❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈
❈❡♥tr♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❈♦♠♣✉t❛çã♦ ❡ ❈♦❣♥✐çã♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦❢✐ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦
❈r✐st✐❛♥♦ ❞❡ ❙♦✉③❛ ❱✐❡✐r❛
❈♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ✉♠
q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ s❡❥❛ ✉♠ tr❛♣é③✐♦
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈
❈❡♥tr♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❈♦♠♣✉t❛çã♦ ❡ ❈♦❣♥✐çã♦
❈♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ✉♠
q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ s❡❥❛ ✉♠ tr❛♣é③✐♦
❈r✐st✐❛♥♦ ❞❡ ❙♦✉③❛ ❱✐❡✐r❛
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❥✉♥t♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡r✲ s✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❈♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ✉♠
q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ s❡❥❛ ✉♠ tr❛♣é③✐♦
❊st❡ ❡①❡♠♣❧❛r ❝♦rr❡s♣♦♥❞❡ à r❡❞❛çã♦ ✜♥❛❧ ❞❛ ❞✐ss❡rt❛çã♦ ❞❡✈✐❞❛♠❡♥t❡ ❝♦r✲ r✐❣✐❞❛ ❡ ❞❡❢❡♥❞✐❞❛ ♣♦r ❈r✐st✐❛♥♦ ❞❡ ❙♦✉③❛ ❱✐❡✐r❛ ❡ ❛♣r♦✈❛❞❛ ♣❡❧❛ ❝♦♠✐ssã♦ ❥✉❧❣❛❞♦r❛✳
❙❛♥t♦ ❆♥❞ré✱ ✷✼ ❞❡ ❆❣♦st♦ ❞❡ ✷✵✶✹✳
Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛ ❖r✐❡♥t❛❞♦r
❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿
✶✳ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛ ✭❖r✐❡♥t❛❞♦r✮ ✲ ❯❋❆❇❈ ✷✳ Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❈â♥❞✐❞♦ ❋❛❧❡✐r♦s
✸✳ Pr♦❢✳ ❉r✳ ❆r♠❛♥❞♦ ❚r❛❧❞✐ ❏ú♥✐♦r
❉❡❝❧❛r❛çã♦ ❞❡ ❛t❡♥❞✐♠❡♥t♦ às ♦❜s❡r✈❛çõ❡s
❊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✱ ✶✼ ❞❡ ❖✉t✉❜r♦ ❞❡ ✷✵✶✹✳
❈r✐st✐❛♥♦ ❞❡ ❙♦✉③❛ ❱✐❡✐r❛ ❆✉t♦r
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♣❡s❛r ❞❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ s❡ ❝♦♠❡t❡r ✐♥❥✉st✐ç❛s ❛♦ ❧✐st❛r ♣❡ss♦❛s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ ❝✉rs♦✱ ❡♠ ❢✉♥çã♦ ❞♦ ❡sq✉❡❝✐♠❡♥t♦ ❞❡ ❛❧❣✉♥s✱ ♥ã♦ ♣♦ss♦ ❞❡✐①❛r ❞❡ ❡①♣r❡ss❛r ♠✐♥❤❛ ❣r❛t✐❞ã♦ ❛✿
• Pr♦❢❡ss♦r ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛✱ ♣♦r s❡r ✉♠ ❡①❡♠♣❧♦ ❞❡ ♣r♦❢❡ss♦r ❛ s❡r s❡❣✉✐❞♦✱ ❝♦♠ ❞❡❞✐❝❛çã♦✱ ❝♦♠♣❡tê♥❝✐❛✱ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ❡ ❝♦♠♣r♦♠❡t✐♠❡♥t♦✳ ◆ã♦ ❤á ♣❛❧❛✈r❛s q✉❡ ❡①♣r❡ss❡♠ t♦❞♦ ♠❡✉ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡ ♥❡♠ ❛ t❛♠❛♥❤❛ ❤♦♥r❛ q✉❡ t✐✈❡ ❡♠ ❛❞q✉✐r✐r ♣❛rt❡ ❞❡ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦❀
• ❘❡♥✐❧③❡ ●❡✐ss ❞❡ ❆❧♠❡✐❞❛✱ ❜❡♠ ♠❛✐s q✉❡ ❛♠✐❣❛✱ ♣♦r t❡r ❢❡✐t♦ ❞❡ s✉❛ ❝❛s❛ ❛ ❡①t❡♥sã♦ ❞❛ ♠✐♥❤❛✱ ♦♥❞❡ ♣✉❞❡ ❡♥❝♦♥tr❛r ♣❛③ ❡ s♦ss❡❣♦ ♣♦r t❛♥t❛s ✈❡③❡s✱ ❞❡s❞❡ ♦s ❡st✉❞♦s ❞❡ ♣r❡♣❛r❛çã♦ ♣❛r❛ ♦ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞❡ ❆❝❡ss♦ ❛té ❜♦❛ ♣❛rt❡ ❞❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦❀
• ❲✐❧❧❛♠s ❏ú♥✐♦r ❖❧✐✈❡✐r❛ ❆♥❞r❛❞❡✱ ❣r❛♥❞❡ ❛♠✐❣♦✱ ♣❡❧♦s ✐♥ú♠❡r♦s ♠♦♠❡♥t♦s ❞❡ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦✱ ❛❝r❡❞✐t❛♥❞♦ s❡♠♣r❡ ♥♦ ♠❡✉ s✉❝❡ss♦✱ ❛ss✐♠ ❝♦♠♦ ❡✉ ❛❝r❡❞✐t♦ ♥♦ s❡✉❀ • ❆♠✐❣♦s ❞❡ ❝✉rs♦✱ ♣♦r ❝♦♠♣❛rt✐❧❤❛r❡♠ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s✳ ❊♠ ❡s♣❡❝✐❛❧ ❛ ▲❛ér❝✐♦
❙❛♥❣✐♦r❛tt♦ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ❛ ▼❛r❝❡❧♦ ▼❡❧♦ ❋❡r♥❛♥❞❡s q✉❡✱ ❝♦♠ s✉❛ s❡✲ r❡♥✐❞❛❞❡ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s✱ t♦r♥♦✉ ♣♦ssí✈❡❧ q✉❡ ♥♦ss♦ ❣r✉♣♦ ❞❡ ❡st✉❞♦s ♣❡r♠❛♥❡❝❡ss❡ ✉♥✐❞♦ ♣♦r t❛♥t♦ t❡♠♣♦❀
✐✐✐
❘❡s✉♠♦
❇❛s❡❛❞♦s ♥♦ ❛rt✐❣♦ ❞❡ ▼✳ ❏♦s❡❢ss♦♥✱ ✜③❡♠♦s ♥♦ss♦s ❡st✉❞♦s ❞♦s q✉❛❞r✐❧át❡r♦s ❝♦♥✈❡①♦s✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ❞♦s tr❛♣é③✐♦s✱ ❡ ❜✉s❝❛♠♦s ♣♦r ❢✉♥❞❛♠❡♥t❛çõ❡s q✉❡ ♦s ❝❛r❛❝t❡r✐③❛ss❡♠✳ ❊st✉❞❛♠♦s s❡✉s â♥❣✉❧♦s✱ s❡✉s ❧❛❞♦s✱ ❛s ♠❡❞✐❞❛s ❞❡ s✉❛s ár❡❛s ❡ ❡st❛❜❡❧❡❝❡♠♦s r❡❧❛çõ❡s ❡♥tr❡ s❡✉s ❡❧❡♠❡♥t♦s✱ ❧❛♥ç❛♥❞♦ ♠ã♦ ❞❡ ❞✐✈❡rs♦s t❡♦r❡♠❛s✱ ❝♦♠♦ ♦ P♦st✉❧❛❞♦ ❞❡ P❛s❝❤✱ ♠❛s ♥♦ss❛ ♣r✐♥❝✐♣❛❧ ❢✉♥❞❛♠❡♥t❛çã♦ ❡stá ♥♦ P♦st✉❧❛❞♦ ❞❛s P❛r❛❧❡❧❛s✳
P❛❧❛✈r❛s✲❈❤❛✈❡
✐✈
❆❜str❛❝t
❇❛s❡❞ ♦♥ t❤❡ ♣❛♣❡r ♦❢ ▼✳ ❏♦s❡❢ss♦♥ ✇❡ ❤❛✈❡ ❡❧❛❜♦r❛t❡❞ ♦✉r st✉❞✐❡s ♦♥ t❤❡ ❝♦♥✈❡① q✉❛❞r✐❧❛t❡r❛❧s✱ ♠♦r❡ ♣r❡❝✐s❡❧② t❤❡ tr❛♣❡③♦✐❞s✱ ❛♥❞ s❡❛r❝❤❡❞ ❢♦r ❢✉♥❞❛♠❡♥t❛t✐♦♥s ✇❤✐❝❤ ✇♦✉❧❞ ❝❤❛r❛❝t❡r✐③❡ t❤❡♠✳ ❲❡ ❤❛✈❡ st✉❞✐❡❞ t❤❡✐r ❛♥❣❧❡s✱ t❤❡✐r s✐❞❡s✱ t❤❡✐r ❛r❡❛s ❛♥❞ st❛❜✐✲ ❧✐s❤❡❞ r❡❧❛t✐♦♥s t♦ t❤❡✐r ❡❧❡♠❡♥ts✱ ♠❛❦✐♥❣ ✉s❡ ♦❢ s❡✈❡r❛❧ t❤❡♦r❡♠s ❛s t❤❡ P❛s❝❤✬s P♦st✉❧❛t❡✱ ❤♦✇❡✈❡r✱ ♦✉r ♠❛✐♥ ❢✉♥❞❛♠❡♥t❛t✐♦♥ ✐s ♦♥ t❤❡ P❛r❛❧❧❡❧ P♦st✉❧❛t❡✳
❑❡②✇♦r❞s
❙✉♠ár✐♦
✶ ❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s ✹
✷ ❈❛r❛❝t❡r✐③❛çõ❡s ❋✉♥❞❛♠❡♥t❛✐s ✶✻
✷✳✶ ❈❛r❛❝t❡r✐③❛çõ❡s ❞♦s tr❛♣é③✐♦s ❡♥✈♦❧✈❡♥❞♦ tr✐❣♦♥♦♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ ❈❛r❛❝t❡r✐③❛çõ❡s ❞♦s tr❛♣é③✐♦s ❡♥✈♦❧✈❡♥❞♦ ár❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸ ❈❛r❛❝t❡r✐③❛çõ❡s q✉❡ ❡♥✈♦❧✈❡♠ ❧❛❞♦s ❡ ❞✐stâ♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸ Pr♦♣♦st❛ ❞❡ ❆t✐✈✐❞❛❞❡s ✸✺
✸✳✶ ◗✉❡stõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ Pr♦♣♦st❛ ❞❡ ❙♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✹ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✹✶
❇✐❜❧✐♦❣r❛✜❛ ✹✷
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✵✳✶ ❉✐❛❣r❛♠❛ ♣❛r❛ ❛ ♣r✐♠❡✐r❛ ❞❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✵✳✷ ❉✐❛❣r❛♠❛ ♣❛r❛ ❛ s❡❣✉♥❞❛ ❞❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶ ◗✉❛❞r✐❧át❡r♦ ♥ã♦ ❝♦♥✈❡①♦ ❡ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ➶♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ ❙♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✹ P♦♥t♦ D ♣❡rt❡♥❝❡♥t❡ à r❡❣✐ã♦ ✐♥t❡r♥❛ ❞❡ ✉♠ â♥❣✉❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✶✳✺ P♦st✉❧❛❞♦ ❞❡ P❛s❝❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✻ ❆s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ✐♥t❡rs❡❝t❛♠✲s❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✼ ■❧✉str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✭✶✳✷✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✽ ➶♥❣✉❧♦ ❡①t❡r♥♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✾ ❚r❛♣é③✐♦ ❝♦♠ A, B ❧♦ ←→CD. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✶✳✶✵ ❚r❛♣é③✐♦ ABCD✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✶✳✶✶ ❚r❛♣é③✐♦ ❡ s✉❛s ❞✐❛❣♦♥❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶ ➶♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❝♦♥❣r✉❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ ❚r❛♣é③✐♦ ❝♦♠ ❧❛❞♦s AB ❡ CD ♥ã♦ ♣❛r❛❧❡❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷✳✸ ❇✐♠❡❞✐❛♥❛ ❞❡ ✉♠ tr❛♣é③✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✹ S1+S2 =S3+S4✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✺ ❚r❛♣é③✐♦ ❝♦♠ ❜✐♠❡❞✐❛♥❛ ♣❛r❛❧❡❧❛ às ❜❛s❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✻ ◗✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ❞✐✈✐❞✐❞♦ ❡♠ q✉❛tr♦ tr✐â♥❣✉❧♦s ♣❡❧❛s s✉❛s ❞✐❛❣♦♥❛✐s✳ ✳ ✷✸ ✷✳✼ Pr♦❞✉t♦s ❞♦s tr✐â♥❣✉❧♦s ♦♣♦st♦s ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ sã♦ ✐❣✉❛✐s✳ ✳ ✳ ✷✹ ✷✳✽ ❈❛s♦ ✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✾ ❈❛s♦ ✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✈✐✐ ✷✳✶✵ ❈❛s♦ ✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✶ ❈❛s♦ ✹✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✷ ■❧✉str❛çã♦ ❛✉①✐❧✐❛r ♣❛r❛ ♦ ▲❡♠❛ ✭✷✳✸✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✶✸ ■❧✉str❛çã♦ ❛✉①✐❧✐❛r ♣❛r❛ ♦ ▲❡♠❛ ✭✷✳✹✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✶✹ ■❧✉str❛çã♦ ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✭✷✳✹✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✶✺ ▼❡❞✐❛♥❛ ❞❡ ❊✉❧❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✶✻ ❉✐❛❣♦♥❛❧ p❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
■♥tr♦❞✉çã♦
◆❛ ❜✉s❝❛ ❞❡ ✉♠ t❡♠❛ ♣❛r❛ ❡st❡ tr❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❡ r❡s♣❡✐t❛♥❞♦ ❛s ♥♦r♠❛s ❞♦ P❘❖❋▼❆❚ q✉❡ ❞❡t❡r♠✐♥❛♠ ❡♠ s❡✉ ❝❛♣ít✉❧♦ ❱■■■ ✲ ❚❘❆❇❆▲❍❖ ❉❊ ❈❖◆❈▲❯❙➹❖ ❉❊ ❈❯❘❙❖✱ ❆rt✐❣♦ ✷✽✿ ✏❖ ❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❞❡✈❡ ✈❡rs❛r s♦❜r❡ t❡♠❛s ❡s✲ ♣❡❝í✜❝♦s ♣❡rt✐♥❡♥t❡s ❛♦ ❝✉rrí❝✉❧♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❡ q✉❡ t❡♥❤❛♠ ✐♠♣❛❝t♦ ♥❛ ♣rát✐❝❛ ❞✐❞át✐❝❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✑✱ ❞❡❝✐❞✐♠♦s ❡st✉❞❛r ❛ ●❡♦♠❡tr✐❛ ❞♦s tr❛♣é③✐♦s✱ q✉❡ é ✉♠ ❛ss✉♥t♦ ♠✉✐t♦ ❛❜♦r❞❛❞♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❖ ❡♥s✐♥♦ ❞❡ ●❡♦♠❡tr✐❛ ✈❡♠ s♦❢r❡♥❞♦ ❛s ❝♦♥s❡q✉ê♥❝✐❛s ❞❛ ❡①❝❧✉sã♦ ❞❛ ❞✐s❝✐♣❧✐♥❛ ❉❡s❡✲ ♥❤♦ ●❡♦♠étr✐❝♦ ❞♦s ❝✉rrí❝✉❧♦s ❞❛s ❡s❝♦❧❛s ♣ú❜❧✐❝❛s✱ ❢❛t♦ ♦❝♦rr✐❞♦ ♥❛ ❞é❝❛❞❛ ❞❡ ✾✵✳ ❊♠ ❉❡s❡♥❤♦ ●❡♦♠étr✐❝♦ ♦s ❛❧✉♥♦s t✐♥❤❛♠ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❧✐❞❛r ♥❛ ♣rát✐❝❛ ❝♦♠ ♦s ❡♥t❡s ❣❡♦♠étr✐❝♦s ❛tr❛✈és ❞❡ s✉❛s ❝♦♥str✉çõ❡s ❝♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦✱ ♦ q✉❡ ❧❤❡s ♣❡r♠✐t✐❛♠ ♠❛✐♦r ❢❛♠✐❧✐❛r✐❞❛❞❡ ❝♦♠ t❛✐s ❡♥t❡s✳ ❆ ❡①❝❧✉sã♦ ❞❡ t❛❧ ❝♦♠♣♦♥❡♥t❡ r❡t✐r♦✉ ❞♦s ❛❧✉♥♦s ♦ ❛s♣❡❝t♦ ♣rát✐❝♦ ❞❛ ●❡♦♠❡tr✐❛✳
◆❡st❡ tr❛❜❛❧❤♦✱ ❞❛♠♦s ❛❧❣✉♠❛s ❝❛r❛❝t❡r✐③❛çõ❡s ❞❡ tr❛♣é③✐♦s✱ ♦✉ s❡❥❛✱ ❝♦♥❞✐çõ❡s ♥❡❝❡s✲ sár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ s❡❥❛ ✉♠ tr❛♣é③✐♦✳ ▼✉✐t❛s ❞❡st❛s ❝❛r❛❝t❡r✐③❛çõ❡s sã♦ ❛♣r❡s❡♥t❛❞❛s ♥❛s ❛✉❧❛s ❞❡ ●❡♦♠❡tr✐❛ ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❝♦♠♦ ♣r♦♣r✐✲ ❡❞❛❞❡s ❞♦s tr❛♣é③✐♦s✳ ◆❡♠ t♦❞❛s ❛s ❢❡rr❛♠❡♥t❛s q✉❡ ✉s❛♠♦s ❛♦ ❧♦♥❣♦ ❞❛ ❞✐ss❡rt❛çã♦ sã♦ ♣❡rt✐♥❡♥t❡s ❛♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✱ ♠❛s sã♦ ♣❡r❢❡✐t❛♠❡♥t❡ ❛❝❡ssí✈❡✐s ❛♦s ❛❧✉✲ ♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❊st❡ tr❛❜❛❧❤♦ ❞❡✈❡ s❡r✈✐r ❞❡ ✐♥s♣✐r❛çã♦ ♣❛r❛ q✉❡ ♦✉tr♦s ❡♥t❡s ❣❡♦♠étr✐❝♦s s❡❥❛♠ ❡st✉❞❛✲ ❞♦s ❞❡ ♠♦❞♦ s❡♠❡❧❤❛♥t❡✱ ❜✉s❝❛♥❞♦ ♣❡❧❛s ❢✉♥❞❛♠❡♥t❛çõ❡s q✉❡ ♦s ❝❛r❛❝t❡r✐③❡♠✱ t♦r♥❛♥❞♦ ♦ ❛♣r❡♥❞✐③❛❞♦ ♠❛✐s ❝♦♥s✐st❡♥t❡✳
❊s❝❧❛r❡❝❡♠♦s q✉❡ ♦s ❛✉t♦r❡s ❞❡ ❧✐✈r♦s ❞✐❞át✐❝♦s ✉s❛♠ ❞✐❢❡r❡♥t❡s ❞❡✜♥✐çõ❡s ♣❛r❛ ✏tr❛♣é③✐♦✑✳ ❖s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ♦ ❞❡✜♥❡♠ ❝♦♠♦ s❡♥❞♦ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♠
apenas ✉♠ ♣❛r ❞❡ ❧❛❞♦s ♦♣♦st♦s ♣❛r❛❧❡❧♦s✱ ❡♥q✉❛♥t♦ q✉❡ ♦s ❧✐✈r♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ♦
❞❡✜♥❡♠ ❝♦♠♦ s❡♥❞♦ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣❛r ❞❡ ❧❛❞♦s ♦♣♦st♦s ♣❛r❛❧❡❧♦s✳ P❛rt✐❝✉❧❛r♠❡♥t❡✱ ❛ ♦❜r❛ ✧▼❛t❡♠át✐❝❛ ❡ ❘❡❛❧✐❞❛❞❡✧✱ ❞❡ ●❡❧s♦♥ ■❡③③✐ ❡ ❆♥t♦♥✐♦ ▼❛❝❤❛❞♦✱ ❆t✉❛❧ ❊❞✐t♦r❛✱ ❙ã♦ P❛✉❧♦✱ ✷✵✵✺✱ ❞❡✜♥❡✿ ❚r❛♣é③✐♦ é ✉♠ q✉❛❞r✐❧át❡r♦ q✉❡ t❡♠ ❞♦✐s ❧❛❞♦s ♣❛r❛❧❡❧♦s✳ ❖✉tr❛ ❞❡✜♥✐çã♦ é ❞❛❞❛ ❡♠ ❬✺❪✱ q✉❡ ❛❞♦t❛♠♦s ❡♠ ♥♦ss♦ tr❛❜❛❧❤♦✳ ❉❡ ❛❝♦r❞♦
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✷ ❝♦♠ ❛ ♣r✐♠❡✐r❛ ❞❡✜♥✐çã♦ ❞❡ tr❛♣é③✐♦✱ ♦s ♣❛r❛❧❡❧♦❣r❛♠♦s ♥ã♦ sã♦ tr❛♣é③✐♦s✱ ❡♥q✉❛♥t♦ q✉❡✱ ♥♦ s❡❣✉♥❞♦ ❝❛s♦✱ ♦s ♣❛r❛❧❡❧♦❣r❛♠♦s sã♦ tr❛♣é③✐♦s✳
❊sq✉❡♠❛t✐❝❛♠❡♥t❡✱ t❡♠♦s ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s q✉❡ r❡s✉♠❡♠ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦s q✉❛✲ ❞r✐❧át❡r♦s ❝♦♥✈❡①♦s✿
❋✐❣✉r❛ ✵✳✶✿ ❉✐❛❣r❛♠❛ ♣❛r❛ ❛ ♣r✐♠❡✐r❛ ❞❡✜♥✐çã♦
❋✐❣✉r❛ ✵✳✷✿ ❉✐❛❣r❛♠❛ ♣❛r❛ ❛ s❡❣✉♥❞❛ ❞❡✜♥✐çã♦
P❛r❛ ♦ ❡st✉❞♦ ❞♦s tr❛♣é③✐♦s✱ ♦s s❡❣✉✐♥t❡s ❢✉♥❞❛♠❡♥t♦s sã♦ ♥❡❝❡ssár✐♦s✿ ♣❛r❛❧❡❧❛s✱ tr✐✲ â♥❣✉❧♦s ❡ s❡✉s â♥❣✉❧♦s✱ ár❡❛s ❞❡ tr✐â♥❣✉❧♦s✱ â♥❣✉❧♦s ❝♦♠♣❧❡♠❡♥t❛r❡s ❡ s✉♣❧❡♠❡♥t❛r❡s✱ r❡❧❛çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✱ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ ❡♥tr❡ ♦✉tr♦s✳
❈❛♣ít✉❧♦ ✶
❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s
◆♦ss♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ♥❡st❡ tr❛❜❛❧❤♦ é ❡st❛❜❡❧❡❝❡r ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ s❡❥❛ ✉♠ tr❛♣é③✐♦✳ ❯♠❛ ❞❡st❛s ❝♦♥❞✐çõ❡s s❡rá ♦❜t✐❞❛ ❛❞✐❛♥t❡✱ ♥♦ ❚❡♦r❡♠❛ ✭✶✳✺✮✱ ♣á❣✐♥❛ ✶✸✳ ❖✉tr❛s ❛♣❛r❡❝❡rã♦ ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳
❈♦♠❡ç❛♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♥❞♦ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ♣❧❛♥❛ ❡✉❝❧✐❞✐❛♥❛ s♦❜r❡ ♦s q✉❛✐s ♥♦s ❢✉♥❞❛♠❡♥t❛r❡♠♦s✳ P♦r ❡①❡♠♣❧♦✱ ❝♦♥✈❡①✐❞❛❞❡ ❞❡ q✉❛❞r✐❧á✲ t❡r♦✱ ♣❛r❛❧❡❧✐s♠♦ ❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✳ ■♥❞✐❝❛♠♦s ❝♦♠♦ r❡❢❡rê♥❝✐❛ ❜ás✐❝❛ ♦ ❧✐✈r♦ ❞❡ ❊✳▼♦✐s❡ ❬✹❪✱ q✉❡ ❛♣r❡s❡♥t❛ ✉♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❛①✐♦♠át✐❝♦ ❞❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ♥❡✉tr❛✱ ❞❡st❛❝❛♥❞♦ ♦ ♣❛♣❡❧ ❞♦ P♦st✉❧❛❞♦ ❞❛s P❛r❛❧❡❧❛s ✭P♦st✉❧❛❞♦ ✭✶✳✶✮✱ ♣á❣✳ ✻✮ ♥❛ ❣❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛✱ ❝♦♠ ♦ q✉❛❧ ❣❛r❛♥t❡✲s❡ ❛ ✉♥✐❝✐❞❛❞❡ ❞❡ ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ r❡t❛ ♣❛ss❛♥❞♦ ♣♦r ✉♠ ♣♦♥t♦ ❢♦r❛ ❞❛ r❡t❛ ❞❛❞❛✳ ◆♦ ♥♦ss♦ ❝❛s♦✱ ❛❞♠✐t✐r❡♠♦s ❡st❡ ♣♦st✉✲ ❧❛❞♦ ♣❛r❛ ❛❧❝❛♥ç❛r♠♦s ♥♦ss♦s ♦❜❥❡t✐✈♦s✳ ❍á ❛❧❣✉♠❛s ❢♦r♠❛s ❡q✉✐✈❛❧❡♥t❡s ♣❛r❛ ❡♥✉♥❝✐á✲❧♦✱ ❡♥tr❡t❛♥t♦ ♥ós ❧❛♥ç❛r❡♠♦s ♠ã♦ ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s✱ t❡♦r❡♠❛s ❡ ❞❡✜♥✐çõ❡s ♣❛r❛ ❡♥tã♦ ❡st❛❜❡❧❡❝❡r♠♦s ❛ ✈❡rsã♦ q✉❡ t♦♠❛r❡♠♦s ❛q✉✐✳
❆ ♣r✐♥❝í♣✐♦ tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♥✈❡①✐❞❛❞❡ ❞❡ q✉❛❞r✐❧át❡r♦s✳ ❈♦♠❡ç❛✲ ♠♦s ❞❡✜♥✐♥❞♦ q✉❛❞r✐❧át❡r♦s✳
❉❡✜♥✐çã♦ ✶✳✶✳ ❉❛❞♦s q✉❛tr♦ ♣♦♥t♦s A, B, C ❡ D ❝♦♣❧❛♥❛r❡s✱ três ❛ três ♥ã♦ ❝♦❧✐♥❡❛r❡s✱
s❡ ♦s s❡❣♠❡♥t♦s AB✱ BC✱ CD ❡ DA ✐♥t❡rs❡❝t❛♠✲s❡ ❛♣❡♥❛s ❡♠ s❡✉s ❡①tr❡♠♦s✱ s✉❛ ✉♥✐ã♦
é ❝❤❛♠❛❞❛ q✉❛❞r✐❧át❡r♦✳
❉❡✜♥✐çã♦ ✶✳✷✳ ❉❛❞❛ ✉♠❛ r❡t❛ r✱ s❡❥❛♠ H1 ❡ H2 ♦s s❡♠✐♣❧❛♥♦s ❞✐s❥✉♥t♦s ❞❡t❡r♠✐♥❛❞♦s
♣♦r r✳ ❉✐③❡♠♦s q✉❡ ♦s ♣♦♥t♦s A ❡ B ❡stã♦ ❡♠ ❧❛❞♦s ♦♣♦st♦s ❞❡ r✱ ❡ ❞❡♥♦t❛♠♦s ♣♦r A, B
❧♦ r✱ s❡ A∈H1 ❡ B ∈H2 ♦✉ s❡ A∈ H2 ❡ B ∈H1✳ ❉✐③❡♠♦s q✉❡ A ❡ B ❡stã♦ ♥♦ ♠❡s♠♦
❧❛❞♦ ❞❡ r✱ ❡ ❞❡♥♦t❛♠♦s ♣♦r A, B ♠❧ r✱ s❡ A∈H1 ❡ B ∈H1 ♦✉ s❡ A∈H2 ❡ B ∈H2✳
❉❡✜♥✐çã♦ ✶✳✸✳ ❯♠ q✉❛❞r✐❧át❡r♦ é ❝❤❛♠❛❞♦ ❝♦♥✈❡①♦✱ s❡ ❝❛❞❛ ✉♠ ❞❡ s❡✉s ❧❛❞♦s ♣❡rt❡♥❝❡ ❛♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ ❧❛❞♦ ♦♣♦st♦✳
❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱
❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✺ ✲ A ❡ B ❡stã♦ ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r←→CD❀
✲ B ❡ C ❡stã♦ ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r ←→DA❀
✲ C ❡ D ❡stã♦ ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r ←→AB❀
✲ D ❡ A ❡stã♦ ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r ←→BC✳
◆❛ ✜❣✉r❛ ✭✶✳✶✮✱ ✐❧✉str❛♠♦s ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦EF GH ❡ ✉♠ ♥ã♦ ❝♦♥✈❡①♦ABCD✳
◆❡st❡ ❝❛s♦✱A, B ❧♦←→CD.
❋✐❣✉r❛ ✶✳✶✿ ◗✉❛❞r✐❧át❡r♦ ♥ã♦ ❝♦♥✈❡①♦ ❡ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦✳
Pr❡❝✐s❛♠♦s ❛✐♥❞❛ ❞❡ ✉♠❛ ❞❡✜♥✐çã♦ q✉❡ s❡rá ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦✿ ❉❡✜♥✐çã♦ ✶✳✹✳ ❉♦✐s â♥❣✉❧♦s ❝✉❥❛ s♦♠❛ s❡❥❛ ✐❣✉❛❧ ❛ 180◦ sã♦ ❝❤❛♠❛❞♦s â♥❣✉❧♦s s✉♣❧❡✲
♠❡♥t❛r❡s✳
❉❛q✉✐ ♣♦r ❞✐❛♥t❡ ✉s❛r❡♠♦s ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s ❡ r❡♣r❡s❡♥t❛çõ❡s ♥♦s ❡❧❡♠❡♥t♦s ❞♦s q✉❛❞r✐❧át❡r♦s ❝♦♥✈❡①♦s✿
• ❆s ♠❡❞✐❞❛s ❞♦s â♥❣✉❧♦s ∠DAB,∠ABC,∠BCD ❡ ∠CDA ❞♦ q✉❛❞r✐❧át❡r♦ ABCD
s❡rã♦ ❞❡♥♦t❛❞♦s ♣♦r A,b B,b Cb ❡ Db✳
• AB✱ BC✱ CD ❡DA sã♦ ♦s ❧❛❞♦s ❞♦ q✉❛❞r✐❧át❡r♦ ABCD❀
• AC ❡BD sã♦ ❛s ❞✐❛❣♦♥❛✐s ❞♦ q✉❛❞r✐❧át❡r♦ABCD❀
• ❉♦✐s ❧❛❞♦s ❞❡ ✉♠❛ q✉❛❞r✐❧át❡r♦ ABCD sã♦ ❛❞❥❛❝❡♥t❡s s❡ s✉❛ ✐♥t❡rs❡❝çã♦ ♦❝♦rr❡r
♥✉♠❛ ❡①tr❡♠✐❞❛❞❡❀
❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✻
• ❉♦✐s â♥❣✉❧♦s sã♦ ❛❞❥❛❝❡♥t❡s s❡ ❡❧❡s ♣♦ss✉❡♠ ✉♠ ❧❛❞♦ ❝♦♠✉♠❀
• ❉♦✐s â♥❣✉❧♦s ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ ABCD sã♦ ♦♣♦st♦s s❡ ❡❧❡s ♥ã♦ sã♦ ❛❞❥❛❝❡♥t❡s✳
❊♠ r❡❧❛çã♦ ❛♦ ❝♦♥❝❡✐t♦ ❞❡ ♣❛r❛❧❡❧✐s♠♦ ❞❡ r❡t❛s✱ ❛❞♠✐t✐♠♦s ❛ s❡❣✉✐♥t❡ ✈❡rsã♦ ❞♦ P♦s✲ t✉❧❛❞♦ ❞❛s P❛r❛❧❡❧❛s✿
P♦st✉❧❛❞♦ ✶✳✶✳ P♦st✉❧❛❞♦ ❞❛s P❛r❛❧❡❧❛s✳
P♦r ✉♠ ♣♦♥t♦ P ♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛ ✉♠❛ ❞❛❞❛ r❡t❛ r✱ ♣❛ss❛ ✉♠❛ ú♥✐❝❛ r❡t❛ s ♣❛r❛❧❡❧❛
❛ r ✭✜❣✳✭✶✳✷✮✮✳
❋✐❣✉r❛ ✶✳✷✿ ➶♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s✳
❚❛♠❜é♠ t♦♠❛r❡♠♦s ♣♦r ❜❛s❡ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✱ ❝✉❥❛ ♣r♦✈❛ é ❞❛❞❛ ❡♠ ❬✹❪✱ ♣á❣✳ ✶✺✵✳ ❚❡♦r❡♠❛ ✶✳✶✳ ❉❛❞❛s ❞✉❛s r❡t❛s ❡ ✉♠❛ tr❛♥s✈❡rs❛❧✱ ❞♦✐s â♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s sã♦ ❝♦♥❣r✉❡♥t❡s s❡ ❛s r❡t❛s ❢♦r❡♠ ♣❛r❛❧❡❧❛s✳
❆ ♣❛rt✐r ❞♦ P♦st✉❧❛❞♦ ❞❛s P❛r❛❧❡❧❛s t❡♠♦s ❛s s❡❣✉✐♥t❡s ❝♦♥s❡q✉ê♥❝✐❛s✿
• ❉❛❞❛s ❞✉❛s r❡t❛s ♣❛r❛❧❡❧❛s r ❡ s ❡ ✉♠❛ tr❛♥s✈❡rs❛❧ t✱ ♦s ♣❛r❡s ❞❡ â♥❣✉❧♦s ❛❧t❡r♥♦s
✐♥t❡r♥♦s sã♦ ❝♦♥❣r✉❡♥t❡s❀ ❉❡♠♦♥str❛çã♦✳
❙❡❥❛♠ r ❡ s ❞✉❛s r❡t❛s ♣❛r❛❧❡❧❛s ❡ t ❛ tr❛♥s✈❡rs❛❧ ❝♦♠✉♠ ❛ ❡❧❛s✱ ❞❡ ♠♦❞♦ q✉❡ t∩r=P ❡ t∩s =Q✳ ❙❡❥❛♠ A ❡B ♣♦♥t♦s t❛✐s q✉❡A ∈r✱ B ∈s✱ A, B ❧♦t✳ ❙❡❥❛♠ r′ ✉♠❛ r❡t❛ q✉❡ ❝♦♥té♠ ♦ ♣♦♥t♦ P ❡ A′ ∈r′ ❝♦♠ A′, B ❧♦ t✳ ❊①✐st❡ ✉♠❛ ú♥✐❝❛ r❡t❛
r′ ♣♦r P ♣❛r❛ ❛ q✉❛❧ ♦s â♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s sã♦ ❝♦♥❣r✉❡♥t❡s ❡✱ ♣❡❧♦ ❚❡♦r❡♠❛
✶✳✶✱ t❡r❡♠♦s r′ k s✳ ▼❛s✱ ❝♦♠♦ ❤á ✉♠❛ ú♥✐❝❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛ s ♣♦r P✱ ❡♥tã♦ r′ =r
❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✼
• ❉❛❞❛s ❞✉❛s r❡t❛s ❡ ✉♠❛ tr❛♥s✈❡rs❛❧✱ s❡ ❛s ❞✉❛s r❡t❛s sã♦ ♣❛r❛❧❡❧❛s✱ ❡♥tã♦ ♦s ♣❛r❡s ❞❡ â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s sã♦ ❝♦♥❣r✉❡♥t❡s ✭❛ ❞❡♠♦♥str❛çã♦ é ❝♦♠♣❧❡t❛♠❡♥t❡ ❛♥á❧♦❣❛ à ❛♥t❡r✐♦r✮✳
• ❊♠ q✉❛❧q✉❡r tr✐â♥❣✉❧♦ △ABC t❡♠♦sAb+Bb+Cb= 180 ✭✜❣✳ ✭✶✳✸✮✮✳
❋✐❣✉r❛ ✶✳✸✿ ❙♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✳
❉❡♠♦♥str❛çã♦✳
❙❡❥❛♠r kAC ✉♠❛ r❡t❛ t❛❧ q✉❡B, D, E ∈r t❛✐s q✉❡D−B−E✱ ✐st♦ é✱ B ❡stá ❡♥tr❡ D ❡ E✱ ❞❡ ♠♦❞♦ q✉❡D, A ♠❧ ←→BC✳ ❊♥tã♦ ♦s â♥❣✉❧♦s ∠DBC ❡ ∠CBE ❢♦r♠❛♠ ✉♠
♣❛r ❞❡ â♥❣✉❧♦s s✉♣❧❡♠❡♥t❛r❡s✳ ❆ss✐♠✿
DBAb +Bb =DBCb ❡ DBCb +CBEb = 180◦✳
▲♦❣♦✱
DBAb +Bb+CBEb = 180◦✳
▼❛s DBAb ❡ Absã♦ ❛❧t❡r♥♦s ✐♥t❡r♥♦s✱ ❛ss✐♠ ❝♦♠♦CBEb ❡ Cb✳
❊♥tã♦✱
b
A+Bb+Cb= 180◦. ✭✶✳✶✮
❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✽ Pr♦♣♦s✐çã♦ ✶✳✶✳ ❆s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠❛ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ✐♥t❡rs❡❝t❛♠✲s❡ ✭✜❣✳ ✭✶✳✻✮✱ ♣á❣✳ ✾✮✳
P❛r❛ ❛♣r❡s❡♥t❛r♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞❡st❛ Pr♦♣♦s✐çã♦ ♣r❡❝✐s❛r❡♠♦s ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐✲ t♦s✿
❉❡✜♥✐çã♦ ✶✳✺✳ ❙❡❥❛ ✉♠ â♥❣✉❧♦ ∠BAC✳ ❙❡✉ ✐♥t❡r✐♦r✱ ♦✉ s✉❛ r❡❣✐ã♦ ✐♥t❡r♥❛✱ é ❛ ✐♥t❡r✲
s❡❝çã♦ ❞♦ s❡♠✐♣❧❛♥♦ ❞❡✜♥✐❞♦ ♣♦r ←→AB q✉❡ ❝♦♥té♠ ♦ ♣♦♥t♦ C ❝♦♠ ♦ s❡♠✐♣❧❛♥♦ ❞❡✜♥✐❞♦ ♣♦r
←→
AC q✉❡ ❝♦♥té♠ ♦ ♣♦♥t♦ B✳
❋✐❣✉r❛ ✶✳✹✿ P♦♥t♦ D ♣❡rt❡♥❝❡♥t❡ à r❡❣✐ã♦ ✐♥t❡r♥❛ ❞❡ ✉♠ â♥❣✉❧♦✳
❆ss✐♠✱ ✉♠ ♣♦♥t♦D❡♥❝♦♥tr❛✲s❡ ♥♦ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦∠BAC s❡B ❡D❡stã♦ ♥♦ ♠❡s♠♦
s❡♠✐♣❧❛♥♦ ❞❡✜♥✐❞♦ ♣♦r ←→AC ❡ s❡ C ❡ D ❡stã♦ ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❞❡✜♥✐❞♦ ♣♦r←→AB ✭✜❣✳
✭✶✳✹✮✮✳
❆ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡ ❢♦✐ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ●❡♦♠❡tr✐❛ ❛①✐♦♠át✐❝❛✱ ✉♠❛ ✈❡③ q✉❡ ❞❡❧❛ ❢♦r❛♠ ♦❜t✐❞♦s ♠✉✐t♦s ♦✉tr♦s r❡s✉❧t❛❞♦s✳
Pr♦♣♦s✐çã♦ ✶✳✷✳ ✭P♦st✉❧❛❞♦ ❞❡ P❛s❝❤✮✳
❈♦♥s✐❞❡r❡ ✉♠ tr✐â♥❣✉❧♦ △ABC ❡ s❡❥❛ r ✉♠❛ r❡t❛ ❝♦♥t❡♥❞♦ ✉♠ ♣♦♥t♦ P ∈ AC✳ ❊♥✲
tã♦ r ✐♥t❡rs❡❝t❛ AB ♦✉ BC (f ig.(1.5)).
❉❡♠♦♥str❛çã♦✳
❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ r∩ AB = ∅ ❡ r ∩BC = ∅✱ ❡♥tã♦ A, B ♠❧ r✱ ❛ss✐♠ ❝♦♠♦ B, C ♠❧r✳ ❆ss✐♠ s❡♥❞♦✱A, C ♠❧r✱ ♦ q✉❡ é ❛❜s✉r❞♦✱ ❥á q✉❡{P}=r∩AC✳
❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✾
❋✐❣✉r❛ ✶✳✺✿ P♦st✉❧❛❞♦ ❞❡ P❛s❝❤✳
❉❡♠♦♥str❛çã♦✳
❙❡❥❛ ♦ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD ✭✜❣✳ ✭✶✳✻✮✮✳ ❈♦♠♦ A, B ♠❧ DC ❡ ❝♦♠♦ B, C ♠❧
←→
DA✱ ❡♥tã♦ B ♣❡rt❡♥❝❡ à r❡❣✐ã♦ ✐♥t❡r♥❛ ❞♦ â♥❣✉❧♦ ADCb ✳ ❆ss✐♠✱ s❡❣✉❡ ❞♦ P♦st✉❧❛❞♦ ❞❡
P❛s❝❤ ✭Pr♦♣♦s✐çã♦ ✭✶✳✷✮✱ ♣á❣✐♥❛ ✽✮ r❡❧❛t✐✈♦ ❛♦ tr✐â♥❣✉❧♦ △ADC q✉❡ ❛ r❡t❛ s✉♣♦rt❡ ❞❡ BD ✐♥t❡rs❡❝t❛ ❛ ❞✐❛❣♦♥❛❧ AC ❡♠ ✉♠ ♣♦♥t♦ P✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❛ r❡t❛ s✉♣♦rt❡ ❞❡ AC
✐♥t❡rs❡❝t❛ ❛ ❞✐❛❣♦♥❛❧BD ❡♠ ✉♠ ♣♦♥t♦ Q✳ ❈♦♠♦P, Q∈AC←→∩←→BD✱ ❡♥tã♦ P =Q✳
❋✐❣✉r❛ ✶✳✻✿ ❆s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ✐♥t❡rs❡❝t❛♠✲s❡✳
◆♦ss♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ♥❡st❛ ❞✐ss❡rt❛çã♦ sã♦ ♦s tr❛♣é③✐♦s✳ ❏á ♦❜s❡r✈❛♠♦s q✉❡ ❛❧❣✉♥s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦ ❞✐s❝♦r❞❛♠ q✉❛♥t♦ à ❞❡✜♥✐çã♦ ❞❡ tr❛♣é③✐♦s✳ ❆ ❞❡✜♥✐çã♦ ❛ s❡❣✉✐r tr❛③ ❛ ❡s❝♦❧❤❛ q✉❡ ✜③❡♠♦s ❛q✉✐✱ q✉❡ ❡stá ❞❡ ❛❝♦r❞♦ ❝♦♠ ❆✳❈✳ ▼✉♥✐③ ◆❡t♦ ❬✺❪ ❡ ❊✳ ▼♦✐s❡ ❬✹❪✳
❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✶✵ ♣❛r❛❧❡❧♦❣r❛♠♦✳
▼♦str❡♠♦s q✉❡ ♦s tr❛♣é③✐♦s sã♦ q✉❛❞r✐❧át❡r♦s ❝♦♥✈❡①♦s✳ P❛r❛ ✐st♦✱ ♣r❡❝✐s❛♠♦s ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❛✉①✐❧✐❛r❡s✳
❚❡♦r❡♠❛ ✶✳✷✳ ❚♦❞♦ ❧❛❞♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ❝♦♠ ❡①❝❡çã♦ ❞❡ s❡✉s ❡①tr❡♠♦s✱ ❡stá ❝♦♥t✐❞♦ ♥♦ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦ ♦♣♦st♦✳
❈♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ △ABC✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ BC \ {B, C} ⊂ int(∠BAC) ✭✜❣✳
✭✶✳✼✮✮✳
❋✐❣✉r❛ ✶✳✼✿ ■❧✉str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✭✶✳✷✮✳
❉❡♠♦♥str❛çã♦✳
❙❛❜❡♠♦s q✉❡ BC \C ❡stá ❝♦♥t✐❞♦ ♥♦ ❧❛❞♦ ❞❡ ←→AC q✉❡ ❝♦♥té♠ B ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ B, C ❧♦ ←→AC✱ ♦ q✉❡ s❡r✐❛ ❛❜s✉r❞♦✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ BC \B ❡stá ❝♦♥t✐❞♦ ♥♦ ❧❛❞♦ ❞❡ ←→AB
q✉❡ ❝♦♥té♠ C✳ ❊♥tã♦ BC\ {B, C} ⊂int(∠BAC)✳
❚❡♦r❡♠❛ ✶✳✸✳ ❚❡♦r❡♠❛ ❞♦ ➶♥❣✉❧♦ ❊①t❡r♥♦✳
❊♠ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✱ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ â♥❣✉❧♦ ❡①t❡r♥♦ é ❛ s♦♠❛ ❞❛s ♠❡❞✐❞❛s ❞♦s ❞♦✐s â♥❣✉❧♦s ✐♥t❡r♥♦s ♥ã♦ ❛❞❥❛❝❡♥t❡s ❛ ❡❧❡ ✭✜❣✳ ✭✶✳✽✮✮✳
❘❡❢♦r♠✉❧❛♥❞♦✿
❉❛❞♦ ✉♠ tr✐â♥❣✉❧♦ △ABC✱ ❡♥tã♦✿
BCDb =Ab+B.b
❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✶✶
❋✐❣✉r❛ ✶✳✽✿ ➶♥❣✉❧♦ ❡①t❡r♥♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦✳
❉❡♠♦♥str❛çã♦✳
❙❡♥❞♦ A−C−D✱ t❡♠♦s Cb+BCDb = 180◦✱ ❥á q✉❡ sã♦ â♥❣✉❧♦s s✉♣❧❡♠❡♥t❛r❡s✳ ❈♦♠♦
b
A+Bb+Cb= 180◦✱ ❡♥tã♦ Ab+Bb+Cb=Cb+BCDb ✳ ❖✉ s❡❥❛✱
BCDb =Ab+B.b ✭✶✳✷✮
❚❡♦r❡♠❛ ✶✳✹✳ ❚♦❞♦ tr❛♣é③✐♦ é ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦✳ ❘❡❢♦r♠✉❧❛♥❞♦✿
❙❡❥❛ ABCD ✉♠ tr❛♣é③✐♦ ❝♦♠ BC k DA✳ ❊♥tã♦ ABCD é ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦✱
♦✉ s❡❥❛✿
• A, B ♠❧←→CD❀
• B, C ♠❧←→DA❀
• C, D ♠❧←→AB❀
• D, A ♠❧←→BC✳
❉❡♠♦♥str❛çã♦✳
❈♦♠♦ AD k BC✱ ❡♥tã♦ A, D ♠❧ ←→BC ❡ ♦ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠ B, C ❝♦♠ r❡❧❛çã♦ ❛ ←→AD✳
❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✶✷ ❱❛♠♦s s✉♣♦r✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ A, B ❧♦ ←→CD✱ ♦✉ s❡❥❛✱ q✉❡ ♣❡rt❡♥ç❛♠ ❛ s❡♠✐♣❧❛♥♦s
❞✐st✐♥t♦s ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛ r❡t❛ s✉♣♦rt❡ ❞❡ CD✳ ❈♦♠♦ ABCD é ✉♠ q✉❛❞r✐❧át❡r♦✱ ❞❡✈❡
♦❝♦rr❡r ❛ ❡①✐stê♥❝✐❛ ❞❡ X ∈←→CD ❝♦♠ X−C−D✳
❙❛❜❡♠♦s q✉❡ A, C, D ♥ã♦ sã♦ ❝♦❧✐♥❡❛r❡s ✭❞❛❞♦ q✉❡ ABCD é q✉❛❞r✐❧át❡r♦✮✱ ❡♥tã♦✱
❝♦♥s✐❞❡r❡♠♦s ♦ tr✐â♥❣✉❧♦ △ACD✳
❈♦♠♦ X−C −D✱ ❡♥tã♦ ACXb é ❡①t❡r♥♦ ❛♦ tr✐â♥❣✉❧♦ △ACD ❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦
➶♥❣✉❧♦ ❊①t❡r♥♦ ✭❚❡♦r❡♠❛ ✭✶✳✸✮✱ ♣á❣✐♥❛ ✶✵✮✱ t❡♠♦s q✉❡ ACXb = CADb +ADCb ✳ ❆ss✐♠✱ ACX > Cb ADb ✳
❈♦♠♦ A −X −B✱ X ∈ ←→CD✱ ❡♥tã♦ A, B ❧♦ ←→CD✳ P❡❧♦ ❚❡♦r❡♠❛ ✭✶✳✷✮✱ ♣á❣✐♥❛ ✶✵✱ X ∈ int(∠ACB)✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ACXb +XCBb =ACBb ✳ ❈♦♠♦ ABCD é tr❛♣é③✐♦
❝♦♠ AD k BC ❡ ←→AC é tr❛♥s✈❡rs❛❧✱ ❡♥tã♦ ∠CAD ∼= ∠ACB ✭â♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s✮✳
▲♦❣♦ CAD < Ab CX < Ab CBb ✱ ✐♠♣❧✐❝❛♥❞♦ ❡♠ CAD < Ab CBb ✱ ♦ q✉❡ é ❛❜s✉r❞♦✳ ❊♥tã♦A, B
♠❧CD✳
❆♥❛❧♦❣❛♠❡♥t❡✱ ✈❡r✐✜❝❛✲s❡ q✉❡ C, D ♠❧←→AB✳
❋✐❣✉r❛ ✶✳✾✿ ❚r❛♣é③✐♦ ❝♦♠A, B ❧♦ ←→CD.
❊♠❜♦r❛ t❡r♠♦s ❞❡❞✐❝❛❞♦ ♦ ❈❛♣ít✉❧♦ ✷ ♣❛r❛ ❛♣r❡s❡♥t❛r ❛s ❝❛r❛❝t❡r✐③❛çõ❡s ❞❡ tr❛♣é③✐♦ q✉❡ ❝♦♥s✐❞❡r❛♠♦s ❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s✱ ❛ s❡❣✉✐r ❞❡st❛❝❛♠♦s ❛❧❣✉♠❛s✱ ♠✉✐t♦ r❡❧❛❝✐♦♥❛❞❛s à ❞❡✜♥✐çã♦ ❞❡ tr❛♣é③✐♦✳
❆ ♣r✐♠❡✐r❛ ❞❡❧❛s ❡stá ❞❛❞❛ ❛ s❡❣✉✐r ❡ é ó❜✈✐❛✳
❙❡ ❛s r❡t❛s s✉♣♦rt❡s ❞♦s ❧❛❞♦s AB ❡ CD ✭✜❣✳ ✭✶✳✶✵✮✱ ♣á❣✳ ✶✸✮ ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦
❝♦♥✈❡①♦ ABCD ❢♦r♠❛♠ ✉♠ â♥❣✉❧♦ α ❡♥tr❡ s✐✱ ❡♥tã♦ ♦ q✉❛❞r✐❧át❡r♦ é ✉♠ tr❛♣é③✐♦ ❞❡
❧❛❞♦s ♦♣♦st♦s ♣❛r❛❧❡❧♦sAB ❡CD s❡✱ ❡ s♦♠❡♥t❡ s❡✱α ❂ ✵◦.
❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✶✸
❋✐❣✉r❛ ✶✳✶✵✿ ❚r❛♣é③✐♦ ABCD✳
❖ q✉❛❞r✐❧át❡r♦ ABCD é ✉♠ tr❛♣é③✐♦ ❞❡ ❧❛❞♦s ♦♣♦st♦s ♣❛r❛❧❡❧♦s AB ❡ CD s❡✱ ❡ s♦✲
♠❡♥t❡ s❡✱ ABDb ∼=BDCb ✭â♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s✮ ✭✜❣✳ ✭✶✳✶✵✮✮✳
❉❡st❛ ❝♦♥❞✐çã♦ ❝❤❡❣❛♠♦s ❛ ♦✉tr❛ ❡q✉✐✈❛❧❡♥t❡✱ ❢✉♥❞❛♠❡♥t❛❞♦s ♥❛ r❡❧❛çã♦ ❞❡ â♥❣✉❧♦s ❝♦✲ ❧❛t❡r❛✐s ✐♥t❡r♥♦s✿
❚❡♦r❡♠❛ ✶✳✺✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD ❝♦♠ ❧❛❞♦s ♦♣♦st♦s AB ❡ CD ♣❛r❛❧❡❧♦s é
✉♠ tr❛♣é③✐♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱
b
A+Db =Bb+Cb = 180◦. ✭✶✳✸✮
❉❡♠♦♥str❛çã♦✳
➱ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ r❡❧❛çã♦ ❡♥tr❡ â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❡ ♦ ♣❛r❛❧❡❧✐s♠♦ ❞❡ AB
❡ CD✳
❯s❛♥❞♦ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ ❛❧❝❛♥ç❛♠♦s ✉♠❛ ✐♠♣♦rt❛♥t❡ r❡❧❛çã♦ ❡♥tr❡ ♦s s❡❣✲ ♠❡♥t♦s q✉❡ ❢♦r♠❛♠ ❛s ❞✐❛❣♦♥❛✐s ❞♦ tr❛♣é③✐♦ ABCD❞❡ ❧❛❞♦s ♣❛r❛❧❡❧♦s AB ❡CD✱ q✉❡ s❡
✐♥t❡rs❡❝t❛♠ ♥♦ ♣♦♥t♦P ✭✜❣✳ ✭✶✳✶✵✮✱ ♣á❣✳ ✶✸✮✳
△ABP ∼ △CDP (Caso AA),
AP CP =
BP
DP· ✭✶✳✹✮
❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✶✹ ❜❛s❡s ❞♦ tr❛♣é③✐♦✮ ❡ ♦s ♦✉tr♦s ❞♦✐s ❝♦♠ ♠❡s♠❛ ár❡❛ ✭✜❣✳ ✭✶✳✶✶✮✱ ♣á❣✳ ✶✹✮✱ ❝♦♥❢♦r♠❡ ❛ Pr♦♣♦s✐çã♦ ✭✷✳✹✮✳
❋✐❣✉r❛ ✶✳✶✶✿ ❚r❛♣é③✐♦ ❡ s✉❛s ❞✐❛❣♦♥❛✐s✳
Pr♦♣♦s✐çã♦ ✶✳✸✳ ❙❡❥❛♠ ABCD ✉♠ tr❛♣é③✐♦ ❝♦♠ ❧❛❞♦s ♣❛r❛❧❡❧♦s AB ❡ CD ❡ {P} =
AC∩BD✳ ❊♥tã♦✱ ❛s ár❡❛s ❞♦s tr✐â♥❣✉❧♦s △AP D ❡ △BP C sã♦ ❝♦♥❣r✉❡♥t❡s✳
❉❡♠♦♥str❛çã♦✳
❉❡♥♦t❡ ♣♦r SX Y Z ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r ❝♦♠ ✈ért✐❝❡s X✱ Y ❡ Z✳
❈♦♠♦ ♦s tr✐â♥❣✉❧♦s △ABD ❡ △ABC ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ❜❛s❡ AB ❡ ❛ ♠❡s♠❛ ❛❧t✉r❛✱
❡♥tã♦ SABD =SABC✳
▼❛s✱ SABD =SAP D+SABP ❡SABC =SABP +SBCP✳ P♦rt❛♥t♦ SADP =SBCP.
❆s ♠❡s♠❛s r❡❧❛çõ❡s ❛♥t❡r✐♦r❡s ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❡♠ t❡r♠♦s tr✐❣♦♥♦♠étr✐❝♦s✳ P❛r❛ ♦ tr❛♣é③✐♦ABCD ❞❡ ❧❛❞♦s ♣❛r❛❧❡❧♦sAB ❡CD✱ t❡♠♦s✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✭✶✳✺✮✱ ♣á❣✐♥❛ ✶✸✱ q✉❡✿
(
s❡♥ Ab=s❡♥ Db ❡ s❡♥ Cb=s❡♥ B,b
❝♦s Ab=✲ ❝♦s Db ❡ ❝♦s Cb =✲ ❝♦s B.b
▲♦❣♦✱
❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✶✺ ❡
❝♦s Ab·❝♦sCb = ❝♦s Bb·❝♦sD.b ✭✶✳✻✮
P♦r ✭✶✳✺✮ ❡ ✭✶✳✻✮ ❝❤❡❣❛♠♦s ❛✿
s❡♥ Ab·s❡♥ Cb+❝♦sAb·❝♦s Cb =s❡♥ Bb·s❡♥ Db+❝♦sBb·❝♦s Db✳
❖✉ s❡❥❛✱
❝♦s(Ab−Cb) =❝♦s (Bb−Db). ✭✶✳✼✮
❘❡s✉♠✐♥❞♦✿
Pr♦♣♦s✐çã♦ ✶✳✹✳ ❙❡ ABCD ❢♦r ✉♠ tr❛♣é③✐♦ ❞❡ ❧❛❞♦s ♦♣♦st♦s AB ❡ CD✱ ❡♥tã♦ ❝♦s (Ab−
b
❈❛♣ít✉❧♦ ✷
❈❛r❛❝t❡r✐③❛çõ❡s ❋✉♥❞❛♠❡♥t❛✐s
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❝♦♥❞✐çõ❡s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ♦s tr❛♣é③✐♦s✱ ✉t✐❧✐③❛♥❞♦✲s❡ ❞❡ r❡❧❛çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✱ ❝á❧❝✉❧♦ ❞❡ ár❡❛ ❞❡ tr✐â♥❣✉❧♦s✱ ♠❡❞✐❞❛s ❞❡ ❧❛❞♦s ❡ ❞✐stâ♥❝✐❛s✳
❯t✐❧✐③❛r❡♠♦s ❛ ♥♦t❛çã♦ SX Y Z ♣❛r❛ ♥♦s r❡❢❡r✐r à ár❡❛ S ❞♦ tr✐â♥❣✉❧♦△XY Z✳
❉❛❞♦ ♦ tr❛♣é③✐♦ ABCD ❞❡ ❜❛s❡s ♣❛r❛❧❡❧❛s AB ❡ CD ✭✜❣✳ ✭✶✳✶✵✮✱ ♣á❣✳ ✶✸✮✱ s✉❛ ár❡❛ S ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ♣❡❧❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s tr✐â♥❣✉❧♦s △ABD ❡ △BCD✱ r❡s♣❡❝t✐✈❛✲
♠❡♥t❡✱ ♦✉ s❡❥❛✱
S =SABD+SBCD.
❈♦♠♦ ♦s tr✐â♥❣✉❧♦s △ABD ❡ △BCD tê♠ ❛ ♠❡s♠❛ ❛❧t✉r❛ h r❡❧❛t✐✈❛ ❛♦s ❧❛❞♦sAB ❡ CD✱ ❛ss✐♠✿
S=AB·h· 1
2 +CD·h· 1 2·
❙❡♥❞♦ AB =a ❡ CD =b✱ t❡♠♦s q✉❡✿
S= a+b
2 ·h. ✭✷✳✶✮
▲❡♠❜r❛♠♦s q✉❡ ❛ ár❡❛ SABC ❞♦ tr✐â♥❣✉❧♦ △ABC t❛♠❜é♠ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ♣❡❧❛
❡①♣r❡ssã♦ ❛❜❛✐①♦✱ q✉❡ ♥♦s s❡rá út✐❧ ♥❛ ♣ró①✐♠❛ s❡çã♦✿
SABC =
s❡♥ Bb·a·c
2 , ✭✷✳✷✮
❡♠ q✉❡ a = AB✱ c = BC ❡ Bb é ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ✐♥t❡r♥♦ ∠ABC ❞♦ tr✐â♥❣✉❧♦
△ABC✳
❈❆P❮❚❯▲❖ ✷✳ ❈❆❘❆❈❚❊❘■❩❆➬Õ❊❙ ❋❯◆❉❆▼❊◆❚❆■❙ ✶✼
✷✳✶ ❈❛r❛❝t❡r✐③❛çõ❡s ❞♦s tr❛♣é③✐♦s ❡♥✈♦❧✈❡♥❞♦ tr✐❣♦♥♦✲
♠❡tr✐❛
❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ♥♦s ❞á ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ tr❛♣é③✐♦s ❛ ♣❛rt✐r ❞❛ s♦♠❛ ❞♦s ❝♦ss❡♥♦s ❞❡ â♥❣✉❧♦s ❛❞❥❛❝❡♥t❡s ❞♦ q✉❛❞r✐❧át❡r♦✳
❚❡♦r❡♠❛ ✷✳✶✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD é ✉♠ tr❛♣é③✐♦ ❞❡ ❧❛❞♦s ♣❛r❛❧❡❧♦s AB ❡ CD s❡✱ ❡ s♦♠❡♥t❡ s❡✿
❝♦s Ab+❝♦s Db = 0 e ❝♦s Bb+❝♦s Cb= 0. ✭✷✳✸✮
❋✐❣✉r❛ ✷✳✶✿ ➶♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❝♦♥❣r✉❡♥t❡s✳
❉❡♠♦♥str❛çã♦✳
❙✉♣♦♥❤❛ q✉❡ ♦ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ABCD s❡❥❛ ✉♠ tr❛♣é③✐♦ ❝♦♠ AB kCD✳ ❆ss✐♠✱
❞♦ ♣❛r❛❧❡❧✐s♠♦ ❡♥tr❡ AB ❡ CD✱ s❡❣✉❡ q✉❡ Ab❡ Db sã♦ s✉♣❧❡♠❡♥t❛r❡s✱ ❝♦♥❢♦r♠❡ ✐❧✉str❛❞♦
♥❛ ✜❣✉r❛ ✭✷✳✶✮✳ ▲♦❣♦✿
❝♦s Ab+❝♦s Db =❝♦s Ab+❝♦s(180◦−Ab) = ❝♦sAb−❝♦s Ab= 0.
❆♥❛❧♦❣❛♠❡♥t❡✱ ❝♦s Bb+❝♦sCb = 0.
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ ♦s â♥❣✉❧♦sA,b B,b Cb❡Db ❞♦ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ABCD
s❛t✐s❢❛ç❛♠ ❛ r❡❧❛çã♦ ✭✷✳✸✮✳
❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ABCD♥ã♦ s❡❥❛ ✉♠ tr❛♣é③✐♦ ❞❡ ❧❛❞♦s ♣❛r❛❧❡❧♦sAB ❡CD✱
❝♦♥❢♦r♠❡ ✜❣✉r❛ ✭✷✳✷✮✳ ❉❡st❛ ❢♦r♠❛✱ Ab+Db 6= 180◦ ❡ Bb +Cb 6= 180◦✳ ❙❡♠ ♣❡r❞❛ ❞❡
❈❆P❮❚❯▲❖ ✷✳ ❈❆❘❆❈❚❊❘■❩❆➬Õ❊❙ ❋❯◆❉❆▼❊◆❚❆■❙ ✶✽
❋✐❣✉r❛ ✷✳✷✿ ❚r❛♣é③✐♦ ❝♦♠ ❧❛❞♦sAB ❡CD ♥ã♦ ♣❛r❛❧❡❧♦s✳
❈♦♠♦ABCDé ✉♠ q✉❛❞r✐át❡r♦ ❝♦♥✈❡①♦✱ ❡♥tã♦0◦ <A,b D <b 180◦ ❡ t❡♠♦s q✉❡ ❛ ❢✉♥çã♦
f(x) = ❝♦sxé ❞❡❝r❡s❝❡♥t❡ ❡♠(0◦,180◦)✳ ▲♦❣♦✱ ❝♦sAb+❝♦sDb =❝♦sAb−❝♦s(180−Db)<0✱
♦ q✉❡ ❝♦♥tr❛❞✐③ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ ❝♦s Ab+❝♦sDb = 0✳
P♦rt❛♥t♦✱ ABCD é ✉♠ tr❛♣é③✐♦ ❞❡ ❧❛❞♦s ♣❛r❛❧❡❧♦sAB ❡ CD✳
❆ s❡❣✉✐♥t❡ Pr♦♣♦s✐çã♦ ❡st❛❜❡❧❡❝❡ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ♦s â♥❣✉❧♦s ❞❡ ✉♠ tr❛♣é③✐♦ ✉t✐❧✐③❛♥❞♦✲ s❡ ❛ ❢✉♥çã♦ ❝♦t❛♥❣❡♥t❡✳
Pr♦♣♦s✐çã♦ ✷✳✶✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCDé ✉♠ tr❛♣é③✐♦ ❝♦♠ ❧❛❞♦s ♣❛r❛❧❡❧♦s AB
❡ CD s❡✱ ❡ s♦♠❡♥t❡ s❡✿
❝♦t Ab+❝♦t Db =❝♦t Bb+❝♦t Cb = 0.
❆ ❞❡♠♦♥str❛çã♦ é ❛♥á❧♦❣❛ à ❛♥t❡r✐♦r✱ ❥á q✉❡ ❛ ❢✉♥çã♦ ❝♦t❛♥❣❡♥t❡ é ❞❡s❝r❡s❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ (0◦,180◦) ❡ ❝♦t ① ❂ ✲❝♦t ✭✶✽✵◦−①).
❚❡♦r❡♠❛ ✷✳✷✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD é ✉♠ tr❛♣é③✐♦ ❝♦♠ ❧❛❞♦s ♣❛r❛❧❡❧♦s AB ❡ CD s❡✱ ❡ s♦♠❡♥t❡ s❡✿
t❛♥ (Ab
2)·t❛♥ (
b
D
2) =t❛♥ (
b
B
2)·t❛♥ (
b
C 2) = 1.
❉❡♠♦♥str❛çã♦✳
❈♦♠♦ Ab+Db = 180◦✱ ❝♦♥❢♦r♠❡ ❛ r❡❧❛çã♦ ✭✶✳✸✮✱ ♣á❣✐♥❛ ✶✸✳ ❊♥tã♦ Ab 2 +
b
D
2 = 90
◦✱ ♦✉
s❡❥❛✱ Ab
2 ❡
b
D
❈❆P❮❚❯▲❖ ✷✳ ❈❆❘❆❈❚❊❘■❩❆➬Õ❊❙ ❋❯◆❉❆▼❊◆❚❆■❙ ✶✾
t❛♥ (Ab 2) =
1
❝♦t(A2b)
= 1
❝♦t(90◦− Db
2)
= 1
t❛♥(D2b)
=❝♦t (Db 2)✳
❡
t❛♥ (Db 2) =
1
❝♦t(Db2)
= 1
❝♦t(90◦− Ab
2)
= 1
t❛♥(A2b)
=❝♦t (Ab 2)✳
❆❧é♠ ❞✐ss♦✱ ♣♦r ❞❡✜♥✐çã♦ ❞❡ ❝♦t❛♥❣❡♥t❡✱ t❡♠♦s q✉❡✿ t❛♥ (Ab
2)·❝♦t (
b
A 2) = 1✳
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱
t❛♥ (Ab
2)·t❛♥ (
b
D 2) = 1✳
❆♥❛❧♦❣❛♠❡♥t❡✱ t❡♠♦s✿
t❛♥ (Bb
2)·t❛♥ (
b
C 2) = 1.
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ ♦ q✉❛❞r✐❧át❡tr♦ ABCD ♥ã♦ s❡❥❛ ✉♠ tr❛✲
♣é③✐♦ ❞❡ ❜❛s❡s ♣❛r❛❧❡❧❛sAB ❡CD✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ s✉♣♦♥❤❛ q✉❡Ab+D >b 180◦
❡ Bb+C <b 180◦✳ ❆ss✐♠✱ (Ab
2) + (
b
D
2)>90◦✳ ▲♦❣♦ t❛♥ (
b
A 2 +
b
D 2✮❁✵.
❈❛❧❝✉❧❛♥❞♦✲s❡ t❛♥ (A 2 +
b
D
2)<0✱ t❡♠♦s✿
0>t❛♥ (A2b+
b
D 2) =
t❛♥ (Ab
2) +t❛♥ (
b
D 2)
1−t❛♥ (Ab
2)·t❛♥ (
b
D 2)
. ✭✷✳✹✮
❈♦♠♦ ♦s â♥❣✉❧♦s Ab
2 ❡
b
D
2 sã♦ ❛❣✉❞♦s✱ ❡♥tã♦ ♦ ♥✉♠❡r❛❞♦r ❞❡ ✭✷✳✹✮ é ♣♦s✐t✐✈♦✳ ▲♦❣♦
❞❡✈❡♠♦s t❡r ♦ ❞❡♥♦♠✐♥❛❞♦r ❞❡ ✭✷✳✹✮ ♥❡❣❛t✐✈♦✳ ❖✉ s❡❥❛ t❛♥ Ab
2 ·t❛♥
b
D 2 >1✳
P♦r ❤✐♣ót❡s❡✱Bb+C <b 180◦✳ ❊♥tã♦✱ Bb
2 +
b
C
2 <90◦✱ ❧♦❣♦ t❛♥(
b
B 2 +
b
C
2)>0✳ ❈❛❧❝✉❧❛♥❞♦✲s❡
t❛♥ (Bb2 +
b
C
2)✱ t❡♠♦s✿
0<t❛♥ (Bb 2 +
b
C 2) =
t❛♥ (Bb
2) +t❛♥ (
b
C 2)
1−t❛♥ (Bb
2)·t❛♥ (
b
C 2)
. ✭✷✳✺✮
❈♦♠♦ ♦s â♥❣✉❧♦s Bb
2 ❡
b
C
2 sã♦ ❛❣✉❞♦s✱ ❡♥tã♦ ♦ ♥✉♠❡r❛❞♦r ❞❡ ✭✷✳✺✮ é ♣♦s✐t✐✈♦✳ ▲♦❣♦
❞❡✈❡♠♦s t❡r ♦ ❞❡♥♦♠✐♥❛❞♦r ❞❡ ✭✷✳✺✮ t❛♠❜é♠ ♣♦s✐t✐✈♦✱ ♦ q✉❡ ♦❝♦rr❡ q✉❛♥❞♦ t❛♥ (Bb 2)·
❈❆P❮❚❯▲❖ ✷✳ ❈❆❘❆❈❚❊❘■❩❆➬Õ❊❙ ❋❯◆❉❆▼❊◆❚❆■❙ ✷✵ ❆ss✐♠✱ t❛♥ (Ab
2)·t❛♥ (
b
D
2) 6= t❛♥ (
b
B
2)·t❛♥ (
b
C
2)✱ ♦ q✉❡ ❝♦♥tr❛r✐❛ ❛ ❤✐♣ót❡s❡✳ P♦rt❛♥t♦✱
ABCD é ✉♠ tr❛♣é③✐♦ ❞❡ ❧❛❞♦s ♣❛r❛❧❡❧♦sAB ❡CD✳
✷✳✷ ❈❛r❛❝t❡r✐③❛çõ❡s ❞♦s tr❛♣é③✐♦s ❡♥✈♦❧✈❡♥❞♦ ár❡❛s
◆❡st❛ s❡çã♦✱ ♦❜t❡r❡♠♦s q✉❛tr♦ ❝❛r❛❝t❡r✐③❛çõ❡s ❞❡ tr❛♣é③✐♦s ❛ ♣❛rt✐r ❞♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s✳ ◆✉♠❛ ❞❡❧❛s s❡rá ✉t✐❧✐③❛❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❜✐♠❡❞✐❛♥❛✱ ❞❡✜♥✐❞♦ ❛ s❡❣✉✐r✿
❉❡✜♥✐çã♦ ✷✳✶✳ ❉❛❞♦ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD✱ ✉♠❛ ❜✐♠❡❞✐❛♥❛ é ✉♠ s❡❣♠❡♥t♦
❞❡ r❡t❛ ❝♦♠ ❡①tr❡♠✐❞❛❞❡s ♥♦s ♣♦♥t♦s ♠é❞✐♦s ❞❡ ❞♦✐s ❧❛❞♦s ♦♣♦st♦s ❞❡ss❡ q✉❛❞r✐❧át❡r♦✳ Pr♦♣♦s✐çã♦ ✷✳✷✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD é ✉♠ tr❛♣é③✐♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✉♠❛
❜✐♠❡❞✐❛♥❛ ♦ ❞✐✈✐❞❡ ❡♠ ❞♦✐s q✉❛❞r✐❧át❡r♦s ❞❡ ♠❡s♠❛ ár❡❛ ✭✜❣✳ ✭✷✳✸✮✮✳
❋✐❣✉r❛ ✷✳✸✿ ❇✐♠❡❞✐❛♥❛ ❞❡ ✉♠ tr❛♣é③✐♦✳
❉❡♠♦♥str❛çã♦✳
❈♦♥s✐❞❡r❡♠♦s q✉❡ ♦ tr❛♣é③✐♦ ABCD t❡♥❤❛ ❧❛❞♦s ♣❛r❛❧❡❧♦s AB ❡ CD ❡ s❡❥❛♠ M1 ❡ M2
♦s ♣♦♥t♦s ♠é❞✐♦s ❞❡ AB ❡ CD✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ ❜✐♠❡❞✐❛♥❛ M1M2 ❞✐✈✐❞❡ ♦ tr❛♣é③✐♦
ABCD ✭✜❣✳ ✭✷✳✸✮✮ ❡♠ ❞♦✐s tr❛♣é③✐♦s ❞❡ ❜❛s❡s ❞❡ ♠❡s♠❛ ♠❡❞✐❞❛ ❡ ♠❡s♠❛ ❛❧t✉r❛ h✳
❈❛❧❝✉❧❛♥❞♦✲s❡ ❛s ár❡❛sS1 ❡S2 ❞♦s tr❛♣é③✐♦s AM1M2D ❡M1BCM2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱
t❡♠♦s q✉❡
S1 =
AM1+DM2
2 ·h=
BM1+CM2
❈❆P❮❚❯▲❖ ✷✳ ❈❆❘❆❈❚❊❘■❩❆➬Õ❊❙ ❋❯◆❉❆▼❊◆❚❆■❙ ✷✶ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛♠ABCD✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦✱M1❡M2 ♦s ♣♦♥t♦s ♠é❞✐♦s ❞❡
AB ❡ CD✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡h ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ M1 ❡ M2✳ ❉❡♥♦t❡♠♦s ♣♦r S1 =SADM1✱
S2 =SDM1M2✱S3 =SCM1M2 ❡ S4 =SBCM1✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✭✷✳✹✮✳
❖❜s❡r✈❡♠♦s q✉❡ S2 = S3✱ ♣♦✐s ♦s tr✐â♥❣✉❧♦s △DM1M2 ❡ △CM1M2 tê♠ ❜❛s❡s ❞❡
♠❡s♠❛ ♠❡❞✐❞❛ ❡ tê♠ ❛ ♠❡s♠❛ ❛❧t✉r❛✳ ▼❛s✱ ♣♦r ❤✐♣ót❡s❡✱
S1+S2 =S3+S4.
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ S1 = S4✳ ▼❛s✱ ❛s ❜❛s❡s ❞♦s tr✐â♥❣✉❧♦s △AM1D ❡ △BM1C tê♠
❛ ♠❡s♠❛ ♠❡❞✐❞❛✳ ❆ss✐♠✱ ❡st❡s tr✐â♥❣✉❧♦s tê♠ ❛ ♠❡s♠❛ ❛❧t✉r❛✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞♦s ♣♦♥t♦s C ❡D à r❡t❛ ←→AB ❝♦✐♥❝✐❞❡♠✳ P♦rt❛♥t♦✱ CD kAB ❡ ABCD é ✉♠ tr❛♣é③✐♦✳
❋✐❣✉r❛ ✷✳✹✿ S1+S2 =S3+S4✳
❖❜s❡r✈❛çã♦ ✷✳✶✳ ❆ Pr♦♣♦s✐çã♦ ✭✷✳✷✮✱ ♣á❣✐♥❛ ✷✵✱ ❛♣❧✐❝❛✲s❡ ❛♦ ❝❛s♦ ❞❛ ❜✐♠❡❞✐❛♥❛ ❝♦♠ ❡①tr❡♠✐❞❛❞❡s ♥♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s ♣❛r❛❧❡❧♦s ❞♦ tr❛♣é③✐♦✳ P♦r ❡①❡♠♣❧♦✱ s❡ M1 ❡
M2 ❢♦r❡♠ ♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s AD ❡ BC✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞♦ tr❛♣é③✐♦ ABCD✱
t❛✐s q✉❡ AD ∦ BC✳ ❊♥tã♦ M1M2 =
AB+CD
2 ✱ ♣♦✐s ❛ ❜✐♠❡❞✐❛♥❛ M1M2 é ❛ ❜❛s❡ ♠é❞✐❛ ❞♦
tr❛♣é③✐♦ABCD✱ ❞❡ ♠♦❞♦ q✉❡ABM2M1 ❡ M1M2CD s❡❥❛♠ tr❛♣é③✐♦s✳ ❆ss✐♠✱ ❞❡♥♦t❛♥❞♦✲
s❡ ♣♦r S1 ❡ S2 ❛s ár❡❛s ❞♦s tr❛♣é③✐♦s ABM2M1 ❡ M1M2CD✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r h ❛
❛❧t✉r❛ ❞♦ tr❛♣é③✐♦ ABCD✱ a=AB✱ b=CD✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✭✷✳✺✮✱ t❡♠♦s q✉❡✿
M1M2 =
a+b
2 ❡ S1 =
(a+ 3b)·h
8 6=
(3a+b)·h
8 =S2✳
Pr♦♣♦s✐çã♦ ✷✳✸✳ ❙❡ ❛s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD s❡ ✐♥t❡rs❡❝t❛♠ ❡♠ P✱ ❡♥tã♦ ABCD é ✉♠ tr❛♣é③✐♦ ❞❡ ❧❛❞♦s ♣❛r❛❧❡❧♦s AB ❡ CD s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛s ár❡❛s
❈❆P❮❚❯▲❖ ✷✳ ❈❆❘❆❈❚❊❘■❩❆➬Õ❊❙ ❋❯◆❉❆▼❊◆❚❆■❙ ✷✷
❋✐❣✉r❛ ✷✳✺✿ ❚r❛♣é③✐♦ ❝♦♠ ❜✐♠❡❞✐❛♥❛ ♣❛r❛❧❡❧❛ às ❜❛s❡s✳
❉❡♠♦♥str❛çã♦✳
❉❡♥♦t❡♠♦s ♣♦r hX Y Z ❛ ❛❧t✉r❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❞❡ ✈ért✐❝❡s X✱ Y ❡Z✳
SAP D =SBP C ⇔SAP D+SABP =SBP C +SABD ⇔SABD =SABC ⇔hABD =hABC,
✈✐st♦ q✉❡ ♦s tr✐â♥❣✉❧♦s △ABD ❡ △ABC tê♠ ❛ ♠❡s♠❛ ❜❛s❡ AB✳
▼❛s✱ ❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱AB kCD✱ q✉❡✱ ♣♦r s✉❛ ✈❡③✱ ♦❝♦rr❡
s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD ❢♦r ✉♠ tr❛♣é③✐♦ ❝♦♠ ❧❛❞♦s ♣❛r❛❧❡❧♦sAB
❡ CD✳
❚❡♦r❡♠❛ ✷✳✸✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD é ✉♠ tr❛♣é③✐♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ♣r♦✲
❞✉t♦s ❞❛s ár❡❛s ❞♦s tr✐â♥❣✉❧♦s ❢♦r♠❛❞♦s ♣♦r ✉♠❛ ❞✐❛❣♦♥❛❧ ❢♦r ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞❛s ár❡❛s ❞♦s tr✐â♥❣✉❧♦s ❢♦r♠❛❞♦s ♣❡❧❛ ♦✉tr❛ ❞✐❛❣♦♥❛❧ ✭✜❣✳ ✭✷✳✻✮✱ ♣á❣✳ ✷✸✮✳
❉❡♠♦♥str❛çã♦✳
❙❡❥❛P ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛s ❞✐❛❣♦♥❛✐sAC❡BD❞♦ q✉❛❞r✐❧át❡r♦ABCD✳ ❉❡♥♦t❡♠♦s
♣♦r S1 =SAP D✱S2 =SCDP✱S3 =SBCP ❡ S4 =SABP✳ ❖❜s❡r✈❡♠♦s q✉❡ SABD =S1+S4✱
SBCD =S2+S3✱ SACD =S1+S2 ❡SABC =S3+S4✳ ❆ss✐♠✱
(S1+S4)·(S2+S3) = (S1+S2)·(S3+S4)⇔S1 ·S2+S3·S4 =S1·S4+S2·S3 ⇔ (S1−S3)·(S2−S4) = 0⇔S1 =S3 ou S2 =S4.
❈❆P❮❚❯▲❖ ✷✳ ❈❆❘❆❈❚❊❘■❩❆➬Õ❊❙ ❋❯◆❉❆▼❊◆❚❆■❙ ✷✸
❋✐❣✉r❛ ✷✳✻✿ ◗✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ❞✐✈✐❞✐❞♦ ❡♠ q✉❛tr♦ tr✐â♥❣✉❧♦s ♣❡❧❛s s✉❛s ❞✐❛❣♦♥❛✐s✳
▲❡♠❛ ✷✳✶✳ ❆s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ♦ ❞✐✈✐❞❡ ❡♠ q✉❛tr♦ tr✐â♥❣✉❧♦s ♣❛r❛ ♦s q✉❛✐s ♦ ♣r♦❞✉t♦ ❞❛s ár❡❛s ❞❡ ❞♦✐s tr✐â♥❣✉❧♦s ♦♣♦st♦s é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞❛s ár❡❛s ❞♦s ♦✉tr♦s ❞♦✐s tr✐â♥❣✉❧♦s✳
❉❡♠♦♥str❛çã♦✳
❈♦♥s✐❞❡r❡ ♦ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD✳ ❙❡❥❛♠ {P} = AC ∩BD✱ S1 = SAP D✱ S2 =
SCDP✱ S3 = SBCP✱ ❙4 = SABP✱ θ ♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ❡♥tr❡ DP ❡ AP✱ AP = p✱ DP = r✱
CP =q ❡BP =s ✭❝♦♠♦ ♥❛ ✜❣✉r❛ ✭✷✳✼✮✮✳ ❆ss✐♠✿
S1·S3 =
p·r·s❡♥ θ
2 ·
q·s·s❡♥ θ
2 =
p·q·r·s·s❡♥2θ
4 ,
S2·S4 =
q·r·s❡♥ (180◦−θ)
2 ·
p·s·s❡♥ (180◦−θ)
2 =
p·q·r·s·s❡♥2
θ
4 .
▲♦❣♦✱
S1·S3 =S2·S4✳
❖✉tr❛ ♠❛♥❡✐r❛ ♣❛r❛ s❡ ❞❡♠♦♥str❛r ❡st❡ ❧❡♠❛ s❡♠ ♦ ❛✉①í❧✐♦ ❞❛ tr✐❣♦♥♦♠❡tr✐❛ é ✉s❛r ❛s ❛❧t✉r❛s ❞♦s tr✐â♥❣✉❧♦s r❡❧❛t✐✈♦s ❛ ✉♠❛ ú♥✐❝❛ ❞✐❛❣♦♥❛❧✳ ❆ss✐♠✱ s❡ h1 é ❛ ❛❧t✉r❛ ❞♦s
tr✐â♥❣✉❧♦s △ADP ❡ △ABP r❡❧❛t✐✈❛ à ❞✐❛❣♦♥❛❧ BD ❡ s❡ h2 é ❛ ❛❧t✉r❛ ❞♦s tr✐â♥❣✉❧♦s
❈❆P❮❚❯▲❖ ✷✳ ❈❆❘❆❈❚❊❘■❩❆➬Õ❊❙ ❋❯◆❉❆▼❊◆❚❆■❙ ✷✹
❋✐❣✉r❛ ✷✳✼✿ Pr♦❞✉t♦s ❞♦s tr✐â♥❣✉❧♦s ♦♣♦st♦s ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ sã♦ ✐❣✉❛✐s✳
S1·S3 =
r·h1 2 ·
s·h2 2 =
r·s·h1·h2
4 ,
S2·S4 =
r·h2 2 ·
s·h1 2 =
r·s·h1·h2
4 .
❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ é ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ tr❛♣é③✐♦s ❡♥✈♦❧✈❡♥❞♦ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ár❡❛s ❞❡ ❞♦✐s tr✐â♥❣✉❧♦s ♦♣♦st♦s✱ ♦❜t✐❞♦s ❛ ♣❛rt✐r ❞❛s ❞✐❛❣♦♥❛✐s ❞♦ tr❛♣é③✐♦✳
❚❡♦r❡♠❛ ✷✳✹✳ ❈♦♠ ❛s ♥♦t❛çõ❡s ✐♥tr♦❞✉③✐❞❛s ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✭✷✳✶✮✱ ♣á❣✐♥❛ ✷✸✱ ♦ q✉❛❞r✐❧át❡r♦ t❡♠ ár❡❛ K = (√S2+
√
S4)2 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❡ ❢♦r ✉♠ tr❛♣é③✐♦ ❝✉❥♦s
❧❛❞♦s ♣❛r❛❧❡❧♦s sã♦ ♦s ❧❛❞♦s ❞♦s tr✐â♥❣✉❧♦s ❝✉❥❛s ár❡❛s sã♦ S2 ❡ S4✳
❉❡♠♦♥str❛çã♦✳
❖ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ✭✜❣✳ ✭✷✳✼✮✱ ♣á❣✳ ✷✹✮ t❡♠ ár❡❛ K =S1+S2+S3+S4✳
P❡❧♦ ▲❡♠❛ ✭✷✳✶✮✱ t❡♠♦s S1·S3 =S2·S4✳ ❊♥tã♦✿
K =S1+S2 +S3+S4+ 2
√
S2·S4−2
√
S1·S3,
K = (√S2+
√
S4)2
+ (√S1−
√
S3)2
.
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ Pr♦♣♦s✐çã♦ ✭✷✳✸✮✱ ♣á❣✐♥❛ ✷✶✱ ♦ q✉❛❞r✐❧át❡r♦ é ✉♠ tr❛♣é③✐♦ ❝✉❥♦s ❧❛❞♦s ♣❛r❛❧❡❧♦s sã♦ ❧❛❞♦s ❞♦s tr✐â♥❣✉❧♦s ❝♦♠ ár❡❛sS2 ❡S4 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ S1 =S3✳ ■st♦ ♦❝♦rr❡
s❡✱ ❡ s♦♠❡♥t❡ s❡✱K = (√S2+
√
❈❆P❮❚❯▲❖ ✷✳ ❈❆❘❆❈❚❊❘■❩❆➬Õ❊❙ ❋❯◆❉❆▼❊◆❚❆■❙ ✷✺
✷✳✸ ❈❛r❛❝t❡r✐③❛çõ❡s q✉❡ ❡♥✈♦❧✈❡♠ ❧❛❞♦s ❡ ❞✐stâ♥❝✐❛s
◆❡st❛ ú❧t✐♠❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❝❛r❛❝t❡r✐③❛çõ❡s ❞❡ tr❛♣é③✐♦s ❡♥✈♦❧✈❡♥❞♦ ❜❛s✐✲ ❝❛♠❡♥t❡ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞❡ s❡✉s ❧❛❞♦s ❡ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞❡ s✉❛s ❞✐❛❣♦♥❛✐s✳
❚❡♦r❡♠❛ ✷✳✺✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD é ✉♠ tr❛♣é③✐♦ ❝♦♠ ❧❛❞♦s ♣❛r❛❧❡❧♦s AB ❡ CD s❡✱ ❡ s♦♠❡♥t❡ s❡✿
AD BC =
s❡♥ Cb
s❡♥ Db. ✭✷✳✻✮
❉❡♠♦♥str❛çã♦✳
❖ q✉❛❞r✐❧át❡r♦ ABCD é ✉♠ tr❛♣é③✐♦ ❝♦♠ ❧❛❞♦s ♣❛r❛❧❡❧♦s AB ❡ CD s❡✱ ❡ s♦♠❡♥t❡ s❡✱
♦s tr✐â♥❣✉❧♦s △ACD ❡ △BCD ✭✜❣✳ ✭✶✳✶✶✮✱ ♣á❣✳ ✶✹✮ tê♠ ♠❡s♠❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛♦ ❧❛❞♦ CD✳ ■st♦ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ SACD =SBCD✳ ❖✉ s❡❥❛✱
1
2·CD·AD·s❡♥ Db = 1
2 ·CD·BC·s❡♥ Cb✳
❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱
AD BC =
s❡♥ Cb
s❡♥ Db. ✭✷✳✼✮
▲❡♠❛ ✷✳✷✳ P❛r❛ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD ❝♦♠ AB = a✱ BC = b✱ CD = c✱ DA=d✱ ❡ ❞✐❛❣♦♥❛✐s ♠❡❞✐♥❞♦ p=AC ❡ q=BD✱ ✈❛❧❡ q✉❡✿
p2+q2 =b2+d2+ 2ac·❝♦s θ, ✭✷✳✽✮
♦♥❞❡ θ é ♦ â♥❣✉❧♦ ❡♥tr❡ ←→AB ❡ ←→CD✳
❉❡♠♦♥str❛çã♦✳
❙❡❥❛ θ ♦ â♥❣✉❧♦ ❡♥tr❡ ♦s ♣r♦❧♦♥❣❛♠❡♥t♦s ❞❡ AB ❡ CD✳ ❙❡ θ = 0✱ ❡♥tã♦ AB k CD✱
❞❡ ♠♦❞♦ q✉❡ ♦ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD é ✉♠ tr❛♣é③✐♦✳ ❈❛s♦ θ6= 0✱ ❡①✐st❡ ✉♠ ♣♦♥t♦
J q✉❡ é ❛ ✐♥t❡rs❡❝çã♦ ❞❡ ←→AB ❝♦♠ ←→CD✳
❙❡❥❛ ● ♦ ♣♦♥t♦ t❛❧ q✉❡ CG é ♣❛r❛❧❡❧♦ ❛ AB✱ CG é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ DG ❡ s❡❥❛ E ♦
❈❆P❮❚❯▲❖ ✷✳ ❈❆❘❆❈❚❊❘■❩❆➬Õ❊❙ ❋❯◆❉❆▼❊◆❚❆■❙ ✷✻ ❈♦♠♦ ∠BJ C ❡ ∠DCG sã♦ â♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s ✭♥♦s ❝❛s♦s ✶ ❡ ✹✮ ♦✉ â♥❣✉❧♦s
❝♦rr❡s♣♦♥❞❡♥t❡s ✭♥♦s ❝❛s♦s ✷ ❡ ✸✮ ❢♦r♠❛❞♦s ❡♥tr❡ ❛ tr❛♥s✈❡rs❛❧ ←J C→ ❝♦♠ ❛s ♣❛r❛❧❡❧❛s ←→AB
❡ ←→CG✱ ❡♥tã♦ BJ Cb =DCGb =θ✳
❙❡❥❛ F ∈ ←→AB t❛❧ q✉❡ CF é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ←→AB✳ ❊♥tã♦ GEF C é ✉♠ r❡tâ♥❣✉❧♦✱ ❞❡
♠♦❞♦ q✉❡ EF =GC =c·❝♦s θ✱ DG=c·s❡♥ θ✳
▲♦❣♦✱ s❡♥❞♦ h = CF ❡ x = AE✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s ♣♦ss✐❜✐❧✐❞❛❞❡s ❡ s✉❛s r❡s♣❡❝t✐✈❛s
r❡♣r❡s❡♥t❛çõ❡s ♥❛s ✜❣✉r❛s ✭✷✳✽✮✱ ✭✷✳✾✮✱ ✭✷✳✶✵✮ ❡ ✭✷✳✶✶✮✿
• ED=h−c·s❡♥ θ✱ F B =a−c·❝♦s θ−x✱ ♣❛r❛ ♦ ❈❛s♦ ✶ ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ✭✷✳✽✮✱
❋✐❣✉r❛ ✷✳✽✿ ❈❛s♦ ✶✳
• ED=h+c·s❡♥ θ✱ F B =a−c·❝♦s θ−x✱ ♣❛r❛ ♦ ❈❛s♦ ✷ ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ✭✷✳✾✮✱
