❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙❊❘●■P❊
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❚❊❈◆❖▲❖●■❆
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲
❚r❛♥s❢♦r♠❛çõ❡s ●❡♦♠étr✐❝❛s ♥♦ P❧❛♥♦
P❛✉❧♦ ❆r❛ú❥♦ ❞❛ ❙✐❧✈❛
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙❊❘●■P❊
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❚❊❈◆❖▲❖●■❆
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲
P❛✉❧♦ ❆r❛ú❥♦ ❞❛ ❙✐❧✈❛
❚r❛♥s❢♦r♠❛çõ❡s ●❡♦♠étr✐❝❛s ♥♦ P❧❛♥♦
❚r❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❡r❣✐♣❡ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✭P❘❖❋▼❆❚✮✳
❖❘■❊◆❚❆❉❖❘✿ Pr♦❢✳ ❉r✳ ❏✳ ❆♥❞❡rs♦♥ ❱❛❧❡♥ç❛ ❈❛r❞♦s♦
❈❖✲❖❘■❊◆❚❆❉❖❘✿ Pr♦❢✳ ❉r✳ ❘✐❝❛r❞♦ P✐♥❤❡✐r♦ ❞❛ ❈♦st❛
❊st❡ ❡①❡♠♣❧❛r ❝♦rr❡s♣♦♥❞❡ à ✈❡rsã♦ ✜♥❛❧ ❞❛ ❞✐ss❡rt❛çã♦ ❞❡❢❡♥❞✐❞❛ ♣❡❧♦ ❛❧✉♥♦ P❛✉❧♦ ❆r❛ú❥♦ ❞❛ ❙✐❧✈❛✱ ♦r✐❡♥t❛❞❛ ♣❡❧♦ Pr♦❢✳ ❉r✳ ❏♦sé ❆♥❞❡rs♦♥ ❱❛❧❡♥ç❛ ❈❛r❞♦s♦ ❡ ♦r✐❡♥t❛❞❛ ♣❡❧♦ Pr♦❢✳ ❉r✳ ❘✐❝❛r❞♦ P✐♥❤❡✐r♦ ❞❛ ❈♦st❛✳
FICHA CATALOGHÁFICA ELABOHADA PELA BIBLIOTECA CENTHAL
UNIVEHSIDADE FEDEHAL DE SEHGIPE
Silva, Paulo Araújo da
S586t Transformações geométricas no plano / Paulo Araújo da Silva ; orientador José Anderson Valença Cardoso. – São Cristóvão, 2016. 68 f. ; il.
Dissertação (mestrado Profissional em Matemática) – Universidade Federal de Sergipe, 2016.
O
1 Transformação geométrica. 2. Homotetias. 3. Cisalhamento. 4. Isometria. 5. Morfismos. 6. Combinação linear. 7. Convexa. I. Cardoso, José Anderson Valença, orient. II. Título
❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛♦s ♠❡✉s ♣❛✐s ✭✐♥ ♠❡♠♦r✐❛♠✮✱ ♣❡ss♦❛s q✉❡ ♥ã♦ t✐✈❡r❛♠ ❡st✉❞♦✱ ♠❛s ✜③❡r❛♠ ❞❡ t✉❞♦ ♣❛r❛ q✉❡ s❡✉s ✜❧❤♦s t✐✈❡ss❡♠✱ à ♠✐♥❤❛ ❡s♣♦s❛ ●✐❧✈❛♥❡✐❞❡ ●♦♠❡s ▲✐♠❛ ❆r❛ú❥♦ ❡ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ✜❧❤♦s ▲❡♦♥❛r❞♦ ▲✐♠❛ ❆r❛ú❥♦ ❡ P❛✉❧❛ ▲❛r✐ss❛ ▲✐♠❛ ❆r❛ú❥♦✱ ♣❛r❛ q✉❡ ❡❧❡s t♦♠❡♠ ❝♦♠♦ ❡①❡♠♣❧♦ ❞❡ ❞❡❞✐❝❛çã♦✱ ❢♦rç❛ ❞❡ ✈♦♥t❛❞❡ ❡ ❛♠♦r ❛ ♣r♦✜ssã♦ ❞❡ ♣r♦❢❡ss♦r q✉❡ ❛❥✉❞❛ ❛ ❢♦r♠❛r t♦❞❛s ❛s ♦✉tr❛s ♣r♦✜ssõ❡s✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛ ❉❡✉s✱ ♦ ●r❛♥❞❡ ❆rq✉✐t❡t♦ ❞♦ ❯♥✐✈❡rs♦✱ q✉❡ ♠❡ ❞❡✉ ❙❛ú❞❡✱ ❙❛❜❡❞♦r✐❛ ❡ ❙❡❣✉r❛♥ç❛ ♣❛r❛ q✉❡ ❝♦♥s❡❣✉✐ss❡ ❝♦♥❝❧✉✐r ❡st❡ tr❛❜❛❧❤♦✳ ❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ❏♦sé ❆♥❞❡rs♦♥ ❱❛❧❡♥ç❛ ❈❛r❞♦s♦ q✉❡ s❡ ❞✐s♣ôs ❛ ♠❡ ♦r✐❡♥t❛r ❡ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢✳ ❉r✳ ❘✐❝❛r❞♦ P✐♥❤❡✐r♦ ❞❛ ❈♦st❛ ♣♦r s✉❛ ❝♦♥tr✐❜✉✐çã♦ ♥♦ ♠❡✉ ❚❈❈✳ ➚ ♠✐♥❤❛ ❡s♣♦s❛ ●✐❧✈❛♥❡✐❞❡ ❡ ♠❡✉s ✜❧❤♦s ▲❡♦♥❛r❞♦ ▲✐♠❛ ❡ P❛✉❧❛ ▲❛r✐ss❛ q✉❡ ❡♥t❡♥❞❡r❛♠ ♠✐♥❤❛ ❛✉sê♥❝✐❛ ♥♦s ✜♥❛✐s ❞❡ s❡♠❛♥❛ ❞✉r❛♥t❡ ❡ss❡s ❞♦✐s ❛♥♦s ❡ ♠❡✐♦✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛ ❙❇▼ ♣❡❧❛ ❜r✐❧❤❛♥t❡ ✐❞é✐❛ ❞❡ ❝r✐❛çã♦ ❞♦ ❝✉rs♦ ❡ ❡♠ ❡s♣❡❝✐❛❧ à ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ q✉❡ ❢♦✐ ❡ss❡♥❝✐❛❧ ♥♦ ❞❡❝♦rr❡r ❞♦ ❝✉rs♦✳
❘❡s✉♠♦
◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢❛③❡♠♦s ✉♠ ❡st✉❞♦ s♦❜r❡ tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ♥♦ ♣❧❛♥♦✱ ❡①♣❧♦r❛♥❞♦ ❝❛r❛❝t❡ríst✐❝❛s ❣❡♦♠étr✐❝❛s ❡ ❛❧❣é❜r✐❝❛s✳ ❆ r❡❧❛çã♦ ❡♥tr❡ ❛ ❣❡♦♠❡tr✐❛ ❡ ❛ á❧❣❡❜r❛ é r❡s♣♦♥sá✈❡❧ ♣♦r ❡①tr❛♦r❞✐♥ár✐♦s ♣r♦❣r❡ss♦s ♥❛ ♠❛t❡♠át✐❝❛ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ ✐♥✐❝✐❛❧ é ❛♣r❡s❡♥t❛r ❛❧❣✉♠❛s ❞❛s ♣r✐♥❝✐♣❛✐s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s✱ ❛ ❡①❡♠♣❧♦ ❞❛s ❍♦♠♦t❡t✐❛s✱ ❞❛s ❚r❛♥s❧❛çõ❡s✱ ❞❡ ❈✐s❛❧❤❛♠❡♥t♦s✱ ❞❛s ❙✐♠❡tr✐❛s✱ ❞❛s ❘♦t❛çõ❡s✱ ❞❛s ❘❡✢❡①õ❡s✱ ❞❛s ■s♦♠❡tr✐❛s✱ ❡t❝✳✱ ❞❡ ❢♦r♠❛ ✐♥t✉✐t✐✈❛ ❡ ✐❧✉str❛♥❞♦ ❝♦♠ ❡①❡♠♣❧♦s s✐♠♣❧❡s✳ ❊♠ s❡❣✉✐❞❛ ❡①♣❧♦r❛♠♦s ❝❛r❛❝t❡ríst✐❝❛s ❛❧❣é❜r✐❝❛s ❡❧❡♠❡♥t❛r❡s q✉❡ ♣❡r♠✐t❡♠ tr❛t❛r ❡ ❣❡♥❡r❛❧✐③❛r ♦ ❡st✉❞♦ ❞❡ tr❛♥s❢♦r♠❛çõ❡s✳ ❆♣r❡s❡♥t❛♠♦s ❛✐♥❞❛ ♦s ❝♦♥❝❡✐t♦s ❞❡ ▼♦r✜s♠♦s ❡ ❉❡❢♦r♠❛çõ❡s ❞❡ ✐♠❛❣❡♥s ✉t✐❧✐③❛♥❞♦ ♥♦çõ❡s✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♠♦ ❈♦♠❜✐♥❛çã♦ ▲✐♥❡❛r ❈♦♥✈❡①❛✳
P❛❧❛✈r❛s ❈❤❛✈❡s✿ ❚r❛♥s❢♦r♠❛çã♦ ●❡♦♠étr✐❝❛✱ ❍♦♠♦t❡t✐❛s✱ ❚r❛♥s❧❛çõ❡s✱ ❘♦t❛çõ❡s✱ ❈✐s❛❧❤❛♠❡♥t♦✱ ■s♦♠❡tr✐❛✱ ▼♦r✜s♠♦s✱ ❉❡❢♦r♠❛çõ❡s✱ ❈♦♠❜✐♥❛çã♦ ▲✐♥❡❛r ❈♦♥✈❡①❛
❙✉♠ár✐♦
❘❡s✉♠♦ ✈✐✐
■♥tr♦❞✉çã♦ ✶
✶ ❍♦♠♦t❡t✐❛s ❡ ❈♦♥tr❛çõ❡s ♥♦ P❧❛♥♦ ✸
✶✳✶ ❈♦♥tr❛çã♦ ❛ ✉♠ P♦♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❈♦♥tr❛çã♦ ❛ ✉♠❛ ❘❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✷ ❱❡t♦r❡s ❡ ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ♥♦ P❧❛♥♦ ✶✸ ✷✳✶ ❈♦♦r❞❡♥❛❞❛s ❡ ❞✐stâ♥❝✐❛ ♥❛ r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ❉✐stâ♥❝✐❛ ❡♥tr❡ ♣♦♥t♦s ♥♦ ♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸ ❱❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸✳✶ ❙❡❣♠❡♥t♦s ❖r✐❡♥t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸✳✷ ❙✐st❡♠❛ ❞❡ ❈♦♦r❞❡♥❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✹ ❚r❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✺ ●❡♦♠❡tr✐❛ ❞❛s ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✻ ❚r❛♥s❢♦r♠❛çã♦ ❞♦ ●❛t♦ ❞❡ ❆r♥♦❧❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✸ ❉❡❢♦r♠❛çõ❡s ❡ ▼♦r✜s♠♦s ●❡♦♠étr✐❝♦s ✺✵
✸✳✶ ❉❡❢♦r♠❛çõ❡s ❡ ▼♦r✜s♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✷ ▼♦r✜s♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✾
■♥tr♦❞✉çã♦
❆s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ❢❛③❡♠ ♣❛rt❡ ❞❛ ❤✐stór✐❛ ❞❛ ❤✉♠❛♥✐❞❛❞❡✱ ❤á ♠❛✐s t❡♠♣♦ ❞♦ q✉❡ s❡ ♣♦ss❛ ✐♠❛❣✐♥❛r✳ ❯♠❛ ❞❛s ♣r✐♠❡✐r❛s ❡✈✐❞ê♥❝✐❛s ❛♣❛r❡❝❡ ♥❛ ♣✐♥t✉r❛ r✉♣❡str❡ ❞♦ sít✐♦ ❞❡ ❊❧ ❇✉❡② ♥❛ ❇♦❧í✈✐❛✳
❋✐❣✉r❛ ✶✿ P✐♥t✉r❛ ❘✉♣❡str❡
◆❡st❡ tr❛❜❛❧❤♦ ✜③❡♠♦s ✉♠ ❡st✉❞♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ♥♦ ♣❧❛♥♦ R2 s♦❜ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❣❡♦♠étr✐❝♦ ❡ ❛❧❣é❜r✐❝♦✱ t❡♥❞♦ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♦❜t❡r ♦ s✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦ ❞❛s ♠❛tr✐③❡s✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ▲✐♠❛[✺]✱ ✏❆ ✐♥t❡r❝♦♥❡①ã♦ ❡♥tr❡ ❣❡♦♠❡tr✐❛ ❡ á❧❣❡❜r❛ r❡s✉❧t❛♥t❡ ❞❡ss❡ ♣♦♥t♦ ❞❡ ✈✐st❛ ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣♦r ❡①tr❛♦r❞✐♥ár✐♦s ♣r♦❣r❡ss♦s ♥❛ ♠❛t❡♠át✐❝❛ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s✑✳
❘❡❢❡r❡♥t❡ ❛♦ ❡♥s✐♥♦ ❞❛s ♠❛tr✐③❡s✱ ❉❛♥t❡ [✹] ❞❡❢❡♥❞❡ q✉❡ ❛s ♠❛tr✐③❡s ❞❡✈❡♠ s❡r ❡①♣❧♦r❛❞❛s ♥ã♦ só ❝♦♠♦ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦✱ ♠❛s ❝♦♠♦ ❝ó❞✐❣♦ ❞❡ ✐♠❛❣❡♥s✱ t❛❜❡❧❛ ❞❡ ❞✉♣❧❛ ❡♥tr❛❞❛✳ P♦r ❡①❡♠♣❧♦✱ ✉♠❛ ✐♠❛❣❡♠ ❞❡ r❡s♦❧✉çã♦ 600×800 t❡♠ 600⋅800 = 480000 ♣✐①❡❧s ❞✐str✐❜✉í❞♦s ❡♠ 600 ❧✐♥❤❛s ❡ 800 ❝♦❧✉♥❛s✳ ◗✉❛♥❞♦ ✉♠
♣r♦❣r❛♠❛ ❣rá✜❝♦ ❛❧t❡r❛ ❛ ♣♦s✐çã♦✱ r❡✢❡t❡✱ r♦t❛❝✐♦♥❛ ♦✉ ♠✉❞❛ ❛ ❡s❝❛❧❛ ❞❛ ✐♠❛❣❡♠✱ ♥❛ ✈❡r❞❛❞❡ ❡stá ♠✉❞❛♥❞♦ ❛ ♣♦s✐çã♦ ❞♦s ♣✐①❡❧s q✉❡ ❛ ❢♦r♠❛♠✳ ■ss♦ t✉❞♦ é ❢❡✐t♦ ♣♦r ♦♣❡r❛çõ❡s ❞❡ ♠❛tr✐③❡s✱ ❡♠ ❝♦♠♣✉t❛çã♦ ❣rá✜❝❛ é ♦ q✉❡ s❡ ❝❤❛♠❛ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ●❡♦♠étr✐❝❛s ♥♦ ♣❧❛♥♦✳ ❇❛s✐❝❛♠❡♥t❡✱ ❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ♥♦ ♣❧❛♥♦ sã♦ q✉❛tr♦✿ r♦t❛çã♦✱ r❡✢❡①ã♦✱ ❡s❝❛❧❛ ❡ tr❛♥s❧❛çã♦❀ ▲✐♠❛ [✺] ❝♦♥❝♦r❞❛ q✉❡ ✉♠❛ ❞❛s ❥✉st✐✜❝❛t✐✈❛s ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ♠❛tr✐③❡s sã♦ ❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ❡ q✉❡ ❛s ♠❡s♠❛s tr❛r✐❛♠ ✉♠ s✐❣♥✐✜❝❛❞♦ às ♦♣❡r❛çõ❡s ❡♥tr❡ ♠❛tr✐③❡s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦✳ P♦ré♠ ♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❡♠ s✉❛ ♠❛✐♦r✐❛✱ ♥ã♦ ❛❜♦r❞❛♠ ♠❛tr✐③❡s ❛ss♦❝✐❛♥❞♦✲❛s ❛ tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s✳
❙❡❣✉♥❞♦ ❆♥t♦♥[✶]❛ ♣r✐♥❝✐♣❛❧ ❛♣❧✐❝❛çã♦ ❞❡ ❞❡❢♦r♠❛çõ❡s ❡ ♠♦r✜s♠♦s t❡♠ s✐❞♦ ❛ ♣r♦❞✉çã♦ ❞❡ ❡❢❡✐t♦s ❡s♣❡❝✐❛✐s ♥♦ ❝✐♥❡♠❛✱ ♥❛ t❡❧❡✈✐sã♦ ❡ ♥❛ ♣r♦♣❛❣❛♥❞❛✳ ◆♦ ❡♥t❛♥t♦✱ t❛♠❜é♠ s✉r❣✐r❛♠ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s ❝✐❡♥tí✜❝❛s ❡ t❡❝♥♦❧ó❣✐❝❛s ♣❛r❛ ❡st❛s té❝♥✐❝❛s✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❛ss✐stê♥❝✐❛ à ❝✐r✉r❣✐❛ ♣❧ást✐❝❛ ❡ ❞❡ r❡❝♦♥str✉çã♦✱ ❛ ✐♥✈❡st✐❣❛çã♦ ❞❡ ✈❛r✐❛çõ❡s ♥♦ ♣r♦❥❡t♦ ❞❡ ✉♠ ♣r♦❞✉t♦ ❡ ♦ ✏❡♥✈❡❧❤❡❝✐♠❡♥t♦✑ ❞❡ ❢♦t♦❣r❛✜❛s ❞❡ ♣❡ss♦❛s ❞❡s❛♣❛r❡❝✐❞❛s ♦✉ s✉s♣❡✐t❛s ❞❛ ♣♦❧í❝✐❛✳ ■♠❛❣✐♥❡ ✉♠ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ ♦rt♦❣♦♥❛❧
❝♦♠ ❛ ♦r✐❣❡♠ ♥♦ ❝❡♥tr♦ ❞❛ t❡❧❛ ❞♦ ♠♦♥✐t♦r ✭s✉♣♦st❛ ♣❧❛♥❛✮ ❡ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r
(X, Y)=P✳
P♦r ♠❡✐♦ ❞❡ ✉♠ ♣r♦❞✉t♦ ❞❡ ♠❛tr✐③❡s ❞❛ ❢♦r♠❛✿
( a bc d )( XY )=( axcx++dyby )=( XY′′ )
❖❜t❡♠♦s ✉♠ ♥♦✈♦ ♣♦♥t♦(X′, Y′)=P′✱ t❛❧ q✉❡✿ X′=ax+by ❡ Y′=cx+dy✳
❆s r❡❧❛çõ❡s ❡♥tr❡ ♠❛tr✐③❡s ❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ♣♦ss✐❜✐❧✐t❛♠ ❛ ✐♥t❡❣r❛çã♦ ❡♥tr❡ ❣❡♦♠❡tr✐❛ ❡ á❧❣❡❜r❛✱ ❝❛♠♣♦s ❞✐st✐♥t♦s ❞❛ ♠❛t❡♠át✐❝❛✱ ♠❛s✱ ❛ss✐♠ ❝♦♠♦ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ sã♦ ✐❣♥♦r❛❞♦s ♣♦r ♠✉✐t♦s ♣r♦❢❡ss♦r❡s ❞❡ ♠❛t❡♠át✐❝❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳
P❛r❛ ❛❧❝❛♥ç❛r♠♦s ♦ ♦❜❥❡t✐✈♦ ♣r♦♣♦st♦✱ ✐♥✐❝✐❛r❡♠♦s ♦ ❡st✉❞♦ ♣❡❧❛s ❝♦♥tr❛çõ❡s ❛ ✉♠ ♣♦♥t♦ ♦✉ ❛ ✉♠❛ r❡t❛✱ ❝♦♥❢♦r♠❡ ❙❤❡r✈❛t♦✈ [✼] ♦♥❞❡ ♦❜t❡r❡♠♦s ✉♠❛ ❛♠♣❧✐❛çã♦ ♦✉ ✉♠❛ r❡❞✉çã♦ ❞❛ ✜❣✉r❛ ♦r✐❣✐♥❛❧✱ ❛tr❛✈és ❞❡ ✉♠ ❢❛t♦rk✱ ♦✉ s❡❥❛✱ ✉♠❛ s❡♠❡❧❤❛♥ç❛
q✉❡ ♣r❡s❡r✈❛ ♦✉ ✐♥✈❡rt❡ ♦ ♣♦s✐❝✐♦♥❛♠❡♥t♦ ❞❛ ✜❣✉r❛✳ ◆♦ ❈❛♣ít✉❧♦ ✷✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❬✶✱ ✷✱ ✸✱ ✻✱ ✹❪✱ ❢❛❧❛r❡♠♦s s♦❜r❡ ♦ ❡st✉❞♦ ❞♦s ✈❡t♦r❡s ❝♦♠ s✉❛s r❡♣r❡s❡♥t❛çõ❡s ❡ ♦♣❡r❛çõ❡s ♥♦ ♣❧❛♥♦✱ ❡♥✈♦❧✈❡♥❞♦ tr❛♥s❧❛çã♦ ❞❡ r♦t❛çã♦ ❡ ✈❡r❡♠♦s ❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s ❡♠R2 ❝♦♠ ♦ ❡st✉❞♦ ❞❛s ♠❛tr✐③❡s ❡ ❛ ❣❡♦♠❡tr✐❛ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s✳ ❋✐♥❛❧✐③❛♥❞♦ ❝♦♠ ♦ ❈❛♣ít✉❧♦ ✸ ✈❡r❡♠♦s ❛s tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ❞♦ ❣❛t♦ ❞❡ ❆r♥♦❧❞ ♥♦ q✉❛❞r❛❞♦ ✉♥✐tár✐♦ ❡ ❞❡❢♦r♠❛çõ❡s ❡ ♠♦r✜s♠♦s ❞❡ ✐♠❛❣❡♥s ❡♠ R2✱ ❜❡♠ ❝♦♠♦ ♦s ♠♦❞❡❧♦s ❜ás✐❝♦s ❞❡ ❞❡❢♦r♠❛çõ❡s ❝♦♠♦✿ ❝✐s❛❧❤❛♠❡♥t♦ ♣✉r♦✱ ❝✐s❛❧❤❛♠❡♥t♦ s✐♠♣❧❡s✱ r♦t❛çã♦ ❡ ♠✉❞❛♥ç❛ ❞❡ ár❡❛✳ ❆ ❢♦r♠❛ ♠❛✐s ❛❞❡q✉❛❞❛ ♣❛r❛ ♦ tr❛t❛♠❡♥t♦ ❞❡ss❛ q✉❡stã♦ é ❛tr❛✈és ❞♦ ❝á❧❝✉❧♦ ♠❛tr✐❝✐❛❧✱ ❡♥✈♦❧✈❡♥❞♦ ♠❛tr✐③❡s q✉❛❞r❛❞❛s ❞❡ ♦r❞❡♠2✳
❈❛♣ít✉❧♦ ✶
❍♦♠♦t❡t✐❛s ❡ ❈♦♥tr❛çõ❡s ♥♦ P❧❛♥♦
◆♦ ♣r❡s❡♥t❡ ❝❛♣ít✉❧♦ ❜✉s❝❛♠♦s ❛♣r❡s❡♥t❛r ❞❡ ❢♦r♠❛ ✐♥t✉✐t✐✈❛ ❝♦♥❝❡✐t♦s ❣❡♦♠étr✐❝♦s q✉❡ s❡rã♦ ❢♦r♠❛❧✐③❛❞♦s ❡ ❡st❡♥❞✐❞♦s ❡♠ t❡r♠♦s ❛❧❣é❜r✐❝♦s ❡❧❡♠❡♥t❛r❡s ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳
✶✳✶ ❈♦♥tr❛çã♦ ❛ ✉♠ P♦♥t♦
◆❛ s♦❧✉çã♦ ❞❡ ❝❡rt♦s ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s✱ ✉s❛✲s❡✱ ❝♦♠ ❢r❡q✉ê♥❝✐❛✱ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡♥♦♠✐♥❛❞❛ ❈♦♥tr❛çã♦ ❛ ✉♠ P♦♥t♦ ♦✉ ❍♦♠♦t❡t✐❛✳ ◆❛ ❝♦♥tr❛çã♦ ❛♦ ♣♦♥t♦ O ✭❝❤❛♠❛❞♦ ❞❡ ❝❡♥tr♦ ❞❡ ❝♦♥tr❛çã♦✮✱ r❡❛❧✐③❛❞❛ ❝♦♠ ♦ ❢❛t♦r ❞❡ ❝♦♥tr❛çã♦ k✱ s✐❣♥✐✜❝❛ q✉❡ ❝❛❞❛ ♣♦♥t♦ A ❞❡ ✉♠ ♣❧❛♥♦ ♣❛ss❛ ❛♦ ♣♦♥t♦ A′
♥❛ r❡t❛ ❞❡t❡r♠✐♥❛❞❛ ♣♦rO ❡ A✱ s❡♥❞♦
k =∣OA
′∣ ∣OA∣,
♦♥❞❡ ♣♦r ∣OA∣ ❡st❛♠♦s r❡♣r❡s❡♥t❛♥❞♦ ❛ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ O ❛♦ ♣♦♥t♦ A ✭✈❡❥❛
❋✐❣✉r❛ ✶✳✶✮✳
❋✐❣✉r❛ ✶✳✶✿ ❈♦♥tr❛çã♦
❙❡ ♦ ❢❛t♦r ❞❡ ❝♦♥tr❛çã♦k é ♠❛✐♦r q✉❡1✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✷✮✱ ❡♥tã♦ ∣OA′∣>∣OA∣✳ ◆❡ss❡
❝❛s♦✱ ❛ tr❛♥s❢♦r♠❛çã♦ ♣♦❞❡r✐❛ s❡ ❝❤❛♠❛r ✏❡①♣❛♥sã♦ ❛♦ ♣♦♥t♦O✑✳
❋✐❣✉r❛ ✶✳✷✿ ❊①♣❛♥sã♦
◆❛ ❤♦♠♦t❡t✐❛ ❛♦ ♣♦♥t♦O✱ ❝❛❞❛ ✜❣✉r❛F s❡ tr❛♥s❢♦r♠❛ ❡♠ ♦✉tr❛F′✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✸✮ s❡♠❡❧❤❛♥t❡ à ♦r✐❣✐♥❛❧✱ ❝♦♠ ♦ ❝❡♥tr♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ♥♦ ♣♦♥t♦ O ❡ ❢❛t♦r ❞❡
s❡♠❡❧❤❛♥ç❛ ✐❣✉❛❧ ❛ k✳ ❙❡ k <1 ❛ ♥♦✈❛ ✜❣✉r❛ t❡♠ s✉❛ ár❡❛ r❡❞✉③✐❞❛ ❡ q✉❛♥❞♦ k>1
❛ s✉❛ ár❡❛ s❡rá ❛✉♠❡♥t❛❞❛✳
❋✐❣✉r❛ ✶✳✸✿ ❋✐❣✉r❛ ❙❡♠❡❧❤❛♥t❡
❈♦♥tr❛çã♦ ❞❡ ❘❡t❛s ❛ ✉♠ P♦♥t♦
◆♦ ❝❛s♦ ❞❡ ❝♦♥tr❛çã♦ ❞❡ ✉♠❛ r❡t❛✱ ❛♦ ❝♦♥tr❛í✲❧❛ tr❛♥s❢♦r♠❛✲s❡ ❛ ♠❡s♠❛ ❡♠ ♦✉tr❛ r❡t❛ ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✹✮✳
❋✐❣✉r❛ ✶✳✹✿ ❘❡t❛
❆❧é♠ ❞✐ss♦✱ r❡t❛s ♣❛r❛❧❡❧❛s s❡ tr❛♥s❢♦r♠❛♠ ❡♠ r❡t❛s ♣❛r❛❧❡❧❛s ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✺✮
❋✐❣✉r❛ ✶✳✺✿ ❘❡t❛s P❛r❛❧❡❧❛s
❈♦♥tr❛çã♦ ❞❡ ❈✐r❝✉♥❢❡rê♥❝✐❛s ❛ ✉♠ P♦♥t♦
◗✉❛♥❞♦ s❡ ❢❛③ ❛ ❝♦♥tr❛çã♦ ❛ ✉♠ ♣♦♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❝♦♥✈❡rt❡✲s❡ ❛ ♠❡s♠❛ ❡♠ ✉♠❛ ♦✉tr❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✻✮✳
❈♦♥tr❛çã♦ ❛ ✉♠ P♦♥t♦ ❡ ➪r❡❛
◆✉♠❛ ❤♦♠♦t❡t✐❛✱ t♦❞♦s ♦s s❡❣♠❡♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ ❞✐♠✐♥✉❡♠ ♦✉ ❛✉♠❡♥t❛♠ ♣♦r ✉♠❛ r❡❧❛çã♦ ❝♦♥st❛♥t❡ k✳ ❆s ár❡❛s ❞❡ t♦❞❛s ❛s ✜❣✉r❛s t❛♠❜é♠ ❞✐♠✐♥✉❡♠ ♦✉
❋✐❣✉r❛ ✶✳✻✿ ❈♦♥tr❛çã♦ ❞❡ ✉♠❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❛ ✉♠ P♦♥t♦
❛✉♠❡♥t❛♠ ♣♦r ✉♠❛ r❡❧❛çã♦ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛ k2 ✭q✉❛❞r❛❞♦ ❞♦ ❢❛t♦r ❞❡ ❝♦♥tr❛çã♦✮✳ ❆♥t❡s ❞❡ ✐❧✉str❛r ♦ ❝♦♥❝❡✐t♦ ❡♠ ✉♠❛ ✜❣✉r❛ q✉❛❧q✉❡r✱ ✈❛♠♦s ❢❛③❡r ✉♠❛ ❜r❡✈❡ ❛♥á❧✐s❡ ❞♦ ❝❛s♦ ❞❡ ✉♠❛ ✜❣✉r❛ r❡t❛♥❣✉❧❛r✳
❊①❡♠♣❧♦ ✶✳✶✳ ❖❜t❡♥❤❛ ❛ ✜❣✉r❛ tr❛♥s❢♦r♠❛❞❛ ❞♦ r❡tâ♥❣✉❧♦ ABCD✱ ❝♦♠ ❝❡♥tr♦ O
❡ r❛③ã♦ ✸✳
❋✐❣✉r❛ ✶✳✼✿ ➪r❡❛ ❞❡ ❋✐❣✉r❛
❈♦♠ ♦ ❝❛s♦ ❞❡ ✜❣✉r❛ r❡t❛♥❣✉❧❛r ❡♥t❡♥❞✐❞♦ ♣♦❞❡♠♦s ✐❧✉str❛r ♦ ❝❛s♦ ❞❡ ✉♠❛ ✜❣✉r❛ ❛r❜✐trár✐❛✳ ❈♦♠ ❡❢❡✐t♦✱ s❡❥❛F ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛✳ ❆♥❛❧✐s❡♠♦s ✉♠❛ r❡tí❝✉❧❛ ❝♦♠♣♦st❛
♣♦r ✉♠ ❝❡rt♦ ♥ú♠❡r♦ ❞❡ q✉❛♥❞r❛❞♦s ♣❡q✉❡♥♦s ❞❡ ♠❡s♠❛ ár❡❛ ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✽✮✳ ❆
❋✐❣✉r❛ ✶✳✽✿ ❋✐❣✉r❛ ❉❡❝♦♠♣♦st❛ ❡♠ ◗✉❛❞r❛❞♦s
ár❡❛ ❞❡F é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ q✉❛❞r❛❞♦s q✉❡ s❡ ❡♥❝♦♥tr❛♠ ❞❡♥tr♦
❞❛ ✜❣✉r❛ F✱ ♠✉❧t✐♣❧✐❝❛❞♦ ♣❡❧❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦✳ ❖ ❡rr♦ s❡rá t❛♥t♦ ♠❡♥♦r q✉❛♥t♦
♠❛✐♦r ❢♦r ♦ ♥ú♠❡r♦ ❞❡ q✉❛❞r❛❞♦s ♥❛ r❡tí❝✉❧❛✳ ❚♦♠❛♥❞♦ q✉❛❞r❛❞♦s s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦s✱ ♣♦❞❡♠♦s t♦r♥❛r ♦ ❡rr♦ tã♦ ♠❡♥♦r q✉❛♥t♦ s❡ ❞❡s❡❥❛✳ ◆❛ ❤♦♠♦t❡t✐❛✱ ❛ r❡tí❝✉❧❛ ❞❡ q✉❛♥❞r❛❞♦s s❡ tr❛♥s❢♦r♠❛ ❡♠ ✉♠❛ ♥♦✈❛ r❡tí❝✉❧❛ ❞❡ q✉❛❞r❛❞♦s✱ ❡♥q✉❛♥t♦ q✉❡ ❛ ✜❣✉r❛F s❡ tr❛♥s❢♦r♠❛ ♥❛ ✜❣✉r❛F′✱ ❞❡♥tr♦ ❞❛ q✉❛❧ ❤❛✈❡rá t❛♥t♦s q✉❛❞r❛❞♦s ❞❛ ♥♦✈❛ r❡tí❝✉❧❛ ✭♠❡♥♦r❡s ❡♠ ár❡❛ s❡k <1 ❡ ♠❛✐♦r❡s✱ s❡ k>1✮ q✉❛♥t♦s ❝♦♥tí♥❤❛♠
♥❛ ✜❣✉r❛ ♦r✐❣✐♥❛❧ F✳ ❆ ár❡❛ ❞❛ ♥♦✈❛ ✜❣✉r❛F′
é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ q✉❛❞r❛❞♦s ❝♦♥t✐❞♦s ♥❡❧❛✱ ♠✉❧t✐♣❧✐❝❛❞♦ ♣❡❧❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦✳ ▼❛s ❛ ár❡❛ ❞❡ ❝❛❞❛ q✉❛❞r❛❞♦ ♥♦✈♦ é ✐❣✉❛❧ ❛ ❞♦ q✉❛❞r❛❞♦ ♦r✐❣✐♥❛❧✱ ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r k2✱ ❞❛❞♦ q✉❡ ❝❛❞❛ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦✱ ♥❛ ❝♦♥tr❛çã♦ ❛♦ ♣♦♥t♦✱ é ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r k✱ ❝♦♥❢♦r♠❡ ✐❧✉str❛❞♦
♥♦ ❊①❡♠♣❧♦ ✶✳✶✳ P♦r ✐ss♦ t❡♠♦s q✉❡✿
✄
✂ ❆ ár❡❛ ❞❡F ✁
′
é ✐❣✉❛❧ ❛ ár❡❛ ❞❡F ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦rk2✳
❊①❡♠♣❧♦ ✶✳✷✳ ❖ ❡❢❡✐t♦ ❣❡♦♠étr✐❝♦ ❞❡ ✉♠❛ ❤♦♠♦t❡t✐❛ é ❞❡ ❛♠♣❧✐❛çã♦ ♦✉ r❡❞✉çã♦ ❞❡ ✜❣✉r❛s✳ ❈♦♠✉♠❡♥t❡ tr❛t❛♠♦s ♦ ❝❛s♦ k >0✱ ❡ ❝❤❛♠❛♠♦s ♦ ❝❛s♦ k <0 ❞❡ ❤♦♠♦t❡t✐❛
✐♥✈❡rs❛✳ ❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ q✉❛♥❞♦ ∣k∣>1✱ ❛ ✜❣✉r❛ ❤♦♠♦tét✐❝❛ é ✉♠❛ ❛♠♣❧✐❛çã♦
❞❛ ✜❣✉r❛ ✐♥✐❝✐❛❧❀ q✉❛♥❞♦ ∣k∣=1✱ ❛ ✜❣✉r❛ ❤♦♠♦tét✐❝❛ é ❝♦♥❣r✉❡♥t❡ à ✜❣✉r❛ ✐♥✐❝✐❛❧❀ ❡
q✉❛♥❞♦ ∣k∣<1✱ ❛ ✜❣✉r❛ ❤♦♠♦tét✐❝❛ é ✉♠❛ r❡❞✉çã♦ ❞❛ ✜❣✉r❛ ✐♥✐❝✐❛❧✳
❖ ♣❡♥tá❣♦♥♦ VA′B′C′D′ é ✉♠❛ ❛♠♣❧✐❛çã♦ ❞♦ ♣❡♥tá❣♦♥♦ VABCD✱ ❝♦♠ r❛③ã♦
k>1✳
❋✐❣✉r❛ ✶✳✾✿ ❆♠♣❧✐❛çã♦ ❞❡ P❡♥tá❣♦♥♦
❊①❡♠♣❧♦ ✶✳✸✳ ❊①❛♠✐♥❡♠♦s ❛ r❡s♦❧✉çã♦ ❞♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ q✉❡ ✐❧✉str❛ ❛ ❛♣❧✐❝❛çã♦ ❞❛ ❝♦♥tr❛çã♦ ❛♦ ♣♦♥t♦✿
✄
✂ ✁
❉❛❞♦ ✉♠ tr✐â♥❣✉❧♦ ABC✱ ✐♥s❝r❡✈❡r ♥❡❧❡ ✉♠ r❡tâ♥❣✉❧♦ á✉r❡♦ t❡♥❞♦ ♦ s❡✉ ❧❛❞♦
♠❛✐♦r ❝♦♥t✐❞♦ ♥♦ ❧❛❞♦ BC✳
❙♦❧✉çã♦✿ ❈♦♥s✐❞❡r❛✲s❡ ✉♠ tr✐â♥❣✉❧♦ ABC✳ ❈♦♥stró✐✲s❡ ✉♠ q✉❛❞r❛❞♦ P QRS ❡♠
s❡✉ ✐♥t❡r✐♦r✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ❧❛❞♦ P Q ❡st❡❥❛ ❝♦♥t✐❞♦ ♥♦ ❧❛❞♦ BC ❞♦ tr✐â♥❣✉❧♦ ❡ ♦
✈ért✐❝❡S♣❡rt❡♥ç❛ ❛♦ ❧❛❞♦AB✳ ❙❡❥❛M ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦P Q✳ ❈♦♠ ❝❡♥tr♦
❡♠ M ❡ r❛✐♦ M R✱ tr❛ç❛✲s❡ ✉♠ ❛r❝♦ ❡♥❝♦♥tr❛♥❞♦ BC ♥♦ ♣♦♥t♦ N✳ ❙❡❥❛ O ♦ ♣♦♥t♦
❋✐❣✉r❛ ✶✳✶✵✿ ❘❡tâ♥❣✉❧♦ ■♥s❝r✐t♦
❞❡ ✐♥t❡rs❡çã♦ ❞❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛BC ♣❛ss❛♥❞♦ ♣♦r S❝♦♠ ❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛BC✱
♣❛ss❛♥❞♦ ♣♦rN✳ ❖ r❡tâ♥❣✉❧♦ P N OS é ✉♠ r❡tâ♥❣✉❧♦ á✉r❡♦ ✭♣♦r ❝♦♥str✉çã♦✮✳ P❛r❛
✐♥s❝r❡✈❡r ❡ss❡ r❡tâ♥❣✉❧♦ ♥♦ tr✐â♥❣✉❧♦ ABC✱ ❝♦♥s✐❞❡r❛✲s❡ ❛ s❡♠✐✲r❡t❛ ❝♦♠ ♦r✐❣❡♠
❡♠ B ♣❛ss❛♥❞♦ ♣♦rO✱ s❡♥❞♦ V ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛ s❡♠✐✲r❡t❛ ❝♦♠ AC✳ P❛r❛
❡♥❝♦♥tr❛r ♦s ♦✉tr♦s ✈ért✐❝❡s ❞♦ r❡tâ♥❣✉❧♦ ❝♦♥s✐❞❡r❛✲s❡ ❛ ❤♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦B❡ ❢❛t♦r
❞❡ ❝♦♥tr❛çã♦ ∣BV∣
∣BO∣✳ ❖ r❡tâ♥❣✉❧♦ XY W V é ♦ r❡tâ♥❣✉❧♦ á✉r❡♦ ✐♥s❝r✐t♦ ♥♦ tr✐â♥❣✉❧♦
ABC✳
✶✳✷ ❈♦♥tr❛çã♦ ❛ ✉♠❛ ❘❡t❛
◆❛ ❣❡♦♠❡tr✐❛✱ ❡♠ ❞✐✈❡rs❛s s✐t✉❛çõ❡s é ❝♦♥✈❡♥✐❡♥t❡ ❛♣❧✐❝❛r ✉♠❛ tr❛♥s❢♦r♠❛çã♦ q✉❡ ❝❤❛♠❛♠♦s ❞❡ ✏❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛✑✳ ◆❛ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛ ✭❞✐❣❛♠♦s l✱
❞❡♥♦♠✐♥❛❞❛ ❞❡ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✮✱ ❝♦♠ ❢❛t♦r ❞❡ ❝♦♥tr❛çã♦ k✱ ❝❛❞❛ ♣♦♥t♦ A ❞♦
♣❧❛♥♦ ♣❛ss❛ ❛♦ ♣♦♥t♦ A′ ❞♦ r❛✐♦ P A✱ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❡✐①♦✱ s❡♥❞♦
k=∣P A
′∣ ∣P A∣,
♦✉ s❡❥❛✱ ∣P A∣=k∣P A∣ ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✶✶✱(a)✮✳
❋✐❣✉r❛ ✶✳✶✶✿ ❈♦♥tr❛çã♦ ❛ ✉♠❛ ❘❡t❛
◗✉❛♥❞♦ k>1✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✶✶✱ (b)✮✱ t❡♠♦s
∣P A′∣>∣P A∣,
❡ ♥❡ss❡ ❛ tr❛♥s❢♦r♠❛çã♦ ♣♦❞❡r✐❛ s❡ ❝❤❛♠❛r ❞❡ ✏❡①♣❛♥sã♦ ❛♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✑✳ ❚♦❞♦s ♦s ♣♦♥t♦s ❞♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✱ ❞✉r❛♥t❡ ❛ ❝♦♥tr❛çã♦✱ ♣❡r♠❛♥❡❝❡♠ ❡♠ s❡✉s ❧♦❝❛✐s ❞❡ ♦rí❣❡♠ ✭♥ã♦ ♠✉❞❛♠ ❞❡ ♣♦s✐çã♦✮✳
❆ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛ t❡♠ ♣r♦♣r✐❡❞❛❞❡s ❛♥á❧♦❣❛s à ❞❛ ❝♦♥tr❛çã♦ ❛ ✉♠ ♣♦♥t♦✳
❈♦♥tr❛çã♦ ❞❡ ❘❡t❛s ❛ ✉♠❛ ❘❡t❛
❆ ♣r✐♠❡✐r❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ ♣♦❞❡♠♦s ❛♣r❡s❡♥t❛r é ❛ s❡❣✉✐♥t❡✿
✄
✂ ◆❛ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛✱ ❝❛❞❛ r❡t❛ s❡ tr❛♥s❢♦r♠❛ ❡♠ ♦✉tr❛ r❡t❛✳ ✁
P❛r❛ ❥✉st✐✜❝❛r ❡ss❛ ❛✜r♠❛çã♦✱ ❝♦♠❡❝❡♠♦s ❝♦♠ ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✳ ❙❡ ✉♠❛ r❡t❛ l é ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦ ❡ d é ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡
❡st❛s r❡t❛s✱ ❡♥tã♦ ❛ r❡t❛ l s❡ tr❛♥s❢♦r♠❛ ♥❛ r❡t❛ l′✱ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✱ s❡♥❞♦ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ l′ ❡ ♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦ ✐❣✉❛❧ ❛k⋅d ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✶✷✮✳
❋✐❣✉r❛ ✶✳✶✷✿ ❈♦♥tr❛çã♦ ❞❡ r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ x
❆❣♦r❛✱ s✉♣♦♥❤❛♠♦s q✉❡ l ♥ã♦ s❡❥❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦ ❡ ❞❡s✐❣♥❡♠♦s
♣♦rO✱ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❡ l ❝♦♠ ♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦ ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✶✸✮✳
❋✐❣✉r❛ ✶✳✶✸✿ ❈♦♥tr❛çã♦ ❞❡ r❡t❛ ♥ã♦ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦x
❈♦♠♦ ❝♦♥tr❛çã♦ ♦s ♣♦♥t♦s ❞♦s ❡✐①♦s ♥ã♦ s❡ ❛❧t❡r❛♠✱ ♦ ♣♦♥t♦ O ♣❡r♠❛♥❡❝❡ ❡♠
s❡✉ ❧✉❣❛r✳ ❙❡❥❛♠
✭✐✮ A ✉♠ ♣♦♥t♦ ❛r❜✐trár✐♦ ♥❛ r❡t❛ l ✭❞✐❢❡r❡♥t❡ ❞❡ O✮❀
✭✐✐✮ A ❛ ♣♦s✐çã♦ ♦❝✉♣❛❞❛ ♣❡❧♦ ♣♦♥t♦ A ❛♣ós s♦❢r❡r ✉♠❛ ❝♦♥tr❛çã♦ ❛♦ ❡✐①♦ ❞❡
❝♦♥tr❛çã♦ ❡ ✭✐✐✐✮ k =∣P A′∣
∣P A∣.
❊s❝♦❧❤❛♠♦s ♦✉tr♦ ♣♦♥t♦ B ♥❛ r❡t❛ l✳ ❙❡ B′ é ✉♠ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ ❛ r❡t❛ OA′
❡ ❛ ♣❡r♣❡♥❞✐❝✉❧❛r BQ✱ tr❛ç❛❞❛ ❞❡s❞❡ ♦ ♣♦♥t♦B ❛♦ ❡✐①♦ ❝♦♥tr❛çã♦✱ ❡♥tã♦
∣B′Q∣ ∣BQ∣ =
∣A′P∣ ∣AP∣ =k,
✐ss♦ ❣❛r❛♥t✐❞♦ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❚❛❧❡s ♦✉ ♣❡❧❛ s❡♠❡❧❤❛ç❛ ❞♦s tr✐â♥❣✉❧♦sOQB❡OP A✱ OQB′❡OP A′✳ ❆ss✐♠ s❡♥❞♦✱ ✈❡♠♦s q✉❡ ♥❛ ❝♦♥tr❛çã♦ ❛♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✱ ♦ ♣♦♥t♦
B ♣❛ss❛ ❛♦ ♣♦♥t♦B′✳ ❈♦♠♦B é ✉♠ ♣♦♥t♦ ❛r❜✐trár✐♦ ❞❛ r❡t❛l✱ ❡st❛ ú❧t✐♠❛ ♣❛ss❛✱ ♥❛ ❝♦♥tr❛çã♦ ❛♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✱ à r❡t❛OA′✱ ❞❡s✐❣♥❛❞❛ ♥❛t✉r❛❧♠❡♥t❡ ♣♦rl′✳ P♦rt❛♥t♦✱ t❡♠♦s ❛ ❥✉st✐✜❝❛t✐✈❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡✳
❖❜s❡r✈❛çã♦ ✶✳✹✳ ❙❡ θ ❡ θ′ sã♦ ♦s â♥❣✉❧♦s ❢♦r♠❛❞♦s ♣❡❧❛s r❡t❛s l ❡ l′ ❝♦♠ ♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♥tã♦ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❛ t❛♥❣❡♥t❡ ♦❜t❡♠♦s ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✶✸✮✿
tg(θ)=∣P A
′∣ ∣P O∣ =
k⋅∣P A∣
∣P O∣ =k⋅
∣P A∣
∣P O∣ =k⋅tg(θ
′).
❯♠❛ s❡❣✉♥❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛ é ❛ s❡❣✉✐♥t❡✿
✄
✂ ✁
◆❛ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛✱ r❡t❛s ♣❛r❛❧❡❧❛s s❡ tr❛♥s❢♦r♠❛♠ ❡♠ r❡t❛s ♣❛r❛❧❡❧❛s✳
P❛r❛ ✈❡r✐✜❝❛r ❛ ✈❛❧✐❞❛❞❡ ❞❛ ♣r♦♣r✐❡❞❛❞❡✱ s❡❥❛♠l ❡mr❡t❛s ♣❛r❛❧❡❧❛s✳ ❆♣ós s♦❢r❡r❡♠
✉♠❛ ❝♦♥tr❛çã♦ ❛♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✱ ❛s r❡t❛s l ❡ m ♣❛ss❛♠ ❛ l′ ❡ m′✳ ❙✉♣♦♥❞♦ q✉❡l′ ❡m′ t✐✈❡ss❡♠ ✉♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠✱ ✐ss♦ r❡s✉❧t❛r✐❛ q✉❡ l ❡ m t❛♠❜é♠ t❡r✐❛♠ ✉♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠ ✭❝♦♥tr❛❞✐çã♦✮✳ ▲♦❣♦l′ ❡ m′ sã♦ t❛♠❜é♠ r❡t❛s ♣❛r❛❧❡❧❛s ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✶✹✮✳
❋✐❣✉r❛ ✶✳✶✹✿ ❈♦♥tr❛çã♦ ❞❡ ❘❡t❛s P❛r❛❧❡❧❛s ❛♦ ❊✐①♦ ❞❡ ❈♦♥tr❛çã♦
❯♠❛ t❡r❝❡✐r❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛ é✿
✄
✂ ✁
◆❛ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛✱ ❛ r❡❧❛çã♦ ❡♥tr❡ s❡❣♠❡♥t♦s ❞✐s♣♦st♦s ❡♠ ✉♠❛ r❡t❛ ♣❡r♠❛♥❡❝❡ ❝♦♥st❛♥t❡ ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✶✺✮✳
❋✐❣✉r❛ ✶✳✶✺✿ ❘❡❧❛çã♦ ❞❡ s❡❣♠❡♥t♦s ❞❡ ❘❡t❛ ❛♣ós ❈♦♥tr❛çã♦ ❖❜s❡r✈❡ q✉❡ ❛♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❚❛❧❡s ♥❛ ❋✐❣✉r❛ ✶✳✶✺✮✱ t❡r❡♠♦s s❡♠♣r❡✿
∣AB∣ ∣BC∣ =
∣A′B′∣
∣B′C′∣.
❈♦♥tr❛çã♦ ❞❡ ❈✐r❝✉♥❢❡rê♥❝✐❛s ❛ ✉♠❛ ❘❡t❛
❆♦ ❝♦♥trár✐♦ ❞❛ ❤♦♠♦t❡t✐❛✱ ❛ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛ ♥ã♦ tr❛♥s❢♦r♠❛ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡♠ ♦✉tr❛✳ ❘❡❛❧✐③❛❞❛ ✉♠❛ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛✱ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ s❡ tr❛♥s❢♦r♠❛ ❡♠ ✉♠❛ ❊❧✐♣s❡ ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✶✻✮✳
❋✐❣✉r❛ ✶✳✶✻✿ ❈♦♥tr❛çã♦ ❞❡ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❛ ✉♠❛ ❘❡t❛✱ ♦❜té♠✲s❡ ❊❧✐♣s❡✳
❈♦♥tr❛çã♦ ❞❡ ❋✐❣✉r❛ ❛ ✉♠❛ ❘❡t❛
◆❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛✱ ✉♠❛ ✜❣✉r❛F s❡ tr❛♥s❢♦r♠❛ ❡♠ ♦✉tr❛
✜❣✉r❛F′✱ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ s❡♠❡❧❤❛♥t❡ ❛F✳ ◆❛ ❋✐❣✉r❛ ✶✳✶✼ t❡♠♦s ✐❧✉str❛❞❛ ✉♠❛ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛ ❝♦♠✱ ♣♦r ❡①❡♠♣❧♦✱ ❢❛t♦r ❞❡ ❝♦♥tr❛çã♦k=1/3 ❞❡ ♠♦❞♦ q✉❡
P1A′1
P1A1
= P1A
′ 2
P1A2
=...= 1
3.
❋✐❣✉r❛ ✶✳✶✼✿ ❈♦♥tr❛çã♦ ❞❡ ❋✐❣✉r❛ ❛ ✉♠❛ ❘❡t❛
❈♦♥tr❛çã♦ ❛ ✉♠❛ ❘❡t❛ ❡ ➪r❡❛
❈♦♠ ✉♠ r❛❝✐♦❝í♥✐♦ s❡♠❡❧❤❛♥t❡ ❛♦ ❞♦ ❝❛s♦ ❞❡ ❝♦♥tr❛çã♦ ❛ ✉♠ ♣♦♥t♦✱ ♣♦❞❡♠♦s ❛♥❛❧✐s❛r ♦ ❝❛s♦ ❞❛ r❡❧❛çã♦ ❞❛ ár❡❛ ❞❡ ✉♠❛ ✜❣✉r❛ ❣❡♦♠étr✐❝❛ ❛♣ós ❡①❡❝✉t❛❞❛ ❛ tr❛♥s❢♦r♠❛çã♦✳ ❊ss❛ r❡❧❛çã♦ é ❛ s❡❣✉✐♥t❡✿
✄
✂ ✁
◆❛ ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛✱ ✉♠❛ ✜❣✉r❛F′♦❜t✐❞❛ ❞❛ ❝♦♥tr❛çã♦ ❞❡ ✉♠❛ ✜❣✉r❛
F t❡♠ ❛ ár❡❛ ✐❣✉❛❧ ❛ ár❡❛ ❞❡F ♠✉❧t✐♣❧✐❝❛❞❛ ♣❡❧♦ ❢❛t♦r ❞❡ ❝♦♥tr❛çã♦ k✳
❆♥t❡s ❞❡ ❛♣r❡s❡♥t❛r ✉♠❛ ❥✉st✐✜❝❛t✐✈❛ ❤❡✉ríst✐❝❛ ❞❛ r❡❧❛çã♦ ❞❡ ár❡❛ ❞❡ ✉♠❛ ✜❣✉r❛ q✉❛❧q✉❡r ♦❜t✐❞❛ ♣♦r ❝♦♥tr❛çã♦ ❛ ✉♠❛ r❡t❛✱ ✈❛♠♦s ♣r✐♠❡✐r♦ ♦❜s❡r✈❛r ♦ ❝❛s♦ ❞❡ ✉♠ r❡tâ♥❣✉❧♦✳
❊①❡♠♣❧♦ ✶✳✺✳ ❖❜s❡r✈❡ q✉❡ ❛ ✜❣✉r❛ ♦❜t✐❞❛ ♣❡❧❛ ❝♦♥tr❛çã♦ à r❡t❛ AB✱ ❝♦♠ ❢❛t♦r ❞❡
❝♦♥tr❛çã♦k =3✱ ❞♦ r❡tâ♥❣✉❧♦ ABCD é ♦ r❡tâ♥❣✉❧♦ A′B′C′D′✳ ❆❧é♠ ❞✐ss♦✱ ♥♦t❡ q✉❡
❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦A′B′C′D′ é 3 ✈❡③❡s ❛ ár❡❛ ❞❡ ABCD ✭✈❡❥❛ ❋✐❣✉r❛ ✶✳✶✽✮✳
❋✐❣✉r❛ ✶✳✶✽✿ ➪r❡❛ ❞❡ ❘❡tâ♥❣✉❧♦ ❛♣ós ❈♦♥tr❛çã♦
❱❛♠♦s ❛❣♦r❛ ❡①❛♠✐♥❛r ♦ ❝❛s♦ ❞❡ ✉♠❛ ✜❣✉r❛F ❡ ❛ r❡tí❝✉❧❛ ❝♦♠♣♦st❛ ❞❡ q✉❛❞r❛❞♦s
❝♦♠ ♠❡s♠❛ ár❡❛ ✭❋✐❣✉r❛ ✶✳✶✾✮✳ ❖ r❛❝✐♦❝í♥✐♦ ♥ã♦ ❞✐❢❡r❡ ❞❛q✉❡❧❡ ♠♦str❛❞♦ ♥♦ ❝❛s♦ ❞❛ ❤♦♠♦t❡t✐❛ ✭❝♦♥tr❛çã♦ ❛ ✉♠ ♣♦♥t♦✮✱ r❡❛❧✐③❛❞♦ ❝♦♠ ♦ ❢❛t♦r ❞❡ ❝♦♥tr❛çã♦ k✱ ❡♠
q✉❡ ❛s ár❡❛s ❡r❛♠ ♠✉❧t✐♣❧✐❝❛❞❛s ♣♦r k2✳ ◆♦t❡ q✉❡ ❛ ár❡❛ ❞❡
F é ❛♣r♦①✐♠❛❞❛♠❡♥t❡
✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ q✉❛❞r❛❞♦s ❞❡♥tr♦ ❞❡ F ♠✉❧t✐♣❧✐❝❛❞♦ ♣❡❧❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦✳
❈♦♥s✐❞❡r❡♠♦s q✉❡ ✉♠❛ ❞❛s ❞✐r❡çõ❡s ❞❛s ❧✐♥❤❛s ❞❛ r❡tí❝✉❧❛ s❡❥❛ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✳ ❆♣ós ❛ ❝♦♥tr❛çã♦✱ ❛ r❡tí❝✉❧❛ ❞❡ q✉❛❞r❛❞♦s s❡ tr❛♥s❢♦r♠❛rá ♥❛ r❡tí❝✉❧❛ ❞❡
❋✐❣✉r❛ ✶✳✶✾✿ ❋✐❣✉r❛ ❉❡❝♦♠♣♦st❛ ❡♠ ◗✉❛❞r❛❞♦s
r❡tâ♥❣✉❧♦s✳ ❆s ár❡❛s ❞♦s r❡tâ♥❣✉❧♦s sã♦ ✐❣✉❛✐s ❛♦ ♣r♦❞✉t♦ ❞❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ♣❡❧♦ ❢❛t♦r ❞❡ ❝♦♥tr❛çã♦k✳ ◆♦t❡ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❞♦✐s ❧❛❞♦s ♣❛r❛❧❡❧♦s ❞♦ q✉❛❞r❛❞♦
♥ã♦ ✈❛r✐❛♠ ✭♦s q✉❡ sã♦ ♣❛r❛❧❡❧♦s ❛♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✮ ❡ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦s ♦✉tr♦s ❞♦✐s ❧❛❞♦s ✭♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛♦ ❡✐①♦ ❞❡ ❝♦♥tr❛çã♦✮ ✜❝❛♠ ♠✉❧t✐♣❧✐❝❛❞♦s ♣♦rk✳
❈❛♣ít✉❧♦ ✷
❱❡t♦r❡s ❡ ❚r❛♥s❢♦r♠❛çõ❡s ▲✐♥❡❛r❡s
♥♦ P❧❛♥♦
◆❡ss❡ ❝❛♣ít✉❧♦✱ ✐♥tr♦❞✉③✐r❡♠♦s ❝♦♦r❞❡♥❛❞❛s ♥❛ r❡t❛ ❡ ♥♦ ♣❧❛♥♦✱ ♣❛r❛ r❡♣r❡s❡♥t❛r ♣♦♥t♦s ♣♦r ♠❡✐♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ❆ ❧✐♥❣✉❛❣❡♠ ❜ás✐❝❛ q✉❡ ✉t✐❧✐③❛r❡♠♦s ❝♦♥t✐♥✉❛ ❝♦♠ ❛ ❛♣r❡s❡♥t❛çã♦ ❞♦s ✈❡t♦r❡s ♥♦ ♣❧❛♥♦ ❡ ❞❡ s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s✳ ❆ r❡♣r❡s❡♥t❛çã♦ ❞♦s ♣♦♥t♦s ♣♦r s✉❛s ❝♦♦r❞❡♥❛❞❛s t♦r♥❛ ♣♦ssí✈❡❧ r❡s♦❧✈❡r ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s✱ ❡ ♦ ✉s♦ ❞❡ ✈❡t♦r❡s ♣❡r♠✐t❡ ♦ ❡st✉❞♦ ❞❡ ✈ár✐♦s ❝♦♥❝❡✐t♦s ❣❡♦♠étr✐❝♦s ❞❡ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s ❡ ❞✐r❡t❛✳
P❛r❛ ✐ss♦✱ ❛❞♠✐t✐r❡♠♦s q✉❡ ♦ ❧❡✐t♦r t❡♥❤❛ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ❛①✐♦♠❛s ❡ ❞♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❡❧❡♠❡♥t❛r❡s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ P❧❛♥❛✱ r❡❧❛t✐✈♦s ❛♦s s❡✉s ❡❧❡♠❡♥t♦s ❜ás✐❝♦s✿ r❡t❛s ❡ ♣❧❛♥♦s✳ ❱❛♠♦s r❡✈❡r ❛❧❣✉♥s ❛①✐♦♠❛s ❡ r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ q✉❡ s❡rã♦ út❡✐s ♥❛ ❝♦♥str✉çã♦ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✿
❼ ♣♦r ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ♣❛ss❛ ✉♠❛ ú♥✐❝❛ r❡t❛ ✭❛①✐♦♠❛ ❞❡ ✐♥❝✐❞ê♥❝✐❛✮❀ ❼ ❞❛❞♦s ✉♠❛ r❡t❛ r ❡ ✉♠ ♣♦♥t♦ P ♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛ r✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ r❡t❛
♣❛r❛❧❡❧❛ à r❡t❛ r q✉❡ ♣❛ss❛ ♣♦rP ✭❛①✐♦♠❛ ❞❛s ♣❛r❛❧❡❧❛s✮❀
❼ ❞❛❞♦s ✉♠ ♣♦♥t♦P ❡ ✉♠❛ r❡t❛ r✱ ❡①✐st❡ ❛♣❡♥❛s ✉♠❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛r q✉❡
♣❛ss❛ ♣♦r P❀
❼ ♣♦r três ♣♦♥t♦s ❞♦ ❡s♣❛ç♦ ♥ã♦ s✐t✉❛❞♦s ♥✉♠❛ ♠❡s♠❛ r❡t❛ ♣❛ss❛ ✉♠ ú♥✐❝♦ ♣❧❛♥♦ ✭❛①✐♦♠❛ ❞❛ ✐♥❝✐❞ê♥❝✐❛✮✳
❆❧é♠ ❞❡ss❡s✱ ✉t✐❧✐③❛r❡♠♦s ✈ár✐♦s ♦✉tr♦s r❡s✉❧t❛❞♦s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ ❝♦♠♦ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ ▲❡✐ ❞♦s ❈♦ss❡♥♦s✱ ♦s ❝❛s♦s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s ❡t❝✳
❆ ♣❛rt✐r ❞♦s ❡❧❡♠❡♥t♦s ❜ás✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ P❧❛♥❛ ❡ ❞♦s ❛①✐♦♠❛s ❞❡ ♦r❞❡♠✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❞♦✐s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s✿
✞
✝
☎
✆
❙❡❥❛♠ A❡B ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s✳ ❖ s❡❣♠❡♥t♦ ❞❡ r❡t❛AB é ♦ ❝♦♥❥✉♥t♦
❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦s A ❡ B ❡ ♣❡❧♦s ♣♦♥t♦s C ❡♥tr❡A ❡ B✱ ❡ ❛ s❡♠✐r❡t❛
Ð⇀
AB é ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦ s❡❣♠❡♥t♦ AB ❡ ♣♦r t♦❞♦s ♦s ♣♦♥t♦s D
t❛✐s q✉❡ B ❡stá ❡♥tr❡ A ❡D✳
❋✐❣✉r❛ ✷✳✶✿ P♦♥t♦ ❉ ♥❛ s❡♠✐r❡t❛ ❆❇
P❛r❛ ✐♥✐❝✐❛r♠♦s ♥♦ss♦ ❡st✉❞♦✱ ❞❡✈❡♠♦s ❧❡♠❜r❛r q✉❡✱ ♥❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ❘❡❛❧✱ ✜①❛❞❛ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦✱ ❛ ❝❛❞❛ ♣❛r ❞❡ ♣♦♥t♦sA ❡B ❝♦rr❡s♣♦♥❞❡
✉♠ ♥ú♠❡r♦ r❡❛❧✱ ❞❡♥♦♠✐♥❛❞♦ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s A ❡ B ♦✉ ❝♦♠♣r✐♠❡♥t♦
❞♦ s❡❣♠❡♥t♦AB✱ ❡ ❞❡s✐❣♥❛❞♦ ♣♦rd(A, B)♦✉∣AB∣✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ q✉❡ s❛t✐s❢❛③ às
s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✶✳ d(A, B)≥0❀
✷✳ d(A, B)=0⇐⇒A=B❀
✸✳ d(A, B)=d(B, A)❀
✹✳ d(A, B)≤d(A, C)+d(C, B)✭❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✮
✺✳ d(A, B)=d(A, C)+d(C, B)⇐⇒A, B ❡ C sã♦ ❝♦❧✐♥❡❛r❡s ❡C ❡stá ❡♥tr❡A❡B✳
❋✐♥❛❧♠❡♥t❡✱ ♣r❡❝✐s❛♠♦s ❧❡♠❜r❛r q✉❡ ❞❛❞♦s ✉♠❛ s❡♠✐r❡t❛ ÐÐ⇀CD ❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ λ>0✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦ F ♣❡rt❡♥❝❡♥t❡ ❛ ÐÐ⇀CD t❛❧ q✉❡ ∣CF∣=λ✳
✷✳✶ ❈♦♦r❞❡♥❛❞❛s ❡ ❞✐stâ♥❝✐❛ ♥❛ r❡t❛
❙❡❥❛♠ r ✉♠❛ r❡t❛ ❡ Ð⇀OA ✉♠❛ s❡♠✐r❡t❛ ❞❡ r ❝♦♠ ♦r✐❣❡♠ ♥✉♠ ♣♦♥t♦ ❡s❝♦❧❤✐❞♦ O
❞❡r✳ ❈♦♥s✐❞❡r❡B ✉♠ ♣♦♥t♦ ❞❡ rt❛❧ q✉❡O ❡stá ❡♥tr❡A ❡B✳ ❆ s❡♠✐r❡t❛ Ð⇀OB é ❞✐t❛
♦♣♦st❛ á s❡♠✐r❡t❛ Ð⇀OA.
❋✐❣✉r❛ ✷✳✷✿ ❆ r❡t❛ r
❆ r❡t❛ r é ♣♦st❛ ❡♠ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s R ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
❼ à ♦r✐❣❡♠ O ❢❛③❡♠♦s ❝♦rr❡s♣♦♥❞❡r ♦ ♥ú♠❡r♦ ✵ ✭③❡r♦✮❀
❼ ❝❛❞❛ X ≠ O✱ ❞❛ s❡♠✐r❡t❛ Ð⇀OA, ❢❛③❡♠♦s ❝♦rr❡s♣♦♥❞❡r ♦ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ x=d(O, X)❀
❼ ❝❛❞❛ ♣♦♥t♦ X ≠ O✱ ❞❛ s❡♠✐r❡t❛ Ð⇀OB✱ ❢❛③❡♠♦s ❝♦rr❡s♣♦♥❞❡r ♦ ♥ú♠❡r♦ r❡❛❧
♥❡❣❛t✐✈♦ x= −d(O, X)✳
❋✐❣✉r❛ ✷✳✸✿ ❈♦♦r❞❡♥❛❞❛s ❞♦s ♣♦♥t♦s ❞❛ r❡t❛ r
❉❡✜♥✐çã♦ ✷✳✶✳ ❖ ♥ú♠❡r♦ r❡❛❧ x q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♣♦♥t♦ X s❡❣✉♥❞♦ ❛
❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❛❝✐♠❛ ❡st❛❜❡❧❡❝✐❞❛ é ❞❡♥♦♠✐♥❛❞❛ ❛ ❝♦♦r❞❡♥❛❞❛ ❞♦ ♣♦♥t♦ X✳
❉❡✜♥✐çã♦ ✷✳✷✳ ❙❡❥❛♠ X ❡ Y ♣♦♥t♦s ❞❛ r❡t❛ r ❝♦♠ ❝♦♦r❞❡♥❛❞❛s x ❡ y✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉✐③❡♠♦s q✉❡ ♦ ♣♦♥t♦Y ❡stá à ❞✐r❡✐t❛ ❞♦ ♣♦♥t♦ X ✭♦✉ q✉❡ ♦ ♣♦♥t♦ X ❡stá à ❡sq✉❡r❞❛ ❞♦ ♣♦♥t♦ Y✮ s❡✱ s♦♠❡♥t❡ s❡✱ x<y✳
❉❡ss❛ ❢♦r♠❛✱ ♦s ♣♦♥t♦s ❞❛ s❡♠✐r❡t❛Ð⇀OA ❞✐st✐♥t♦s ❞❡ O ❡stã♦ à ❞✐r❡✐t❛ ❞❡O ❡ ♦s
♣♦♥t♦s ❞❛ s❡♠✐r❡t❛ ♦♣♦st❛ ❛ Ð⇀OA ❡stã♦ à ❡sq✉❡r❞❛ ❞❡O✳
❆ss✐♠✱ ❛ s❡♠✐r❡t❛ Ð⇀OA ❡st❛❜❡❧❡❝❡ ✉♠ s❡♥t✐❞♦ ❞❡ ♣❡r❝✉rs♦ ♥❛ r❡t❛ r✳ ❯♠❛ r❡t❛
s♦❜r❡ ❛ q✉❛❧ ❢♦✐ ❡s❝♦❧❤✐❞❛ ✉♠❛ s❡♠✐r❡t❛ Ð⇀OA ❞❡♥♦♠✐♥❛❞❛ ❡✐①♦ E ❞❡ ♦r✐❣❡♠ O ❡
❞✐r❡çã♦ ✐♥❞✉③✐❞❛ ♣❡❧❛ s❡♠✐r❡t❛Ð⇀OA✳
Pr♦♣♦s✐çã♦ ✷✳✸✳ ❙❡ x ❡ y sã♦ ❝♦♦r❞❡♥❛❞❛s ❞♦s ♣♦♥t♦s X ❡ Y s♦❜r❡ ♦ ❡✐①♦ E✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♥tã♦ d(X, Y)=∣x−y∣✳
❉❡♠♦♥str❛çã♦✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r ♦ r❡s✉❧t❛❞♦ q✉❛♥❞♦ X = Y ♦✉ X = 0 ♦✉ Y = 0✳
❙✉♣♦♥❤❛♠♦s q✉❡X✱Y ❡O s❡❥❛♠ três ♣♦♥t♦s ❞✐st✐♥t♦s✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱
s✉♣♦♥❤❛♠♦s q✉❡ X ❡stá à ❡sq✉❡r❞❛ ❞❡ Y✱ ✐st♦ é✱ x <y✳ ❚❡♠♦s ❡♥tã♦ três ❝❛s♦s ❛
❝♦♥s✐❞❡r❛r✳
❈❛s♦ ✶✳ ❙❡ X ❡ Y ❡stã♦ à ❞✐r❡✐t❛ ❞❛ ♦r✐❣❡♠✳ ■st♦ é✱ 0<x<y✳ ◆❡st❡ ❝❛s♦✱ X ❡stá
❋✐❣✉r❛ ✷✳✹✿ ❈❛s♦ ✶✳ 0<x<y
❡♥tr❡O ❡ Y✱ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ Y ❡st❛r✐❛ ❡♥tr❡ O ❡ X ❡ d(O, Y)=y s❡r✐❛ ♠❡♥♦r
q✉❡d(O, X)❂ ①✳ ▲♦❣♦✱
d(O, Y)=d(O, X) +d(X, Y) ⇐⇒ y=x+d(X, Y)
⇐⇒ d(X, Y)=y−x=∣y−x∣.
❈❛s♦ ✷✳ ❙❡ X ❡stá à ❡sq✉❡r❞❛ ❞❡O ❡ Y ❡stá à ❞✐r❡✐t❛ ❞❡ O✳ ■st♦ é✱ x<y<O✳ ❉❡
♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✱ ✈❡r✐✜❝❛♠♦s q✉❡Y ❡stá ❡♥tr❡X ❡ O✳ ❆ss✐♠✱
d(X, O)=d(X, Y) +d(X, O) ⇐⇒ −x=d(X, Y) −y
⇐⇒ d(X, Y)=y−x=∣y−x∣.
❈❛s♦ ✸✳ ❙❡X ❡stá à ❡sq✉❡r❞❛ ❞❡ O ❡Y ❡stá à ❞✐r❡✐t❛ ❞❡O✳ ■st♦ é✱ x<0<y✳ ◆❡st❡
❋✐❣✉r❛ ✷✳✺✿ ❈❛s♦ ✷✳ x<y<0
❋✐❣✉r❛ ✷✳✻✿ ❈❛s♦ ✸✳ x<0<y
❝❛s♦✱ Y ❡stá ♥❛ s❡♠✐r❡t❛ OA⇀ ❡ X ❡stá ♥❛ s❡♠✐r❡t❛ ♦♣♦st❛ ❛ OA⇀ ✳ P♦rt❛♥t♦✱ O ❡stá
❡♥tr❡X ❡Y ❡✿
d(X, Y)=d(X, O) +d(O, Y) ⇐⇒d(X, Y)=−x+y=y−x=∣y−x∣.
❈♦♠♦ ❡♠ q✉❛❧q✉❡r ❝❛s♦ ♦❜t❡♠♦s q✉❡d(X, Y)=∣x−y∣✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳
P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✸ t❡♠♦s q✉❡✱ s❡ CD é ✉♠ s❡❣♠❡♥t♦ ❞♦ ❡✐①♦E t❛❧ q✉❡ C ❡stá
à ❡sq✉❡r❞❛ ❞❡ D✱ ❡♥tã♦ ♦ ♣♦♥t♦ X ♣❡rt❡♥❝❡ ❛♦ s❡❣♠❡♥t♦ CD s❡✱ ❡ s♦♠❡♥t❡ s❡✱ c≤x≤d✱ ♦♥❞❡ c✱ d ❡ x sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ C✱ D ❡ X✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ■st♦ é✱
❤á ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ♣♦♥t♦s ❞♦ s❡❣♠❡♥t♦ CD ❡ ♦s ♥ú♠❡r♦s
r❡❛✐s ❞♦ ✐♥t❡r✈❛❧♦[c, d]✿ CD←→[c, d]
❊①❡♠♣❧♦ ✷✳✹✳ ❙❡❥❛♠ X ❡ Y ♣♦♥t♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s x ❡ y ♥♦ ❡✐①♦ E✳ ❊♥tã♦✱ ❛
❝♦♦r❞❡♥❛❞❛ ❞♦ ♣♦♥t♦ ♠é❞✐♦ M ❞♦ s❡❣♠❡♥t♦ XY é m= x+y
2 ✳
❋✐❣✉r❛ ✷✳✼✿ P♦♥t♦ ▼é❞✐♦ ❞♦ s❡❣♠❡♥t♦ ❳❨
❙♦❧✉çã♦✿ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛♠♦s q✉❡ X ❡stá à ❡sq✉❡r❞❛ ❞❡ Y ✭ ❝❛s♦ ❡♠ q✉❡ Y ❡stá
à ❡sq✉❡r❞❛ ❞❡ X s❡ tr❛t❛ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✮✳ ❈♦♠♦ ♦ ♣♦♥t♦ ♠é❞✐♦ M ❡stá ❡♥tr❡X
❡Y✱ t❡♠♦s x<m<y✳ ▲♦❣♦✿
d(M, X)=d(M, Y) ⇐⇒ ∣x−m∣=∣y−m∣
⇐⇒ m−x=y−m
⇐⇒ 2m=x+y
⇐⇒ m=x+y
2
✷✳✷ ❉✐stâ♥❝✐❛ ❡♥tr❡ ♣♦♥t♦s ♥♦ ♣❧❛♥♦
❙❡❥❛♠ P = (a, b) ❡ Q= (c, d) ♣♦♥t♦s ❞♦ ♣❧❛♥♦ Π ❞❛❞♦s ♣❡❧❛s s✉❛s ❝♦♦r❞❡♥❛❞❛s
❡♠ r❡❧❛çã♦ ❛ ✉♠ s✐st❡♠❛ ❞❡ ❡✐①♦s ♦rt♦❣♦♥❛✐s OX ❡ OY ❞❛❞♦s✳ ❙❡❥❛ R=(c, b)✱
❆ ❞✐stâ♥❝✐❛ ❞❡ P ❛ Q✱ q✉❡ ❞❡s✐❣♥❛♠♦sd(P, Q)✱ é ❛ ❤✐♣♦t❡♥✉s❛ P Q❞♦ tr✐â♥❣✉❧♦
r❡tâ♥❣✉❧♦ ∆P QR ❞❡ ❝❛t❡t♦s P R ❡ QR✳ ❙❡♥❞♦ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞❡
✉♠ ❡✐①♦ ♠❡❞✐❞❛ ♣❡❧♦ ♠ó❞✉❧♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❞❛s s✉❛s ❝♦♦r❞❡♥❛❞❛s✱ ❛s ♠❡❞✐❞❛s ❞❡ss❡s ❝❛t❡t♦s sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱∣P R∣=∣a−c∣❡ ∣QR∣=∣b−d∣✳ ❉♦ t❡♦r❡♠❛ ❞❡ ♣✐tá❣♦r❛s✱
♦❜t❡♠♦s✿
d(P, Q)=∣P Q∣=√∣P R∣2+∣QR∣2=√(a−c)2+(b−d)2
❆ss✐♠✱ ❛ ❞✐stâ♥❝✐❛ ❞❡P =(a, b)❛Q=(c, d)é ❛ r❛✐③ q✉❛❞r❛❞❛ ❞❛ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s
❞❛s ❞✐❢❡r❡♥ç❛s ❞❛s ❝♦♦r❞❡♥❛❞❛s ❝♦rr❡s♣♦♥❞❡♥t❡s✳
❋✐❣✉r❛ ✷✳✽✿ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦
❊①❡♠♣❧♦ ✷✳✺✳ ❈❛❧❝✉❧❡ ❛ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦A=(−1,2) ❛♦ ♣♦♥t♦ B =(2,−3)✳
❙♦❧✉çã♦✿ ❚❡♠♦s✿
d(A, B)=√(2−(−1))2+(−
3−2)2=√
9+25=√34
❉❡✜♥✐çã♦ ✷✳✻✳ ❖ ❝ír❝✉❧♦ C ❞❡ ❝❡♥tr♦ ♥♦ ♣♦♥t♦ A ♣❡rt❡♥❝❡♥t❡ ❛ Π ❡ r❛✐♦ r>0 é ♦
❝♦♥❥✉♥t♦ q✉❡ ❝♦♥s✐st❡ ❞♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ Π s✐t✉❛❞♦s à ❞✐stâ♥❝✐❛ r ❞♦ ♣♦♥t♦ A✱ ♦✉
s❡❥❛✿
C={P ∈Π ∣ d(P, A)=r}
❙❡A=(a, b)♥✉♠ s✐st❡♠❛ ❞❡ ❡✐①♦s ♦rt♦❣♦♥❛✐sOX ❡OY ♥♦ ♣❧❛♥♦Π✱P =(x, y)∈ C
d(P, A)=r⇐⇒d(P, A)2 =r2
⇐⇒(x−a)2
+(y−b)2 =r2
.
❆ss✐♠✱ ❛ss♦❝✐❛♠♦s ❛♦ ❝ír❝✉❧♦C ❛ ❡q✉❛çã♦(x−a)2+(y−b)2=r2✱ q✉❡ r❡❧❛❝✐♦♥❛ ❛ ❛❜s❝✐ss❛ ❝♦♠ ❛ ♦r❞❡♥❛❞❛ ❞❡ ❝❛❞❛ ✉♠ ❞❡ s❡✉s ♣♦♥t♦s✳ Pr♦♣r✐❡❞❛❞❡s ❣❡♦♠étr✐❝❛s ❞♦ ❝ír❝✉❧♦ sã♦ ❞❡❞✉③✐❞❛s ♣♦r ♠ét♦❞♦s ❛❧❣é❜r✐❝♦s ❡st✉❞❛♥❞♦ s✉❛ ❡q✉❛çã♦✳
❊①❡♠♣❧♦ ✷✳✼✳ ❉❡t❡r♠✐♥❡ ♦ ❝❡♥tr♦ ❡ ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ❞❛❞♦ ♣❡❧❛ ❡q✉❛çã♦✿
C∶x2
+y2
−4x+6y=0
❋✐❣✉r❛ ✷✳✾✿ ❈ír❝✉❧♦ ❈ ❞❡ ❝❡♥tr♦ ❆
❙♦❧✉çã♦✿ ❈♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦s✱ ♦❜t❡♠♦s✿
x2
−4x+y2
+6y=0
(x2−4x+4)+(y2+6y+9) =0+4+9
(x−2)2+(y+3)2=13
P♦rt❛♥t♦✱ ♦ ❝ír❝✉❧♦C t❡♠ ❝❡♥tr♦ ♥♦ ♣♦♥t♦A(2,−3)❡ r❛✐♦ r=√13✳
✷✳✸ ❱❡t♦r❡s
✷✳✸✳✶ ❙❡❣♠❡♥t♦s ❖r✐❡♥t❛❞♦s
P❛r❛ ✐♥✐❝✐❛r ❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ ♣r❡s❡♥t❡ s❡çã♦✱ ✈❛♠♦s ❛ss✉♠✐r q✉❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❝♦♥s✐❞❡r❛❞♦s ✭♣♦♥t♦s✱ r❡t❛s✱ ❡t❝✳✮✱ ♣❡rt❡♥❝❡♠ ❛ ✉♠ ♣❧❛♥♦ ✜①♦✳
❈♦♥s✐❞❡r❡ A ❡ B ♣♦♥t♦s ♥♦ ♣❧❛♥♦✳ ◆♦ s❡❣♠❡♥t♦ AB✱ ♦ ♣♦♥t♦ A é ❝❤❛♠❛❞♦
♦r✐❣❡♠ ❡ ♦ ♣♦♥t♦B ❡①tr❡♠✐❞❛❞❡✳ ▼❡s♠♦ q✉❡ ♦s s❡❣♠❡♥t♦sAB ❡BA r❡♣r❡s❡♥t❡♠ ♦
♠❡s♠♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞♦ ♣❧❛♥♦ ✭♦s ♣♦♥t♦s ❞❛ r❡t❛ q✉❡ ♣❛ss❛♠ ♣♦rA ❡ B q✉❡
❡stã♦ ❡♥tr❡A ❡B✱ ✐♥❝❧✉✐♥❞♦ A❡B✮✱ ❛ s✉❛ ♦r✐❡♥t❛çã♦ ✭✐st♦ é✱ ♦ s❡♥t✐❞♦ ❞❡ ♣❡r❝✉rs♦✮
é ❝♦♥trár✐❛ ✭♦✉ ♦♣♦st❛✱ ✈❡❥❛ ❋✐❣✉r❛ ✷✳✶✵✮✳
❋✐❣✉r❛ ✷✳✶✵✿ ❙❡❣♠❡♥t♦s ❖r✐❡♥t❛❞♦s
❉❡✈❡♠♦s ✜①❛r ❡ ❞✐st✐♥❣✉✐r ❜❡♠ ❞♦✐s ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s✿ ❛ ❞✐r❡çã♦ ❡ ♦ s❡♥t✐❞♦ ✭♦✉ ♦r✐❡♥t❛çã♦✮ ❞❡ ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❞✐r❡çã♦ t❡♠ ❛♣r❡s❡♥t❛çã♦ ♠❛✐s ♦❜❥❡t✐✈❛ ❞♦ q✉❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ s❡♥t✐❞♦✳
❉❡✜♥✐çã♦ ✷✳✽ ✭❉✐r❡çã♦✮✳ ❆ ❞✐r❡çã♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ é ❞❛❞❛ ♣❡❧❛ r❡t❛ q✉❡ ♦ ❝♦♥té♠✳ ❉✐③❡♠♦s q✉❡ ❞♦✐s s❡❣♠❡♥t♦s t❡♠ ♠❡s♠❛ ❞✐r❡çã♦ q✉❛♥❞♦ ❛s r❡t❛s q✉❡ ♦s ❝♦♥tê♠ sã♦ ♣❛r❛❧❡❧❛s ✭♦✉ ❝♦✐♥❝✐❞❡♥t❡s✮✳
❋✐❣✉r❛ ✷✳✶✶✿ ▼❡s♠❛ ❉✐r❡çã♦
◆❛ ❋✐❣✉r❛ ✷✳✶✶✱ ♦s s❡❣♠❡♥t♦sAB ❡ CD tê♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦✱ ♣♦✐s ❛s r❡t❛s q✉❡
♦s ❝♦♥tê♠ sã♦ ♣❛r❛❧❡❧❛s✳ ❖s s❡❣♠❡♥t♦s AB ❡ EF tê♠ ♠❡s♠❛ ❞✐r❡çã♦ ♣♦rq✉❡ ❛s
r❡t❛s q✉❡ ♦s ❝♦♥tê♠ sã♦ ❝♦✐♥❝✐❞❡♥t❡s✱ ✐st♦ é✱ ♦s ♣♦♥t♦sA✱ B✱ E ❡F sã♦ ❝♦❧✐♥❡❛r❡s✳
❉❡✜♥✐çã♦ ✷✳✾ ✭❙❡♥t✐❞♦✮✳ P❛r❛ ❡♥t❡♥❞❡r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞♦✐s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s t❡r❡♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦✱ ❝♦♥s✐❞❡r❡♠♦s ❞♦✐s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s AB ❡ CD
❝♦♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ❡ s❡❥❛♠ rAB✱ ❛ r❡t❛ ❞❡t❡r♠✐♥❛❞❛ ♣♦r A ❡ B✱ ❡ sCD✱ ❛
r❡t❛ ❞❡t❡r♠✐♥❛❞❛ ♣♦r C ❡ D✳ ❉❡s❞❡ q✉❡ rAB ❡ sCD sã♦ ♣❛r❛❧❡❧❛s✱ ❛♥❛❧✐s❡♠♦s
s❡♣❛r❛❞❛♠❡♥t❡ ❞♦✐s ❝❛s♦s✳
❛✮ ❙❡ rAB ♣❛r❛❧❡❧❛ ❛ sCD ❡ ♥ã♦ ❝♦✐♥❝✐❞❡♥t❡s✱ ❞✐③❡♠♦s q✉❡
✭✐✮ AB ❡ CD tê♠ ♠❡s♠♦ s❡♥t✐❞♦ q✉❛♥❞♦ ♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛s AC ❡ BC ♥ã♦
tê♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠✳
✭✐✐✮ AB ❡ CD ♣♦ss✉❡♠ s❡♥t✐❞♦s ♦♣♦st♦s ✭❝♦♥trár✐♦s✮ q✉❛♥❞♦ ♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛s AC ❡ BD ♣♦ss✉❡♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠✳
❜✮ ❙❡ rAB ❡ sCD ❝♦✐♥❝✐❞❡♥t❡s✳ ◆❡ss❡ ❝❛s♦ ❝♦♥s✐❞❡r❡ E ✉♠ ♣♦♥t♦ ❢♦r❛ ❞❡ rAB ❡ ♦
s❡❣♠❡♥t♦sEF ❝♦♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ❞❡AB✳
✭✐✮ ❉✐③❡♠♦s q✉❡ AB ❡ CD ♣♦ss✉❡♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦ q✉❛♥❞♦ CD ❡ EF tê♠ ♦
♠❡s♠♦ s❡♥t✐❞♦✳
✭✐✐✮ ❉✐③❡♠♦s q✉❡ AB ❡ CD tê♠ s❡♥t✐❞♦s ♦♣♦st♦s q✉❛♥❞♦ CD ❡ EF tê♠ s❡♥t✐❞♦s
♦♣♦st♦s✳
❯♠ t❡r❝❡✐r♦ ❝♦♥❝❡✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ ✉♠ ✈❡t♦r é ♦ ❝♦♠♣r✐♠❡♥t♦✳
❉❡✜♥✐çã♦ ✷✳✶✵ ✭▼ó❞✉❧♦✮✳ ❖ ♠ó❞✉❧♦ ♦✉ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦ AB✱ é ❛ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ A ❛♦ ♣♦♥t♦ B✳ ❉❡♥♦t❛♠♦s ♣♦r ∣AB∣ ✭♦✉ ♣♦r d(AB)✮✳
❉❡✜♥✐çã♦ ✷✳✶✶ ✭❙❡❣♠❡♥t♦s ❊q✉✐♣♦❧❡♥t❡s✮✳ ❉✐③❡♠♦s q✉❡ ❞♦✐s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s sã♦ ❡q✉✐♣♦❧❡♥t❡s q✉❛♥❞♦ tê♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦✱ ♦ ♠❡s♠♦ s❡♥t✐❞♦ ❡ ♦ ♠❡s♠♦ ♠ó❞✉❧♦✳ ❙❡ ♦s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦sAB ❡ CD sã♦ ❡q✉✐♣♦❧❡♥t❡s✱ ❡s❝r❡✈❡♠♦s AB ≡ CD✳
❋✐❣✉r❛ ✷✳✶✷✿ ❙❡♥t✐❞♦s
❈♦♥✈❡♥çõ❡s
P❛r❛ ❡✈✐t❛r ❛♠❜✐❣✉✐❞❛❞❡s✱ ❢❛r❡♠♦s ❛❧❣✉♠❛s ❝♦♥✈❡♥çõ❡s✿
❼ ❯♠ s❡❣♠❡♥t♦ AB ♦♥❞❡ A =B é ❝❤❛♠❛❞♦ ✉♠ s❡❣♠❡♥t♦ ♥✉❧♦✳ ❖s s❡❣♠❡♥t♦s
♥✉❧♦s tê♠ ♠ó❞✉❧♦ ③❡r♦ ❡ ♥ã♦ tê♠ ❞✐r❡çã♦ ♥❡♠ s❡♥t✐❞♦✳
❼ ❙❡ A é ✉♠ ♣♦♥t♦ ❞♦ ♣❧❛♥♦✱ ❞❡s✐❣♥❛♠♦s ♣♦rAA ♦ s❡❣♠❡♥t♦ ♥✉❧♦ ❞❡ ♦r✐❣❡♠ ❡
❡①tr❡♠✐❞❛❞❡ A✳
❼ ❚♦❞♦s ♦s s❡❣♠❡♥t♦s ♥✉❧♦s sã♦ ❝♦♥s✐❞❡r❛❞♦s ❡q✉✐♣♦❧❡♥t❡s✳
Pr♦♣♦s✐çã♦ ✷✳✶✷✳ ❙❡AB é ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ❡ C é ✉♠ ♣♦♥t♦ ❞♦ ♣❧❛♥♦✱ ❡♥tã♦
❡①✐st❡ ❛♣❡♥❛s ✉♠ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ❝♦♠ ♦r✐❣❡♠ ❡♠ C ❡q✉✐♣♦❧❡♥t❡ ❛ AB.
❉❡♠♦♥str❛çã♦✳ ❉❡✈❡♠♦s ❞❡t❡r♠✐♥❛r ✉♠ ♣♦♥t♦D ♥♦ ♣❧❛♥♦ ❞❡ ♠♦❞♦ q✉❡
AB≡CD.
■ss♦ s✐❣♥✐✜❝❛ q✉❡ ♦s s❡❣♠❡♥t♦s AB ❡ CD ❞❡✈❡♠ t❡r ❛ ♠❡s♠❛ ❞✐r❡çã♦✱ ♦ ♠❡s♠♦
s❡♥t✐❞♦ ❡ ♦ ♠❡s♠♦ ♠ó❞✉❧♦✳ ❉❡♥♦t❡♠♦s ♣♦rr❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦rA❡B✱ ❛♥❛❧✐s❡♠♦s
s❡♣❛r❛❞❛♠❡♥t❡ ♦ q✉❡ ❛❝♦♥t❡❝❡ q✉❛♥❞♦C ♥ã♦ ♣❡rt❡♥❝❡ ❛ r❡t❛r❡ q✉❛♥❞♦ C♣❡rt❡♥❝❡
❛r✳ ❊st✉❞❛r❡♠♦s ❞♦✐s ❝❛s♦s✳
❈❛s♦ ✶✿ C♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛r✳ ◆❡st❡ ❝❛s♦✱ ❡①✐st❡ ❛♣❡♥❛s ✉♠❛ r❡t❛s♣❛r❛❧❡❧❛ ❛rq✉❡
♣❛ss❛ ♣❡❧♦ ♣♦♥t♦C✳ ❙❡❥❛S♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦C❡ r❛✐♦∣AB∣✳ ❆ r❡t❛ q✉❡ ♣❛ss❛ ♣♦rA
❡C ❞✐✈✐❞❡ ♦ ♣❧❛♥♦ ❡♠ ❞♦✐s s❡♠✐✲♣❧❛♥♦s✱ ✉♠ ❞♦s q✉❛✐s✱ q✉❡ ❞❡s✐❣♥❛♠♦sPB✱ ❝♦♥té♠
♦ ♣♦♥t♦ B✳ ❖ ❝ír❝✉❧♦ S ✐♥t❡rs❡❝t❛ s ❡♠ ❡①❛t❛♠❡♥t❡ ❞♦✐s ♣♦♥t♦s ❞✐❛♠❡tr❛❧♠❡♥t❡
♦♣♦st♦s✱ ✉♠ ❞♦s q✉❛✐s✱ q✉❡ ❝❤❛♠❛♠♦sD✱ ❡stá ❝♦♥t✐❞♦ ❡♠ PB✳ P♦rt❛♥t♦✱ ♣❡❧❛ ❢♦r♠❛
❝♦♠♦ ❢♦✐ ♦❜t✐❞♦ ♦ ♣♦♥t♦D✱ ♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ CD é ❡q✉✐♣♦❧❡♥t❡ ❛A✳
❈❛s♦ ✷✿ C ♣❡rt❡♥❝❡♥t❡ ❛r✳ ◆❡st❡ ❝❛s♦✱ ♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦C ❡ r❛✐♦∣AB∣✱ ✐♥t❡rs❡❝t❛
❛ r❡t❛ r ❡♠ ❞♦✐s ♣♦♥t♦s ❞✐❛♠❡tr❛❧♠❡♥t❡ ♦♣♦st♦s✳ ▼❛s✱ ❛♣❡♥❛s ✉♠ ❞❡❧❡s✱ q✉❡
❝❤❛♠❛r❡♠♦s D✱ é t❛❧ q✉❡ AB ❡ CD tê♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦✳ ▲♦❣♦ AB ❡ CD sã♦
❡q✉✐♣♦❧❡♥t❡s✱ ♣♦✐s tê♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦ ❡ ♦s s❡✉s ♠ó❞✉❧♦s sã♦ ✐❣✉❛✐s✳
❋✐❣✉r❛ ✷✳✶✸✿ C ♥ã♦ ♣❡rt❡♥❝❡ ❛r
❋✐❣✉r❛ ✷✳✶✹✿ C ♣❡rt❡♥❝❡ ❛ r
✷✳✸✳✷ ❙✐st❡♠❛ ❞❡ ❈♦♦r❞❡♥❛❞❛s
❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❛ss♦❝✐❛r ❝❛❞❛ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ ❛ ✉♠ ♣❛r ♦r❞❡♥❛❞♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ♣r❡❝✐s❛♠♦s ❞❛ ❣❛r❛♥t✐❛ ❞❡ ✉♠ r❡s✉❧t❛❞♦ s♦❜r❡ ♣♦♥t♦ ♠é❞✐♦ ❞❡ s❡❣♠❡♥t♦✳ ▲❡♠❜r❛♠♦s q✉❡ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ AB é ♦ ♣♦♥t♦ M t❛❧
q✉❡∣AM∣=∣M B∣✳
Pr♦♣♦s✐çã♦ ✷✳✶✸✳ ❖s s❡❣♠❡♥t♦s AB ❡ CD sã♦ ❡q✉✐♣♦❧❡♥t❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ AD
❡ BC ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ♣♦♥t♦ ♠é❞✐♦✳
❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s s❡♣❛r❛r ❛ ❥✉st✐✜❝❛t✐✈❛ ❡♠ ❞♦✐s ❝❛s♦ ♣♦ssí✈❡✐s✿
❈❛s♦ ✶✿ ❆ ❞✐r❡çã♦AB ♥ã♦ ❝♦✐♥❝✐❞❡♥t❡ ❝♦♠ ❛ ❞✐r❡çã♦ ❞❡CD✳ ❙❡AB ≡CD ❡♥tã♦ ♦s
s❡❣♠❡♥t♦s ❡stã♦ ❝♦♥t✐❞♦s ❡♠ r❡t❛s ♣❛r❛❧❡❧❛s ♥ã♦ ❝♦✐♥❝✐❞❡♥t❡s ❡✱ ❝♦♠♦ tê♠ ♦ ♠❡s♠♦ ♠ó❞✉❧♦ ❡ ♦ ♠❡s♠♦ s❡♥t✐❞♦✱ ♦ q✉❛❞r✐❧át❡r♦ ABDC é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡✱ ❛s s✉❛s
❞✐❛❣♦♥❛✐s AD ❡ BC✱ ❝♦rt❛♠✲s❡ ♠✉t✉❛♠❡♥t❡ ❛♦ ♠❡✐♦✱ ♦✉ s❡❥❛✱ ♥♦ ♣♦♥t♦ ♠é❞✐♦ ✭✈❡r
❋✐❣✉r❛ ✷✳✶✺✮✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡AD❡BC tê♠ ♦ ♠❡s♠♦ ♣♦♥t♦ ♠é❞✐♦ ❡♥tã♦ABDC
é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ▲♦❣♦ AB ❡ CD tê♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦✱ ♦ ♠❡s♠♦ ♠ó❞✉❧♦ ❡ ❛
♠❡s♠❛ ❞✐r❡çã♦✳ P♦rt❛♥t♦ AB ≡CD✳
❈❛s♦ ✷✿ ❆ ❞✐r❡çã♦AB ❝♦✐♥❝✐❞❡♥t❡ ❝♦♠ ❛ ❞✐r❡çã♦ ❞❡CD✳ ❈♦♥s✐❞❡r❡r❛ r❡t❛ q✉❡
❝♦♥té♠ ♦s ♣♦♥t♦sA, B, C ❡D♣r♦✈✐❞❛ ❞❡ ✉♠❛ ♦r✐❡♥t❛çã♦ ❡ ✉♠❛ ♦r✐❣❡♠O ❡s❝♦❧❤✐❞❛s
❞❡ ♠♦❞♦ q✉❡ B ❡st❡❥❛ à ❞✐r❡✐t❛ ❞❡ A ✭✈❡r ❋✐❣✉r❛ ✷✳✶✻✮ ❡ s❡❥❛♠ a=∣OA∣✱ b =∣OB∣✱ c=∣OC∣ ❡ d=∣OD∣ ✭♥ú♠❡r♦s ❝❤❛♠❛❞♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞❡ A✱ B✱ C ❡ D ♥❛ r❡t❛ r
❡♠ r❡❧❛çã♦ ❛ ✉♠❛ ✉♥✐❞❛❞❡ ❡s❝♦❧❤✐❞❛✮✳
❙❡AB ≡CD✱ ❡♥tã♦ AB ❡ CD tê♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦ ❡ ∣AB∣=∣CD∣✳ ▲♦❣♦✱
a<b ❡ c<d ❡ b−a=d−c.
❋✐❣✉r❛ ✷✳✶✺✿ P❛r❛❧❡❧♦❣r❛♠♦ ❆❇❈❉
❋✐❣✉r❛ ✷✳✶✻✿ AB ❡ CD ❝♦♠ ❞✐r❡çõ❡s ❝♦✐♥❝✐❞❡♥t❡s
❆ss✐♠✱
b−a=d−c⇐⇒a+d=b+c ⇐⇒ a+d
2 =
b+c
2 . ✭✷✳✶✮
❆❣♦r❛ ♦❜s❡r✈❡ q✉❡ ❛ ❝♦♦r❞❡♥❛❞❛ ❞♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ AD é (a+d)/2 ❡ ❛ ❞❡ BC é
(b+c)/2✱ q✉❡ ♣♦r ✭✷✳✶✮ ❝♦♥❝❧✉í♠♦s s❡r❡♠ ✐❣✉❛✐s✳ P♦rt❛♥t♦✱ AD ❡BC tê♠ ♦ ♠❡s♠♦
♣♦♥t♦ ♠é❞✐♦✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡AD ❡ BC tê♠ ♦ ♠❡s♠♦ ♣♦♥t♦ ♠é❞✐♦✱ ❡♥tã♦
a+d
2 =
b+c
2 .
❆ss✐♠✱
a+d=b+c ⇐⇒ b−a=d−c. ✭✷✳✷✮
❉❡ss❛ ❢♦r♠❛✱ b−a ❡ d−c tê♠ ♠❡s♠♦ s✐♥❛❧ ❡ ♠ó❞✉❧♦s ✐❣✉❛✐s✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡
♦s s❡❣♠❡♥t♦s ❝♦❧✐♥❡❛r❡s AB ❡ CD tê♠ ♦ ♠❡s♠♦ s❡♥t✐❞♦ ❡ ♦ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦
∣AB∣ = ∣CD∣✳ P♦rt❛♥t♦ AB ≡CD✳
❘❡♣r❡s❡♥t❛çã♦ ❈❛rt❡s✐❛♥❛
❈♦♥s✐❞❡r❡♠♦s ✉♠ ♣❧❛♥♦ ❞♦ ❡s♣❛ç♦ ❡ ♥❡❧❡ ❝♦♥s✐❞❡r❡♠♦s ✉♠ ♣♦♥t♦ ✜①❛❞♦ O✱ q✉❡
❝❤❛♠❛r❡♠♦s ❞❡ ❖r✐❣❡♠✱ ❡ ❞✉❛s r❡t❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❡♥tr❡ s✐✱ ✐♥t❡❝❡♣t❛♥❞♦✲s❡ ❡♠O✱
