋✐❣✉r❛ ✸✳✹✿ ◆❛ ✐♥✈❡rsã♦ ❝♦♠ r❡❧❛çã♦ ❛Γ✱ ♦ â♥❣✉❧♦OABˆ é tr❛♥s❢♦r♠❛❞♦ ♥♦ â♥❣✉❧♦OBˆ′A′
❡ ♦ â♥❣✉❧♦ OBAˆ é tr❛♥s❢♦r♠❛❞♦ ♥♦ â♥❣✉❧♦OAˆ′B′✳
❖❜s❡r✈❡ q✉❡ ♥❛ ✜❣✉r❛ ✸✳✹ ♦s ♣♦♥t♦s A ❡ B ❞♦ ▲❡♠❛ ✸✳✸ ❡stã♦ ♥♦ ✐♥t❡r✐♦r ❞❡ Γ✳ ◆♦
❡♥t❛♥t♦✱ ♥❛ ❞❡♠♦♥str❛çã♦ q✉❡ ✜③❡♠♦s ✐♥❞❡♣❡♥❞❡✉ ❞♦ r❛✐♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❥á q✉❡ ♦s tr✐â♥❣✉❧♦s ∆OAB ❡∆OB′A′ s❡rã♦ s❡♠♣r❡ s❡♠❡❧❤❛♥t❡s✳
▲❡♠❛ ✸✳✹✳ ❙❡ X1, X2, ..., Xn sã♦ ♣♦♥t♦s ♣❡rt❡♥❝❡♥t❡s ❛ ✉♠❛ ♠❡s♠❛ s❡♠✐rr❡t❛ ❞❡ ♦r✐❣❡♠ P t❛✐s q✉❡P−X1−X2−...−Xn✱ ❡♥tã♦P−Xn′−Xn′−1−...−X2′−X1′✱ ♦♥❞❡Xi, i′ = 1,2, ..., n✱
é ♦ ✐♥✈❡rs♦ ❞❡ Xi ❝♦♠ r❡❧❛çã♦ ❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ P✳
❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡P Xi·P Xi′ =P Xi+1·P Xi′+1 =r
2✱ ♦✉ s❡❥❛
P Xi =
P Xi+1·P Xi′+1
P X′ i
✳ ❈♦♠♦ ♣♦r ❤✐♣ót❡s❡ t❡♠♦s P Xi < P Xi+1✱ s❡❣✉❡ q✉❡
P Xi+1·P Xi′+1
P X′ i
< P Xi+1✱ ✐♠♣❧✐❝❛♥❞♦
P X′
