P❘❖❋▼❆❚
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P❛r❛♥á ✽✶✺✸✶✲✾✾✵✱ ❈✉r✐t✐❜❛✱ P❘❇r❛③✐❧Pr❡♣r✐♥t P❘❖❋▼❆❚ ✶ ✭✷✵✶✹✮ ✷✾ ❞❡ ❛❣♦st♦✱ ✷✵✶✹
❉✐s♣♦♥í✈❡❧ ✈✐❛ ■◆❚❊❘◆❊❚✿ ❤tt♣✿✴✴✇✇✇✳♠❛t✳✉❢♣r✳❜r
❈ír❝✉❧♦s ♥♦tá✈❡✐s ❛ss♦❝✐❛❞♦s ❛ ✉♠
tr✐â♥❣✉❧♦
♣♦r
❈ír❝✉❧♦s ♥♦tá✈❡✐s ❛ss♦❝✐❛❞♦s ❛ ✉♠
tr✐â♥❣✉❧♦
●❧❡❜✐s♦♥ ❞❡ ❙♦✉③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯❋P❘
✵✶✾✵✽✶✲✾✾✵✱ ❈✉r✐t✐❜❛✱ P❘
❇r❛③✐❧
❡✲♠❛✐❧✿ ♣r♦❢❣❧❡❜❅❣♠❛✐❧✳❝♦♠
✷✾ ❞❡ ❛❣♦st♦ ❞❡ ✷✵✶✹
❘❡s✉♠♦
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❝❧áss✐❝♦s ❛ r❡s♣❡✐t♦ ❞❡ tr✐â♥❣✉❧♦s✱ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❞❡✜♥✐çõ❡s ❡ t❡♦r❡♠❛s✳ ▼♦str❛r ❛s r❡❧❛çõ❡s q✉❡ r❡♣r❡s❡♥t❛♠ ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❋ór♠✉❧❛ ❞❡ ❍❡rã♦✳ ❈♦♥str✉✐r ❡ ❝❛❧❝✉❧❛r ♦s r❛✐♦s ❞♦s ❝ír❝✉❧♦s ❡①✲✐♥s❝r✐t♦s✱ ✐♥s❝r✐t♦ ❡ ❝✐r❝✉♥s❝r✐t♦ ❛ ✉♠ tr✐â♥❣✉❧♦ ❡✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛ r❡❧❛çã♦ ❡①✐st❡♥t❡ ❡♥tr❡ ❛s ♠❡❞✐❞❛s ❞♦s r❛✐♦s ❞❡ss❡s ❝ír❝✉❧♦s✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❈ír❝✉❧♦s t❛♥❣❡♥t❡s ♥♦tá✈❡✐s❀ r❛✐♦s ❞❡ ❝ír❝✉❧♦s ❡①✲✐♥s❝r✐t♦s✱ ✐♥s❝r✐t♦ ❡ ❝✐r❝✉♥s❝r✐t♦✳
❆❜str❛❝t
❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ ♣r❡s❡♥t s♦♠❡ ❝❧❛ss✐❝ ❝♦♥❝❡♣ts ❛❜♦✉t tr✐❛♥✲ ❣❧❡s✱ t❤❡✐r ♣r♦♣❡rt✐❡s✱❞❡✜♥✐t✐♦♥s ❛♥❞ t❤❡♦r❡♠s✳ ❲❡ ♣r♦✈❡ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❛r❡❛ ♦❢ ❛♥② tr✐❛♥❣❧❡ s✉❝❤ ❛s ❍❡r❛♦ ❢♦r♠✉❧❛❀ ✇❡ ❝♦♥str✉❝t ❛♥❞ ❝♦♠♣✉t❡ t❤❡ r❛❞✐✐ ♦❢ ✐♥s❝r✐❜❡❞ ❛♥❞ ❝✐r❝✉♥s❝r✐❜❡❞ ❝✐r❝❧❡s ✱ ❡①❝✐r❝❧❡s✱✐♥ ♣❛rt✐❝✉❧❛r✱t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ r❛❞✐✐ ♦❢ t❤♦s❡✳
❑❡②✇♦r❞s✿ ❘❡♠❛r❦❛❜❧❡ t❛♥❣❡♥t ❝✐r❝❧❡s❀ r❛②s ❡①❝✐r❝❧❡s✱ ✐♥s❝r✐❜❡❞ ❛♥❞ ❝✐r✲ ❝✉♠s❝r✐❜❡❞✳
■♥tr♦❞✉çã♦
❊st❡ é ✉♠ tr❛❜❛❧❤♦ ♦♣♦rt✉♥✐③❛❞♦ ♣❡❧♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋P❘✱ ❛tr❛✈és ❞♦ ♣r♦❣r❛♠❛ P❘❖❋▼❆❚✱ ❝♦♠ ❛ ♦r✐❡♥t❛çã♦ ❞♦ Pr♦❢✳ ❉r✳ ❆❧❞❡♠✐r ❏♦sé ❞❛ ❙✐❧✈❛ P✐♥t♦✳ Pr♦❝✉r❛♠♦s ❞❡♠♦♥str❛r ❝♦♠ ✉♠ ❝❡rt♦ r✐❣♦r ❛❧❣✉♥s ❞♦s t❡♦r❡♠❛s✱ ♣r♦♣♦s✐çõ❡s ❡ ❝♦r♦❧ár✐♦s ❞❛ ❣❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛ ♣❧❛♥❛✱ s❡♥❞♦ ❛♣r❡s❡♥t❛❞♦s ❞❡ ♠❛♥❡✐r❛ ❣r❛❞❛t✐✈❛ ♣❛r❛ s❡r❡♠ ✉t✐❧✐③❛❞♦s ❡♠ t❡♦r❡♠❛ ❞♦s ❝❛♣ít✉❧♦s s✉❜s❡q✉❡♥t❡s✱ ♦ q✉❛❧ ❝✐t❛r❡♠♦s ❛ s❡❣✉✐r✳ ◆♦ ❝❛♣ít✉❧♦ ✶ ✐♥✐❝✐❛♠♦s ❝♦♠ ♦s ♣♦st✉❧❛❞♦s ❞❡ ❞✐stâ♥❝✐❛✱ r❡t❛s✱ ♣❧❛♥♦s ❡ ♠❡❞✐çã♦ ❞❡ â♥❣✉❧♦s✳ ❊♠ s❡❣✉✐❞❛✱ ❞❡✜♥✐♠♦s tr✐â♥❣✉❧♦✱ ❝❧❛ss✐✜❝❛çã♦ ❡♥q✉❛♥t♦ ❛♦s ❧❛❞♦s❀ ♦ ♣♦st✉❧❛❞♦ ♣❛r❛ ❛ ❝♦♥❣rê♥❝✐❛ ❞❡ tr✐â♥❣✉❧♦s ✭▲❆▲✮ ❜❡♠ ❝♦♠♦ ♦s t❡♦r❡♠❛s✿ ALA✱LLL✱LAAo ❡ ♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s
r❡tâ♥❣✉❧♦s❀ ❝✐t❛♠♦s ❛❧❣✉♠❛s ♣r♦♣♦s✐çõ❡s ❝❧áss✐❝❛s✱ ❝♦♠♦ ♦ t❡♦r❡♠❛ ❞❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ â♥❣✉❧♦ ❡①t❡r♥♦ ❡ ✉♠ ❝♦r♦❧á✲ r✐♦ q✉❡ ❥✉♥t❛♠❡♥t❡ ❝♦♠ P♦st✉❧❛❞♦ ❞❛s ♣❛r❛❧❡❧❛s ✭❖ 5.o P♦st✉❧❛❞♦ ❞❡ ❊✉❝❧✐❞❡s✮
♠♦str❛ q✉❡ ♦s â♥❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s sã♦ ✐❣✉❛✐s❀ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ ❞❡✜♥✐♠♦s ❝ír❝✉❧♦✱ ❜✐ss❡tr✐③ ❡ ♠❡❞✐❛tr✐③✳
❏á ♥♦ ❝❛♣ít✉❧♦ ✷✱ ✐♥✐❝✐❛♠♦s ❝♦♠ ♦ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ s♦❜r❡ ♣r♦♣♦r❝✐♦♥❛❧✐✲ ❞❛❞❡ ❡ ❛ r❡❝í♣r♦❝❛✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛♠♦s ♦s t❡♦r❡♠❛s ❞❡ s❡♠❡❧❤❛♥ç❛ ❡♥tr❡ tr✐â♥❣✉❧♦s ✭▲▲▲✱ ❆❆ ❡ ▲❆▲✮✳ ❊♠ s❡❣✉✐❞❛ ❞❡✜♥✐♠♦s ár❡❛✱ s❡✉s ♣♦st✉❧❛❞♦s ❡ ❛s ♣r♦♣♦s✐çõ❡s ❞❡ ár❡❛ ♣❛r❛✿ q✉❛❞r❛❞♦ ❡ r❡tâ♥❣✉❧♦ ✭❡st❡s ❞❡♠♦♥str❛❞♦s ❡①❛✉st✐✈❛✲ ♠❡♥t❡✿ ♣❛r❛ ♦s ♥❛t✉r❛✐s✱ r❛❝✐♦♥❛✐s ❡ r❡❛✐s✮✳ ❈✐t❛♠♦s t❛♠❜é♠ ♣r♦♣♦s✐çõ❡s ♣❛r❛✿ ♣❛r❛❧❡❧♦❣r❛♠♦✱ tr✐â♥❣✉❧♦✱ ❧♦s❛♥❣♦ ❡ tr❛♣é③✐♦✳ Pr♦✈❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ✈✐❛ ár❡❛s ✉t✐❧✐③❛♥❞♦ ✉♠ ❝♦r♦❧ár✐♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ár❡❛ ❞❡ tr✐â♥❣✉❧♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ❞❛r❡♠♦s ✐♥í❝✐♦ ❛♦ ♣r♦♣ós✐t♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ✐♥✐❝✐❛♥❞♦ ❝♦♠ ❛ ❝♦♥s✲ tr✉çã♦ ❞♦ ❝ír❝✉❧♦ ✐♥s❝r✐t♦ ❛ ✉♠ tr✐â♥❣✉❧♦ ABC ❡ ❛ ár❡❛ ❡♠ ❢✉♥çã♦ ❞♦ s❡♠✐♣❡✲
rí♠❡tr♦ ✭p✮ ❡ ♦ r❛✐♦ ✭r✮ ❞❡st❡ ❝ír❝✉❧♦✱ ♦✉ s❡❥❛✱ ➪r❡❛=p·r ✳ ❊♠ s❡❣✉✐❞❛✱ ❝♦♥s✲
tr✉í♠♦s ♦s ❝ír❝✉❧♦s ❡①✲✐♥s❝r✐t♦s ❛ ✉♠ tr✐â♥❣✉❧♦ ABC ❞❡ r❛✐♦s ra, rb ❡ rc ❡ ár❡❛ (p−a)ra = (p−b)rb = (p−c)rc✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ ❝♦♥str✉í♠♦s ♦ ❝ír❝✉❧♦ ❝✐r❝✉♥s✲
❝r✐t♦ ❛♦ tr✐â♥❣✉❧♦ABC ❞❡ r❛✐♦R ❞❡ ár❡❛ abc
4R✳ ❖ ❝á❧❝✉❧♦ ❞❡ ✉♠❛s ❞❛s ❛❧t✉r❛s ❞❡
✉♠ tr✐â♥❣✉❧♦ ❡♠ ❢✉♥çã♦ ❞♦s ❧❛❞♦s✱ ♣❛r❛ ❡♥tã♦ ❞❡❞✉③✐r♠♦s ❛ ❝♦♥❤❡❝✐❞❛ ❢ór♠✉❧❛ ❞❡ ❍❡rã♦✿ pp(p−a)(p−b)(p−c)✳ ❉❡♠♦♥str❛♠♦s t❛♠❜é♠ ❛ ❧❡✐ ❞♦s s❡♥♦s ❡ ❝♦s✲
s❡♥♦s✳ ❊✱ ♣♦r ✜♠ ❞❡♠♦♥str❛♠♦s ❛s ❢ór♠✉❧❛s q✉❡ r❡❧❛❝✐♦♥❛♠ ♦s r❛✐♦s ❞♦s ❝ír❝✉❧♦s ❡①✲✐♥s❝r✐t♦s ❝♦♠ ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ✐♥s❝r✐t♦ 1
r = 1 ra +
1 rb +
1
rc ❡ ❝♦♠ ♦ r❛✐♦s ❞♦s
❝ír❝✉❧♦s✿ 4R=ra+rb+rc −r✳
◆❛ ú❧t✐♠❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡①❡rí❝✐♦ ♣❛r❛ ❝♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ✉s❛♥❞♦ ♦ ●❡♦❣❡❜r❛✱ ✈❡rsã♦ ✹✳✷✳ ❖ ♣r♦♣ós✐t♦ ❞❡st❡ tr❛❜❛❧❤♦ é✱ ❛❧é♠ ❞❡ ❞❛r ê♥❢❛s❡ às ♣r♦♣♦s✐çõ❡s ❡ t❡♦r❡♠❛s ❞❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛✱ ❞✐✈✉❧❣❛r r❡s✉❧t❛❞♦s ✐♥t❡r❡ss❛♥t❡s q✉❡ ❡❧❛✱ ❛ ❣❡♦♠❡tr✐❛✱ ❛✐♥❞❛ ♥♦s ♣r♦♣♦r❝✐♦♥❛✳
✶ P♦st✉❧❛❞♦s✱ ❞❡✜♥✐çõ❡s
◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦s ♣♦st✉❧❛❞♦s ❡ ❞❡✜♥✐çõ❡s q✉❡ ❢✉♥❞❛♠❡♥t❛♠ ❛ ●❡♦♠❡tr✐❛ P❧❛♥❛✳
✶✳✶ ❉✐stâ♥❝✐❛✱ r❡t❛ ❡ ♣❧❛♥♦
P♦st✉❧❛❞♦ ✳✶✳ ✭❖ P♦st✉❧❛❞♦ ❞❛ ❉✐stâ♥❝✐❛✮ ❆ t♦❞♦ ♣❛r ❞❡ ♣♦♥t♦s ❞✐st✐♥t♦s ❝♦r✲ r❡s♣♦♥❞❡ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ ♣♦s✐t✐✈♦✳
❉❡✜♥✐çã♦ ✳✷✳ ❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s é ♦ ♥ú♠❡r♦ ❞❛❞♦ ♣❡❧♦ ♣♦st✉❧❛❞♦ ✳✶ ❙❡ ♦s ❞♦✐s ♣♦♥t♦s sã♦ ❞❡♥♦t❛❞♦s ♣♦rR ❡S✱ ❛ ❞✐stâ♥❝✐❛ s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦rRS✳
❆❞♠✐t✐♠♦s ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡R ❡S s❡r❡♠ ♦ ♠❡s♠♦ ♣♦♥t♦✳ ◆❡st❡ ❝❛s♦✱RS = 0✳ ❆ ❞✐stâ♥❝✐❛ é ❞❡✜♥✐❞❛ ♣❛r❛ ✉♠ ♣❛r ❞❡ ♣♦♥t♦s ❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ♦r❞❡♠ ❡♠ q✉❡
❡ss❡s ♣♦♥t♦s sã♦ ♠❡♥❝✐♦♥❛❞♦s✳ P♦rt❛♥t♦✱ s❡♠♣r❡ t❡♠♦sRS =SR✳
P♦st✉❧❛❞♦ ✳✸✳ ✭❖ ♣♦st✉❧❛❞♦ ❞❛ ❘é❣✉❛✮ ❖s ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛ ♣♦❞❡♠ s❡r ♣♦st♦s ❡♠ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❝♦♠ ♦s ♥ú♠❡r♦s r❡❛✐s✱ ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡
✭✶✮ ❛ ❝❛❞❛ ♣♦♥t♦ ❞❛ r❡t❛ ❝♦rr❡s♣♦♥❞❡ ❡①❛t❛♠❡♥t❡ ✉♠ ♥ú♠❡r♦ r❡❛❧❀ ✭✷✮ ❛ ❝❛❞❛ ♥ú♠❡r♦ r❡❛❧ ❝♦rr❡s♣♦♥❞❡ ❡①❛t❛♠❡♥t❡ ✉♠ ♣♦♥t♦ ❞❛ r❡t❛❀ ❡
✭✸✮ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s q✉❛✐sq✉❡r é ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❞♦s ♥ú♠❡r♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✳
P♦st✉❧❛❞♦ ✳✹✳ ✭❖ P♦st✉❧❛❞♦ ❞❛ ❝♦❧♦❝❛çã♦ ❞❛ ❘é❣✉❛✮ ❉❛❞♦s ❞♦✐s ♣♦♥t♦s R ❡ S
♥✉♠❛ r❡t❛✱ ♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞♦ ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡ ❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ R s❡❥❛ ③❡r♦ ❡ ❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ S s❡❥❛ ♣♦s✐t✐✈❛✳
❉❡✜♥✐çã♦ ✳✺✳ ❇ ❡stá ❡♥tr❡ A ❡ C s❡ A✱ B ❡ C sã♦ ♣♦♥t♦s ❞✐st✐♥t♦s ❞❡ ✉♠❛ r❡t❛
❡ AC =AB+BC✳
❖ s✐❣♥✐✜❝❛❞♦ ❞❛ ♣❛❧❛✈r❛ s❡ é s❡r ❡q✉✐✈❛❧❡♥t❡ ❛✳
P♦st✉❧❛❞♦ ✳✻✳ ✭❖ P♦st✉❧❛❞♦ ❞❛ ❘❡t❛✮ P❛r❛ ❝❛❞❛ ♣❛r ❞❡ ♣♦♥t♦s ❞✐st✐♥t♦s ❡①✐st❡ ❡①❛t❛♠❡♥t❡ ✉♠❛ r❡t❛ q✉❡ ♦s ❝♦♥tê♠✳
❋✐❣✉r❛ ✶✿ ❆ r❡t❛ AB←→
❉❡✜♥✐çã♦ ✳✼✳ P❛r❛ ❞♦✐s ♣♦♥t♦s q✉❛✐sq✉❡rA ❡B✱ ♦ s❡❣♠❡♥t♦ AB é ♦ ❝♦♥❥✉♥t♦ ❞❡
t♦❞♦s ♦s ♣♦♥t♦s q✉❡ ❡stã♦ ❡♥tr❡ A ❡ B✳ ❖s ♣♦♥t♦s A ❡ B sã♦ ❝❤❛♠❛❞♦s ❡①tr❡♠♦s
❞❡ AB✳
❉❡✜♥✐çã♦ ✳✽✳ ❖ ♥ú♠❡r♦ AB é ❝❤❛♠❛❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ AB✳
❯♠❛ ✜❣✉r❛ ♣❛r❛ ❛ s❡♠✐rr❡t❛ AB−→✿
❚❡♦r❡♠❛ ✳✾✳ ❙❡❥❛ AB−→ ✉♠❛ s❡♠✐rr❡t❛ ❡ x ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦✳ ❊♥tã♦ ❡①✐st❡
❡①❛t❛♠❡♥t❡ ✉♠ ♣♦♥t♦ P ❞❡
−→
AB t❛❧ q✉❡ AP =x✳
❉❡♠♦♥str❛çã♦✳
P❡❧♦ ♣♦st✉❧❛❞♦ ❞❡ ❈♦❧♦❝❛çã♦ ❞❛ ❘é❣✉❛✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ r❡t❛ AB←→ ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡ ❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ A s❡❥❛ ✵ ❡ AP = |x−0|= |x|= x✳ ❈♦♠♦ s♦♠❡♥t❡ ✉♠ ♣♦♥t♦ ❞❛ s❡♠✐rr❡t❛ ❡♠ ❝♦♦r❞❡♥❛❞❛ x✱ s♦♠❡♥t❡ ✉♠ ♣♦♥t♦ ❞❛ s❡♠✐rr❡t❛ ❡stá ❛ ✉♠❛ ❞✐stâ♥❝✐❛ x ❞❡A✳
❉❡✜♥✐çã♦ ✳✶✵✳ ❯♠ ♣♦♥t♦ B é ❞✐t♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ AC s❡ B ❡stá
❡♥tr❡ A ❡ C ❡ AB =BC✳
❚❡♦r❡♠❛ ✳✶✶✳ ❚♦❞♦ ♦ s❡❣♠❡♥t♦ t❡♠ ❡①❛t❛♠❡♥t❡ ✉♠ ♣♦♥t♦ ♠é❞✐♦✳
❉❡♠♦♥str❛çã♦✳ ◗✉❡r❡♠♦s ✉♠ ♣♦♥t♦ s❛t✐s❢❛③❡♥❞♦ ❛s ❞✉❛s ❝♦♥❞✐çõ❡s✿ AB +BC =AC
❡ AB =BC✳ ❘❡❧❛❝✐♦♥❛♥❞♦ ❡st❛s ❞✉❛s ❡q✉❛çõ❡s✱ t❡r❡♠♦s✿ AB = AC
2 ✳ ▲♦❣♦✱ ❤á
✉♠ ♣♦♥t♦ B ❞❛ s❡♠✐rr❡t❛
−→
AC q✉❡ ❡stá ❛ ✉♠❛ ❞✐stâ♥❝✐❛ AC
2 ✳ ❊♥tã♦ AC t❡♠
❡①❛t❛♠❡♥t❡ ✉♠ ♣♦♥t♦ ♠é❞✐♦✳
P♦st✉❧❛❞♦ ✳✶✷✳ ✭❖ P♦st✉❧❛❞♦ ❞♦ P❧❛♥♦✮ ❚rês ♣♦♥t♦s q✉❛✐sq✉❡r ❝♦♣❧❛♥❛r❡s ❡ ♥ã♦ ❝♦❧✐♥❡❛r❡s ❞❡t❡r♠✐♥❛♠ ✉♠ ♣❧❛♥♦✳
P♦st✉❧❛❞♦ ✳✶✸✳ ✭❖ ♣♦st✉❧❛❞♦ ❞❛ s❡♣❛r❛çã♦ ❞♦ ♣❧❛♥♦✮ ❉❛❞♦s ✉♠❛ r❡t❛ ❡ ✉♠ ♣❧❛♥♦ q✉❡ ❛ ❝♦♥tê♠✱ ♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ ♥ã♦ ♣❡rt❡♥❝❡♠ à r❡t❛ ❢♦r♠❛♠ ❞♦✐s ❝♦♥❥✉♥t♦s t❛✐s q✉❡✿
✭✶✮ ❝❛❞❛ ✉♠ ❞♦s ❝♦♥❥✉♥t♦s é ❝♦♥✈❡①♦✱ ❡ ✭✷✮ s❡ P ♣❡rt❡♥❝❡ ❛ ✉♠ ❞♦s ❝♦♥❥✉♥t♦s ❡ Q ❛♦ ♦✉tr♦✱ ❡♥tã♦ P Q ✐♥t❡r❝❡♣t❛ ❛ r❡t❛✳
✶✳✷ ➶♥❣✉❧♦s
❉❡✜♥✐çã♦ ✳✶✹✳ ➶♥❣✉❧♦ é ❛ ✜❣✉r❛ ❢♦r♠❛❞❛ ♣♦r ❞✉❛s s❡♠✐rr❡t❛s ❞❡ ♠❡s♠❛ ♦r✐❣❡♠✳ ❆s s❡♠✐rr❡t❛s sã♦ ♦s ❧❛❞♦s ❞♦ â♥❣✉❧♦✱ ❡♥q✉❛♥t♦ q✉❡ ❛ ♦r✐❣❡♠ é ♦ ✈ért✐❝❡ ❞❡ss❡ â♥❣✉❧♦✳ ◆❛ ✜❣✉r❛ ✷✱ t❡♠♦s ♦ â♥❣✉❧♦AOBˆ ✱ ♦♥❞❡ ♦s ❧❛❞♦s ❞♦ â♥❣✉❧♦ sã♦ ❛s
s❡♠✐rr❡t❛s OA−→ ❡
−→
OB ❡ ♦ ♣♦♥t♦ O✱ ♦ ✈ért✐❝❡ ❞♦ â♥❣✉❧♦✳ ❯♠ â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣♦r
❋✐❣✉r❛ ✷✿ ❉❡✜♥✐çã♦ ❞❡ â♥❣✉❧♦
❞✉❛s s❡♠✐rr❡t❛s ❞✐st✐♥t❛s ❝♦♥t✐❞❛s ♥✉♠❛ ♠❡s♠❛ r❡t❛ é ❝❤❛♠❛❞♦ ❞❡ â♥❣✉❧♦ r❛s♦ ✭✜❣✉r❛ ✸✮✳
❋✐❣✉r❛ ✸✿ ➶♥❣✉❧♦ r❛s♦ CEDˆ
❉❡✜♥✐çã♦ ✳✶✺✳ ✭■♥t❡r✐♦r ❞❡ ✉♠ â♥❣✉❧♦✮ ❙❡❥❛ ∠BAC ✉♠ â♥❣✉❧♦ ♥✉♠ ♣❧❛♥♦ E✳
❯♠ ♣♦♥t♦ P ❡stá ♥♦ ✐♥t❡r✐♦r ❞♦ ∠BAC s❡ P ❡ B ❡st✐✈❡r❡♠ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❛
r❡t❛ AC←→ ❡ P ❡ C ❡st✐✈❡r❡♠ ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛
←→
AB✳ ❖ ❡①t❡r✐♦r ❞♦ ∠BAC é
♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ❞❡E q✉❡ ♥ã♦ ❡stã♦ ♥♦ â♥❣✉❧♦ ♥❡♠ ♥♦ s❡✉ ✐♥t❡r✐♦r✳
❋✐❣✉r❛ ✹✿ P♦♥t♦ ✐♥t❡r✐♦r ❡ ❡①t❡r✐♦r ❞❡ ✉♠ â♥❣✉❧♦
❆ s❡❣✉✐r ❛♣r❡s❡♥t❛r❡♠♦s ♦s ♣♦st✉❧❛❞♦s ❞❡ ♠❡❞✐❞❛ ❛♥❣✉❧❛r✳
P♦st✉❧❛❞♦ ✳✶✻✳ ✭P♦st✉❧❛❞♦ ❞❛ ♠❡❞✐❞❛ ❞❡ ✉♠ â♥❣✉❧♦✮ ❆ t♦❞♦∠BAC ❝♦rr❡s♣♦♥❞❡
✉♠ ♥ú♠❡r♦ r❡❛❧r ❡♥tr❡ ✵ ❡ ✶✽✵✳
❋✐❣✉r❛ ✺✿ m(∠BAC) =ro
❉❡✜♥✐çã♦ ✳✶✼✳ ❖ ♥ú♠❡r♦ ❞❛❞♦ ♣❡❧♦ P♦st✉❧❛❞♦ ❞❛ ♠❡❞✐❞❛ ❞❡ ✉♠ â♥❣✉❧♦ é ❝❤❛✲ ♠❛❞♦ ❞❡ ♠❡❞✐❞❛ ❞♦ ∠BAC ❡ ❡s❝r❡✈❡♠♦s m(∠BAC) =r ♦✉ (ro
)✳
P♦st✉❧❛❞♦ ✳✶✽✳ ✭P♦st✉❧❛❞♦ ❞❛ ❝♦♥str✉çã♦ ❞❡ â♥❣✉❧♦✮
❙❡❥❛ OB−→ ✉♠❛ s❡♠✐rr❡t❛ ❝♦♥t✐❞❛ ♥❛ ♦r✐❣❡♠ ❞❡ ✉♠ s❡♠✐♣❧❛♥♦ H✳ ❊♥tã♦✱ ♣❛r❛
t♦❞♦ ♥ú♠❡r♦ r❡❛❧r✱ 0≤r ≤180✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ r❡t❛
−→
OB✱ t❛❧ q✉❡m(∠BOA) = r✳
❋✐❣✉r❛ ✻✿ ❈♦♥str✉çã♦ ❞♦ â♥❣✉❧♦∠BOA=r
P♦st✉❧❛❞♦ ✳✶✾✳ ✭P♦st✉❧❛❞♦ ❞❛ ❛❞✐çã♦ ❞❡ â♥❣✉❧♦s✮ ❙❡ C ❡stá ♥♦ ✐♥t❡r✐♦r ❞♦
∠AOB✱ ❡♥tã♦ m(∠AOB) =m(∠AOC) +m(∠COB)✳
❋✐❣✉r❛ ✼✿ ❆❞✐çã♦ ❞❡ â♥❣✉❧♦s
❉❡✜♥✐çã♦ ✳✷✵✳ ❙❡❥❛♠ OB−→ ❡
−→
OA s❡♠✐rr❡t❛s ♦♣♦st❛s ❡
−→
OC ✉♠❛ ♦✉tr❛ s❡♠✐rr❡t❛
q✉❛❧q✉❡r✳ ❊♥tã♦ ♦s â♥❣✉❧♦s ∠COA ❡ ∠COB ❢♦r♠❛♠ ✉♠ ♣❛r ❧✐♥❡❛r✳
❋✐❣✉r❛ ✽✿ P❛r ❧✐♥❡❛r∠COA ❡∠COB
❉❡✜♥✐çã♦ ✳✷✶✳ ❉✐③❡♠♦s q✉❡ ❞♦✐s â♥❣✉❧♦s sã♦ s✉♣❧❡♠❡♥t❛r❡s s❡ ❛ s♦♠❛ ❞❡ s✉❛s ♠❡❞✐❞❛s é ✶✽✵✳
P♦st✉❧❛❞♦ ✳✷✷✳ ❙❡ ❞♦✐s â♥❣✉❧♦s ❢♦r♠❛♠ ✉♠ ♣❛r ❧✐♥❡❛r✱ ❡♥tã♦ ❡❧❡s sã♦ s✉♣❧❡✲ ♠❡♥t❛r❡s✳ ◆❛ ✜❣✉r❛ ✽✱ ∠COA ❡ ∠COB ❢♦r♠❛♠ ✉♠ ♣❛r ❧✐♥❡❛r✳ P♦rt❛♥t♦✱ m(∠COA) +m(∠COB) = 180✳
❉❡✜♥✐çã♦ ✳✷✸✳ ❉✐③❡♠♦s q✉❡ ❞♦✐s â♥❣✉❧♦s sã♦ ♦♣♦st♦s ♣❡❧♦ ✈ért✐❝❡ s❡ ♦s s❡✉s ❧❛❞♦s ❢♦r♠❛♠ ❞♦✐s ♣❛r❡s ❞❡ s❡♠✐rr❡t❛s ♦♣♦st❛s✳
❋✐❣✉r❛ ✾✿ ❖P❱
P♦r ❡①❡♠♣❧♦✱ ♥❛ ✜❣✉r❛ ✾ t❡♠♦s✿ ∠AOC ❡∠DOB sã♦ ♦♣✈✱ ❜❡♠ ❝♦♠♦∠DOA
❡∠BOC✳
❉❡✜♥✐çã♦ ✳✷✹✳ ❉♦✐s â♥❣✉❧♦s sã♦ ❞✐t♦s ❝♦♥❣r✉❡♥t❡s s❡ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛✳ Pr♦♣♦s✐çã♦ ✳✷✺✳ ➶♥❣✉❧♦s ♦♣♦st♦s ♣❡❧♦ ✈ért✐❝❡ sã♦ ❝♦♥❣r✉❡♥t❡s
❉❡♠♦♥str❛çã♦✳ ❉❛ ✜❣✉r❛ ✾ t❡♠♦s q✉❡ ♠✭∠AOC) +m(∠BOC) = 180
♠✭∠BOD) +m(∠BOC) = 180
❘❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛✱ t❡r❡♠♦s✿ m(∠AOC) = m(∠BOD)✳ ❆♥❛❧♦❣❛♠❡♥t❡✱
t❡♠♦sm(∠COB) =m(∠AOD)✳
❉❡✜♥✐çã♦ ✳✷✻✳ ❯♠ â♥❣✉❧♦ ❝✉❥❛ ♠❡❞✐❞❛ é 90o é ❝❤❛♠❛❞♦ ❞❡ â♥❣✉❧♦ r❡t♦✳
❊♥tã♦✱ ♦ s✉♣❧❡♠❡♥t♦ ❞❡ ✉♠ â♥❣✉❧♦ r❡t♦ t❛♠❜é♠ é ✉♠ â♥❣✉❧♦ r❡t♦✳ ◗✉❛♥❞♦ ❞✉❛s r❡t❛s s❡ ✐♥t❡rs❡❝t❛♠✱ s❡ ✉♠ ❞♦s q✉❛tr♦ â♥❣✉❧♦s ❢♦r r❡t♦✱ ❡♥tã♦✱ t♦❞♦s ♦s ♦✉tr♦s s❡rã♦✳ ❖❝♦rr❡♥❞♦ ✐ss♦✱ t❛✐s r❡t❛s sã♦ ❞✐t❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳
✶✳✸ ❚r✐â♥❣✉❧♦s
❈♦♥s✐❞❡r❡ três ♣♦♥t♦s ❆✱ ❇ ❡ ❈ ♥♦ ♣❧❛♥♦✳ ❙❡ ❈ ❡st✐✈❡r s♦❜r❡ ❛ r❡t❛ AB←→✱
❞✐r❡♠♦s q✉❡ ❡ss❡s ♣♦♥t♦s sã♦ ❝♦❧✐♥❡❛r❡s❀ ❝❛s♦ ❝♦♥trár✐♦✱ ❞✐r❡♠♦s q✉❡ sã♦ ♥ã♦ ❝♦❧✐♥❡❛r❡s✳
❋✐❣✉r❛ ✶✵✿ P♦♥t♦s ♥ã♦ ❝♦❧✐♥❡❛r❡s
❉❡✜♥✐çã♦ ✳✷✼✳ ❙❡❥❛♠A✱ B ❡ C ♣♦♥t♦s ♥ã♦ ❝♦❧✐♥❡❛r❡s✳ ❆ r❡✉♥✐ã♦ ❞♦s s❡❣♠❡♥t♦s AB✱ BC ❡ AC é ❝❤❛♠❛❞♦ ❞❡ tr✐â♥❣✉❧♦ ♦ q✉❛❧ r❡♣r❡s❡♥t❛♠♦s ♣♦r ✳ ❖s ♣♦♥t♦s A✱ B ❡ C sã♦ ❝❤❛♠❛❞♦s ❞❡ ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ ❡ ♦s s❡❣♠❡♥t♦s AB✱ BC ❡ AC sã♦ ♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦✳ ❈❛❞❛ tr✐â♥❣✉❧♦ △ABC ❞❡t❡r♠✐♥❛ três â♥❣✉❧♦s✿
∠BAC = ˆA = BACˆ ✱ ∠ABC = ˆB = ABCˆ ❡ ∠ACB = ˆC = ACBˆ q✉❡ sã♦
❝❤❛♠❛❞♦s ❞❡ â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ tr✐â♥❣✉❧♦✳
❋✐❣✉r❛ ✶✶✿ ❯♠ tr✐â♥❣✉❧♦ ❆❇❈
❉❡✜♥✐çã♦ ✳✷✽✳ ✭■♥t❡r✐♦r ❞❡ ✉♠ tr✐â♥❣✉❧♦✮ ❯♠ ♣♦♥t♦ ❡stá ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠ tr✐â♥❣✉❧♦ s❡ ❡❧❡ ❡st✐✈❡r ♥♦ ✐♥t❡r✐♦r ❞❡ t♦❞♦s ♦s â♥❣✉❧♦s ❞♦ tr✐â♥❣✉❧♦✳ ❯♠ ♣♦♥t♦ ❡stá ♥♦ ❡①t❡r✐♦r ❞❡ ✉♠ tr✐â♥❣✉❧♦ s❡ ❡❧❡ ❡st✐✈❡r ♥♦ ♣❧❛♥♦ ❞♦ tr✐â♥❣✉❧♦✱ ♠❛s ♥ã♦ ❡st✐✈❡r ♥♦ tr✐â♥❣✉❧♦ ♥❡♠ ♥♦ s❡✉ ✐♥t❡r✐♦r✳
❋✐❣✉r❛ ✶✷✿ ■♥t❡r✐♦r ❡ ❡①t❡r✐♦r ❞❡ ✉♠ tr✐â♥❣✉❧♦
❈♦♠ r❡❧❛çã♦ ❛ ✉♠ tr✐â♥❣✉❧♦ ABC q✉❛❧q✉❡r✱ ❣❡r❛❧♠❡♥t❡ ❡s❝r❡✈❡♠♦sa =BC✱ b=AC ❡c=AB✱ q✉❡ sã♦ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s s❡✉s ❧❛❞♦s✳
✶✳✹ ❈❧❛ss✐✜❝❛çã♦ ❞❡ tr✐â♥❣✉❧♦s ❡♠ r❡❧❛çã♦ ❛♦s ❧❛❞♦s
❈♦♥s✐❞❡r❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r ABC✱ ❝♦♠ AB✱ AC ❡ BC✱ ♦s ❝♦♠♣r✐♠❡♥✲
t♦s ❞♦s s❡✉s ❧❛❞♦s ✳ ❆ ❝❧❛ss✐✜❝❛ç❛♦ ❡♠ r❡❧❛çã♦ ❛ ❡ss❡s ❧❛❞♦s sã♦✿ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦✿ ❙❡ ❛s ♠❡❞✐❞❛s ❞♦s s❡✉s ❧❛❞♦s ❢♦r❡♠ ✐❣✉❛✐s ❡♥tr❡ s✐✳ ❊♠ sí♠❜♦❧♦s✿
AB =AC =BC✳
tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s✿ ❙❡ ❛♦ ♠❡♥♦s ❞♦✐s ❞❡♥tr❡ ♦s ❧❛❞♦sAB✱AC✱BC❢♦r❡♠ ✐❣✉❛✐s
✭❞♦ ❣r❡❣♦ ✧✐ss♦s❦❡❧♦s✲ ✧♣❡r♥❛s ✐❣✉❛✐s✧✮✳ ❊♠ sí♠❜♦❧♦s AB = BC ♦✉ AB = AC
♦✉AC =BC ✳
tr✐â♥❣✉❧♦ ❡s❝❛❧❡♥♦✿ ❙❡ ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❢♦r❡♠ ❞✐❢❡r❡♥t❡s ❡♥tr❡ s✐✳ ❊♠ sí♠❜♦❧♦sAB 6=AC 6=BC ✭q✉❡ ❡♠ ❣r❡❣♦ s✐❣♥✐✜❝❛ ✧❝❛♣❡♥❣❛✧✮✳
✶✳✺ ❈♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s
❉❡✜♥✐çã♦ ✳✷✾✳ ❉♦✐s tr✐â♥❣✉❧♦s s❡rã♦ ❝♦♥❣r✉❡♥t❡s q✉❛♥❞♦ ♣✉❞❡r♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ s❡✉s ✈ért✐❝❡s ❡✱ ❝♦♠ ✐ss♦✱ ♦s â♥❣✉❧♦s ❡ ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❢♦r❡♠ ✐❣✉❛✐s✳
P♦st✉❧❛❞♦ ✳✸✵✳ ✭❈❛s♦ ▲❆▲ ✲ ▲❛❞♦✲➶♥❣✉❧♦✲▲❛❞♦✮ ❙❡ ❞♦✐s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ ♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣♦r ❡ss❡s ❞♦✐s ❧❛❞♦s ❢♦r❡♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✐❣✉❛✐s ❛ ❞♦✐s ❧❛❞♦s ❞❡ ♦✉tr♦ tr✐â♥❣✉❧♦ ❡ ❛♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣♦r ❡ss❡s ❞♦✐s ❧❛❞♦s✱ ❡♥tã♦ ♦s ❞♦✐s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✳
❋✐❣✉r❛ ✶✸✿ ❖ ♣♦st✉❧❛❞♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s✿ ❝❛s♦ ▲❆▲
❊♠ sí♠❜♦❧♦s✱ t❡♠♦s✿ AB = A′
B′✱
AC = A′
C′ ❡ b
A = Ab′✳ P❡❧♦ ❝❛s♦ ▲❆▲✱ t❡♠♦s△ABC ∼=△A′
B′
C′✳
❚❡♦r❡♠❛ ✳✸✶✳ ✭❝❛s♦ ❆▲❆ ✲ ➶♥❣✉❧♦✲▲❛❞♦✲➶♥❣✉❧♦✮ ❉❛❞♦s ✷ tr✐â♥❣✉❧♦s ABC ❡ A′
B′
C′✳ ❙❡
AB =A′
B′✱ ˆ
A= ˆA′ ❡ Bˆ = ˆB′✱ ❡♥tã♦
△ABC ∼=△A′
B′
C′✳
❋✐❣✉r❛ ✶✹✿ ❞❡♠♦♥str❛çã♦ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s✿ ❝❛s♦ ❆▲❆
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ D∈ BC✱ t❛❧ q✉❡ BD =B′
C′ ✭✜❣✉r❛✿ ✶✹✮✱ ❡♥tã♦✱ ♣♦r
LAL
△ABD ∼= △A′
B′
C′✳ P♦rt❛♥t♦✱
BADˆ = ˆA′
= BACˆ ✱ ❧♦❣♦ D = C ✭♦ ♣♦♥t♦ ❉
❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ♣♦♥t♦ ❈✮✱ ❡✱ ♣❡❧♦ ❝❛s♦ ▲❆▲✱ △ABC ∼=△A′
B′
C′✳
Pr♦♣♦s✐çã♦ ✳✸✷✳ ❊♠ ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s✱ ♦s â♥❣✉❧♦s ❞❛ ❜❛s❡ sã♦ ❝♦♥❣r✉❡♥t❡s✳
❉❡♠♦♥str❛çã♦✳
❙❡❥❛ ✉♠ △ABC✱ ❝♦♠ AB = AC✱ ♣♦✐s t❛❧ tr✐â♥❣✉❧♦ é ✐sós❝❡❧❡s✳ P❡❧❛ ❝♦rr❡s✲
♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦s ✈ért✐❝❡s ❞♦s tr✐â♥❣✉❧♦s✱ t❡♠♦s✿ AB =AC✱Aˆ= ˆA❡AC =AB✳
P❡❧♦ ♣♦st✉❧❛❞♦ ▲❆▲✱ Bˆ = ˆC✳
❋✐❣✉r❛ ✶✺✿ ❚r✐â♥❣✉❧♦ ✐sós❝❡❧❡s ♣❡❧♦ ❝❛s♦ ▲❆▲
Pr♦♣♦s✐çã♦ ✳✸✸✳ ❙❡ ✉♠ tr✐â♥❣✉❧♦ ❆❇❈ t❡♠ ❞♦✐s â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s✱ ❡♥tã♦ ♦ tr✐â♥❣✉❧♦ é ✐sós❝❡❧❡s✳
❋✐❣✉r❛ ✶✻✿ ❚r✐â♥❣✉❧♦ ✐sós❝❡❧❡s ♣❡❧♦ ❝❛s♦ ❆▲❆
❉❡♠♦♥str❛çã♦✳
❙❡❥❛ ✉♠ △ABC✱ ❝♦♠ Bˆ = ˆC✳ ❱❛♠♦s ♠♦str❛r q✉❡ AB = AC✳ P❡❧❛ ❝♦rr❡s✲
♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦s ✈ért✐❝❡s ❞♦s tr✐â♥❣✉❧♦s✱ t❡♠♦s✿ Bˆ = ˆC✱ =CB ❡ Cˆ = ˆB✳ P❡❧♦
t❡♦r❡♠❛ ❆▲❆✱ AB =AC✱ ❧♦❣♦ ♦ tr✐â♥❣✉❧♦ABC é ✐sós❝❡❧❡s✳
❚❡♦r❡♠❛ ✳✸✹✳ ✭❝❛s♦ ▲▲▲−▲❛❞♦−▲❛❞♦−▲❛❞♦✮ ❉❛❞♦s ♦s tr✐â♥❣✉❧♦s△ABC ❡
△A′
B′
C′✳ ❙❡
AB =A′
B′✱
AC =A′
C′ ❡
BC =B′
C′✱ ❡♥tã♦
△ABC ∼=△A′
B′
C′
❋✐❣✉r❛ ✶✼✿ ❝♦♥❣rê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s✿ ❝❛s♦ ▲▲▲
❉❡♠♦♥str❛çã♦✳
❋✐❣✉r❛ ✶✽✿ ❈♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s✿ ❝♦♥str✉çã♦ ❝❛s♦ ▲▲▲
❈♦♥❢♦r♠❡ ✜❣✉r❛ ✶✽✱ ♣♦r ❝♦♥str✉çã♦✱ ❡①✐st❡ ✉♠ ♣♦♥t♦ G✱ t❛❧ q✉❡CBGˆ = ˆB′✳ ❙❡❥❛ D ∈ BG t❛❧ q✉❡ BD = A′
B′✳ ❚❡♠♦s ❡♥tã♦ q✉❡
△BCD ∼= △A′
B′
C′ ♣♦r ▲❆▲✳ ❖❜s❡r✈❛♠♦s ❡♥tã♦ q✉❡ ♦ tr✐â♥❣✉❧♦ABD é ✐sós❝❡❧❡s⇒BADˆ =ADBˆ =α✳
❚❛♠❜é♠ t❡♠♦s q✉❡ ♦ tr✐â♥❣✉❧♦ DAC é ✐sós❝❡❧❡s ⇒ DACˆ =ADCˆ =β ✳ ❚❡♠♦s
❡♥tã♦✱ ♣♦r ▲❆▲✱ q✉❡△ABC ∼=△BCD✳
❚❡♦r❡♠❛ ✳✸✺✳ ✭❈❛s♦ ❡s♣❡❝✐❛❧ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s✮✳ ❈♦♥✲ s✐❞❡r❡ ♦s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s △ABC ❡ △A′
B′
C′✱ r❡tâ♥❣✉❧♦s ❡♠ ˆ
B ❡ Bˆ′✱ r❡s✲ ♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡ AC =A′
C′ ❡
AB =A′
B′✱ ❡♥tã♦
△ABC ∼=△A′
B′
C′ ✳
❉❡♠♦♥str❛çã♦✳
❈♦♥tr✉í♠♦s ♦ ♣♦♥t♦ ❉ ✭✜❣✉r❛✿ ✷✵✮✱ ❞❡ ♠♦❞♦ q✉❡ A′
B′
= DB ❡ B′ˆ
A′C′
= BDCˆ ✳ P♦r ▲❆▲✱ t❡♠♦s q✉❡ △BDC ∼= △A′
B′
C′✳ ❖❜s❡r✈❛♠♦s t❛♠❜é♠ q✉❡ ♦ tr✐â♥❣✉❧♦ ADC é ✐sós❝❡❧❡s✱ ♣♦✐s AC =DC✱ ❡♥tã♦ Aˆ= ˆD✳ P♦r ▲❆▲✱ ❝♦♥❝❧✉í♠♦s
q✉❡△ABC ∼=△A′
B′
C′
❋✐❣✉r❛ ✶✾✿ ❈♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s
❋✐❣✉r❛ ✷✵✿ ❉❡♠♦♥str❛çã♦✿ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s
✶✳✻ Pr♦♣♦s✐çõ❡s ❝❧áss✐❝❛s✱ ♦ ♣♦st✉❧❛❞♦s ❞❛s ♣❛r❛❧❡❧❛s
▲❡♠❛ ✳✸✻✳ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞♦ â♥❣✉❧♦ ❡①t❡r♥♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦✮❊♠ t♦❞♦ tr✐â♥❣✉❧♦✱ ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ❡①t❡r♥♦ é ♠❛✐♦r q✉❡ ❛s ♠❡❞✐❞❛s ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ♥ã♦ ❛❞❥❛❝❡♥t❡s ❛ ❡❧❡
❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ ✉♠ tr✐â♥❣✉❧♦ ABC ❡ M ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ ❧❛❞♦ BC
✭✜❣✉r❛✿ ✷✶✮✳ Pr♦❧♦♥❣✉❡♠♦s ❛ s❡♠✐rr❡t❛AM−→ ❛té ♦ ♣♦♥t♦ ◆✱ t❛❧ q✉❡ AM =M N✱
❡ ♦❜s❡r✈❡ ♦s tr✐â♥❣✉❧♦s AM B ❡ N M C✳ ❚❡♠♦s✱ AM = M N ✭♣♦r ❝♦♥str✉çã♦✮✱ AM Bˆ =NM Cˆ ✭♣r♦♣♦s✐çã♦✿ ✳✷✺✮ ❡BM =CM✳ P♦rt❛♥t♦✱ ♣❡❧♦ ❝❛s♦ ▲❆▲✱ t❡♠♦s
q✉❡ △AM B ∼= △N M C ✱ ❡♥tã♦ ABMˆ = NCMˆ ✳ ▲♦❣♦✱ YCB > Nˆ CMˆ = ABMˆ =ABCˆ ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣r♦✈❛✲s❡ q✉❡YCB > Bˆ ACˆ
❋✐❣✉r❛ ✷✶✿ ❉❡s✐❣✉❛❧❞❛❞❡ ❞♦ â♥❣✉❧♦ ❡①t❡r♥♦
P♦st✉❧❛❞♦ ✳✸✼✳ ✭❖ ♣♦st✉❧❛❞♦ ❞❛s ♣❛r❛❧❡❧❛s✮ ❉❛❞♦s ✉♠❛ r❡t❛r ❡ ✉♠ ♣♦♥t♦A /∈r✱
❡①✐st❡ ✉♠❛ ú♥✐❝❛ r❡t❛ s ♣❛r❛❧❡❧❛ ❛ r q✉❡ ♣❛ss❛ ♣♦r ❆✳
❋✐❣✉r❛ ✷✷✿ ❖ 5o P♦st✉❧❛❞♦ ❞❡ ❊✉❝❧✐❞❡s
❈♦♠ ❡ss❡ ♣♦st✉❧❛❞♦✱ ✈❛♠♦s ♣r♦✈❛r ❛❧❣✉♥s ❞♦s r❡s✉❧t❛❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳ ❈♦♥s✐❞❡r❡ ❞❛❞❛s✱ ♥♦ ♣❧❛♥♦✱ ❛s r❡t❛s r✱ s ❡ t✱ ❝♦♠ t ✐♥t❡rs❡❝t❛♥❞♦ r ❡ s ♥♦s ♣♦♥t♦s ❆ ❡ ❇✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◆❛ ✜❣✉r❛ ✷✷✱ t❡♠♦s ♦s â♥❣✉❧♦s α ❡ β sã♦ ❞✐t♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s✱ ❛♦ ♣❛ss♦ q✉❡ ♦s â♥❣✉❧♦s α ❡ γ sã♦
❞❡♥♦♠✐♥❛❞♦s ❝♦❧❛t❡r❛✐s ✐♥t❡r♥♦s✳
❈♦r♦❧ár✐♦ ✳✸✽✳ ❆ r❡t❛ r é ♣❛r❛❧❡❧❛ ❛ r❡t❛ s ✭r//s✮ s❡✱ ❡✱ s♦♠❡♥t❡ s❡✱ α = β ❡ α+γ = 180o✳
❉❡♠♦♥str❛çã♦✳
■♥✐❝✐❛❧♠❡♥t❡✱ ♥♦t❡ q✉❡✱ ❝♦♠♦ β +γ = 180o✱ t❡♠♦s
α = β ⇔ α+γ = 180o✳
P♦rt❛♥t♦✱ ❜❛st❛ ♣r♦✈❛r♠♦s q✉❡ r||s ⇔ α = β✳ ▼♦str❛r❡♠♦s q✉❡ α = β ⇒ r||s✳
❱❛♠♦s ♣r♦❧♦♥❣❛r ❛s r❡t❛sr ❡ s ✭✜❣✉r❛✿ ✷✷✮ ❞❡ ♠♦❞♦ ❛ ❢♦r♠❛r ♦ tr✐â♥❣✉❧♦ ABP✱
❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ ✷✸✳
❋✐❣✉r❛ ✷✸✿ ❞❡♠♦♥str❛çã♦ ❞❡ P❛r❛❧❡❧✐s♠♦
P❡❧♦ ❧❡♠❛ ✳✸✻✱ ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ α é ♠❛✐♦r ❞♦ q✉❡ β ❡ ♠❛✐♦r ❞♦ q✉❡ BP Aˆ ✱
❝♦♥tr❛❞✐çã♦ ♣♦✐s ❥á tí♥❤❛♠♦s ❛❞♠✐t✐❞♦ q✉❡ α =β✳ ▲♦❣♦✱ α =β ⇒ r//s✳ ❙✉♣♦✲
♥❤❛♠♦s q✉❡ r//s✳ ❊♥tã♦✱ ♣❡❧♦ P♦st✉❧❛❞♦ ❞❛ ♣❛r❛❧❡❧❛s✱ s é ❛ ú♥✐❝❛ r❡t❛ ♣❛r❛❧❡❧❛ r ♣❛ss❛♥❞♦ ♣♦r ❇✱ ❡ ♣♦rt❛♥t♦✱ r//s⇒α=β
❚❡♦r❡♠❛ ✳✸✾✳ ❆ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ✐❣✉❛❧ ❛180o
❋✐❣✉r❛ ✷✹✿ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s
❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦ tr✐â♥❣✉❧♦ ABC ❡ ❛ r❡t❛
−→
XY ♣❛r❛❧❡❧❛ ❛ r❡t❛
−→
BC
♣❛ss❛♥❞♦ ♣♦r ❆ ✭✜❣✉r❛ ✷✹✮✱ t❡♠♦s q✉❡Bˆ =XABˆ ❡Cˆ =YACˆ ✭â♥❣✉❧♦s ❛❧t❡r♥♦s
✐♥t❡r♥♦s✮✳ ❙❡❣✉❡ q✉❡ Aˆ+XABˆ +YACˆ = ˆA+ ˆB+ ˆC = 180o
❚❡♦r❡♠❛ ✳✹✵✳ ✭❚❡♦r❡♠❛ ❞♦ â♥❣✉❧♦ ❡①t❡r♥♦✮ ❊♠ t♦❞♦ tr✐â♥❣✉❧♦✱ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ â♥❣✉❧♦ ❡①t❡r♥♦ é ✐❣✉❛❧ à s♦♠❛ ❞❛s ♠❡❞✐❞❛s ❞♦s ❞♦✐s â♥❣✉❧♦s ✐♥t❡r♥♦ ♥ã♦ ❛❞❥❛❝❡♥t❡s ❛ ❡❧❡✳
❉❡♠♦♥str❛çã♦✳ ◆♦ tr✐â♥❣✉❧♦ ABC ❞❛ ✜❣✉r❛ ✷✹✱ t❡♠♦s q✉❡ Aˆ+ ˆB + ˆC = 180o✱
❝♦♥❢♦r♠❡ t❡♦r❡♠❛ ✳✸✾ ❡ACZˆ + ˆC = 180o ✳ ❚❡♠♦s q✉❡✱ ˆ
A+ ˆB+ ˆC =ACZˆ + ˆC✳
❊♥tã♦✱ ACZˆ = ˆA+ ˆB✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣r♦✈❛✲s❡ ♣❛r❛ ♦s ♦✉tr♦s â♥❣✉❧♦s ❡①t❡r♥♦s✳
❚❡♦r❡♠❛ ✳✹✶✳ ✭▲❛❞♦ ✲ ➶♥❣✉❧♦ ✲ ➶♥❣✉❧♦ ♦♣♦st♦ ✲LAAo✮
❙❡❥❛ ♦s tr✐â♥❣✉❧♦s △ABC ❡ △A′
B′
C′
✳ ❙❡ AB = A′B′✱ Aˆ = ˆA′ ❡ Cˆ = ˆC′✱ ❡♥tã♦ △ABC ∼=△A′
B′
C′ ✳
❋✐❣✉r❛ ✷✺✿ ❈♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s✿ ❝❛s♦LAAo
❉❡♠♦♥str❛çã♦✳ P♦r ❤✐♣ót❡s❡✱ t❡♠♦s q✉❡ Aˆ = ˆA′ ❡ ˆ
C = ˆC′✳ ❱❛♠♦s ♠♦str❛r q✉❡
ˆ
B = ˆB′✳ ❉♦ t❡♦r❡♠❛ ✳✸✾✱ t❡♠♦s ˆ
A + ˆB + ˆC = 180o ❡ ˆ A′
+ ˆB′
+ ˆC′
= 180o✳
❊♥tã♦✱ Bˆ = 180o
− Aˆ −Cˆ = 180o
−Aˆ′
−Cˆ′ = ˆB′✳ ▲♦❣♦✱ ♣❡❧♦ ❝❛s♦ ❆▲❆✱
△ABC ∼=△A′
B′
C′✳
✶✳✼ ❈ír❝✉❧♦✱ ❜✐ss❡tr✐③ ❡ ♠❡❞✐❛tr✐③
❉❡✜♥✐çã♦ ✳✹✷✳ ❉❛❞♦s ✉♠ ♣♦♥t♦ ❖ ❡ ✉♠ r ∈R+✱ ♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r
é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦sP ❞♦ ♣❧❛♥♦ q✉❡ ❡stã♦ à ❞✐stâ♥❝✐❛r ❞❡ O✱ ✐st♦ é✱ t❛✐s q✉❡ OP =r✳
❋✐❣✉r❛ ✷✻✿ ❉❡✜♥✐çã♦ ♣❛r❛ ❝ír❝✉❧♦
❉❛ ✜❣✉r❛ ✷✻✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ✳✹✷✱ t❡♠♦s q✉❡OP =OP′
=OP′′
=r✳
❉❡✜♥✐çã♦ ✳✹✸✳ ✭❇✐ss❡tr✐③✮ ❙❡❥❛ ♦ â♥❣✉❧♦∠AOB✳ ❙❡ ❈ ❡stá ♥♦ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦
∠AOB ❡ m(∠AOP) = m(∠BOP)✱ ❞✐③❡♠♦s q✉❡
−→
OP é ❛ ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦
∠AOB ✭✜❣✉r❛✿ ✷✼✮✳
Pr♦♣♦s✐çã♦ ✳✹✹✳ ❙❡❥❛ AOBˆ ✉♠ â♥❣✉❧♦ ❞❛❞♦✳ ❊♥tã♦✱ P ♣❡rt❡♥❝❡ à ❜✐ss❡tr✐③ ♠
❞❡ AOBˆ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ d(P,
−→
OA) =d(P,
−→
OB)✳
❉❡♠♦♥str❛çã♦✳
❋✐❣✉r❛ ✷✼✿ ❆ ❜✐ss❡tr✐③
❉❛ ✜❣✉r❛ ✷✼✱ ❝♦♥s✐❞❡r❡ ♦s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s△OP A❡△OP B✳ ❚❡♠♦s q✉❡✿ OP ✲ ❧❛❞♦ ❝♦♠✉♠ ❞♦s tr✐â♥❣✉❧♦s✱ AOPˆ = BOPˆ ❡ OAPˆ = OBPˆ = 90o✳ P❡❧♦
❝❛s♦ ▲❆❆o, △OP A ∼= △OP B✱ ❡ ♣♦rt❛♥t♦ P A = d(P,
−→
OA) = P B = d(P,
−→
OB)✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ d(P,
−→
OA) = d(P,
−→
OB) ❡ OP é ❧❛❞♦ ❝♦♠✉♠ ❞♦s tr✐â♥❣✉❧♦s✱
❡♥tã♦ ♣❡❧♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s✱ △OP A ∼=
△OP B✱ ❡♥tã♦ AOPˆ =BOPˆ ❡AO=BO ❡✱ ♣♦rt❛♥t♦✱ P ∈m✳
❉❡✜♥✐çã♦ ✳✹✺✳ ✭▼❡❞✐❛tr✐③✮
❉❛❞♦ ♦ s❡❣♠❡♥t♦ AB ⊂π✳ ❆ ♠❡❞✐❛tr✐③ ❞❡AB é ❛ r❡t❛ m⊂π✱ ♣❡r♣❡♥❞✐❝✉❧❛r
à AB ❡ q✉❡ ♣❛ss❛ ♣❡❧♦ s❡✉ ♣♦♥t♦ ♠é❞✐♦✳
Pr♦♣♦s✐çã♦ ✳✹✻✳ ❆ ♠❡❞✐❛tr✐③ ❞❡ ✉♠ s❡❣♠❡♥t♦✱ ❡♠ ✉♠ ♣❧❛♥♦✱ é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦ ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ ❡q✉✐❞✐st❛♠ ❞❛s ❡①tr❡♠✐❞❛❞❡s ❞♦ s❡❣♠❡♥t♦✳
❉❡♠♦♥str❛çã♦✳
❋✐❣✉r❛ ✷✽✿ ❆ ♠❡❞✐❛tr✐③
❙❡❥❛C ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡AB✱m❛ ♠❡❞✐❛tr✐③ ❞❡AB ✱ ❡ s❡❥❛P ∈m✳ ❙❡P =C✱
❡♥tã♦ t❡r❡♠♦s P A =P B✳ ❙✉♣♦♥❤❛✱ ❡♥tã♦✱ q✉❡ P 6=C✱ ❞❡ ♠♦❞♦ q✉❡ P /∈ AB✳
❚❡♠♦s q✉❡ P C = P C ✭❧❛❞♦ ❡♠ ❝♦♠✉♠✮❀ ∠P CA ∼= ∠P CB = 90o ❡
CA =CB✱
♣♦✐sC é ♣♦♥t♦ ♠é❞✐♦✳ P♦rLAL✱ t❡♠♦s△P CA∼=△P CB✳ P♦rt❛♥t♦✱P A=P B✳
✷ ❖ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ s♦❜r❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✱
➚r❡❛s ❡ ❖ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s
✷✳✶ ❖ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡
❆♥t❡s ❞❡ ❞❡♠♦♥str❛r♠♦s ♦ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧✱ ❞❡♠♦str❛r❡♠♦s ✉♠❛ ♣r♦♣♦✲ s✐çã♦ ♦♥❞❡ ♠♦str❛ ❛ r❛③ã♦ ❡♥tr❡ ár❡❛s ❞❡ ❞♦✐s tr✐â♥❣✉❧♦s ❞❡ ❛❧t✉r❛h✭❛ ♣r♦♣♦s✐çã♦
s♦❜r❡ ár❡❛ ❞❡ tr✐â♥❣✉❧♦ s❡rá ♠♦str❛❞❛ ♥❛ ♣ró①✐♠❛ s❡çã♦✮✳
Pr♦♣♦s✐çã♦ ✳✹✼✳ ❙❡ ❞♦✐s tr✐â♥❣✉❧♦s t❡♠ ❛ ♠❡s♠❛ ❛❧t✉r❛ h✱ ❡♥tã♦ ❛ r❛③ã♦ ❡♥tr❡
s✉❛s ár❡❛s é ✐❣✉❛❧ ❛ r❛③ã♦ ❡♥tr❡ s✉❛s ❜❛s❡s✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ b1 ❡ b2 ❛s ❜❛s❡s✳ ❊♥tã♦
A(ABC) A(P QR) =
1 2b1h 1 2b2h
= b1 b2
✳
❋✐❣✉r❛ ✷✾✿ ❘❛③ã♦ ❡♥tr❡ ár❡❛s ❞❡ tr✐â♥❣✉❧♦s
❚❡♦r❡♠❛ ✳✹✽✳ ✭❖ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ s♦❜r❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✮ ❙❡ ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛ ✉♠ ❧❛❞♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ✐♥t❡r❝❡♣t❛ ♦s ♦✉tr♦s ❞♦✐s ❧❛❞♦s ❡♠ ♣♦♥t♦s ❞✐st✐♥t♦s✱ ❡♥tã♦ ❡❧❛ ❞❡t❡r♠✐♥❛ s❡❣♠❡♥t♦s q✉❡ sã♦ ♣r♦♣♦r❝✐♦♥❛✐s ❛ ❡ss❡s ❧❛❞♦s
❋✐❣✉r❛ ✸✵✿ ❙❡❣♠❡♥t♦s ♣r♦♣♦r❝✐♦♥❛✐s
❖✉ s❡❥❛✱ ♥♦ tr✐â♥❣✉❧♦△ABC s❡❥❛♠ ♦s ♣♦♥t♦sD❡E ♣♦♥t♦s ❞❡ AC ❡BC t❛✐s
q✉❡DE||AB✳ ❊♥tã♦ CA
CD =
CB CE✳
❉❡♠♦♥str❛çã♦✳ ◆♦s tr✐â♥❣✉❧♦s△CDE ❡ △ADE ❝♦♥s✐❞❡r❡♠♦s CD ❡ AD ❝♦♠♦
❜❛s❡s ✭✜❣✉r❛✿ ✸✶✮✳ ❊♥tã♦ ❡ss❡s tr✐â♥❣✉❧♦s t❡♠ ❛ ♠❡s♠❛ ❛❧t✉r❛✱ ♣♦✐s ❛s s✉❛s ❜❛s❡s ❡stã♦ s♦❜ ❛ ♠❡s♠❛ r❡t❛AC←→✳ P♦rt❛♥t♦✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✳✹✼✱ ❛ r❛③ã♦ ❞❡ s✉❛s ár❡❛s
é ✐❣✉❛❧ ❛ r❛③ã♦ ❞❡ s✉❛s ❜❛s❡s✱ ♦✉ s❡❥❛✱
✭✶✮ A(ADE)
A(CDE) = AD CD
❆♥❛❧♦❣❛♠❡♥t❡✱ ♥♦s tr✐â♥❣✉❧♦s △CDE ❡ △BDE ❞❛ ✜❣✉r❛ ✸✶ ❝♦♥s✐❞❡r❡♠♦s CE ❡ BE✳ ❈♦♠♦ ❡ss❡s tr✐â♥❣✉❧♦s ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ❛❧t✉r❛✱ t❡r❡♠♦s✿
✭✷✮ A(BDE)
A(CDE) =
BE CE✳
▼❛s△ADE ❡ △BDE t❡♠ ❛ ♠❡s♠❛ ❜❛s❡ DE✳ ❊❧❡s t❡♠ ❛ ♠❡s♠❛ ❛❧t✉r❛✱ ♣♦✐s
←→
DE ||
←→
AB✭✜❣✉r❛✿ ✸✵✮✳ ▲♦❣♦✱ ♣❡❧♦ ❝♦r♦❧ár✐♦ ✳✻✶✱ t❡♠♦s q✉❡ ✭✸✮A(ADE) =A(BED)✳
❈♦♠❜✐♥❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✶✮✱ ✭✷✮ ❡ ✭✸✮✱ ♦❜t❡♠♦s✿
❋✐❣✉r❛ ✸✶✿ ❚r✐â♥❣✉❧♦s △ADE ❡ △BED t❡♠ ár❡❛s ✐❣✉❛✐s
✭✹✮ AD
CD =
BE CE✳
❆❞✐❝✐♦♥❛♥❞♦ ✶ ❛ ❡q✉❛çã♦ ✭✹✮✱ ❡♠ ❛♠❜♦s ♦s ♠❡♠❜r♦✱ ♦❜t❡♠♦s✿
AD+CD
CD =
BE +CE
CE
AC
CD =
BC CE✳
❱❛♠♦s ❛❣♦r❛ ❡♥✉♥❝✐❛r ❛ r❡❝í♣r♦❝❛ ❞♦ ❚❡♦r❡♠❛ ❛♥t❡r✐♦r
❚❡♦r❡♠❛ ✳✹✾✳ ❙❡ ✉♠❛ r❡t❛ ✐♥t❡r❝❡♣t❛ ❞♦✐s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ ❞❡t❡r♠✐♥❛ s❡❣♠❡♥t♦s ♣r♦♣♦r❝✐♦♥❛✐s ❛ ❡st❡s ❞♦✐s ❧❛❞♦s✱ ❡♥tã♦ ❡❧❛ é ♣❛r❛❧❡❧❛ ❛♦ t❡r❝❡✐r♦ ❧❛❞♦
➱ ❞❛❞♦ ♦ tr✐â♥❣✉❧♦△ABC✳ ❙❡❥❛ D✉♠ ♣♦♥t♦ ❡♥tr❡ A ❡C ❡ s❡❥❛E ✉♠ ♣♦♥t♦
❡♥tr❡B ❡ C✳ ❙❡ AC
CD =
BC
CE✱ ❡♥tã♦
←→
DE ||
←→
AB✳
❉❡♠♦♥str❛çã♦✳
❙❡❥❛ AB←→′ ✉♠❛ r❡t❛ ♣♦r
A✱ ♣❛r❛❧❡❧❛ ❛
←→
DE✱ ✐♥t❡r❝❡♣t❛♥❞♦ CB ❡♠ B′✳ P❡❧♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r
CA
CD =
CB′
CE✳
❈♦♠♦✱ ♣♦r ❤✐♣♦t❡s❡✿ CA
CD =
CB CE
t❡♠♦s CB ′
CE =
CB CE
❋✐❣✉r❛ ✸✷✿ ❆ r❡❝í♣r♦❝❛ ❞♦ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡
❡CB′
=CB✳ P♦rt❛♥t♦✱ B =B′ ❡ ←→
DE ||
←→
AB✳
✷✳✷ ❙❡♠❡❧❤❛♥ç❛ ❡♥tr❡ tr✐â♥❣✉❧♦s
❊♠ ❬✺❪ ♣á❣✐♥❛ ✷✷✱ t❡♠♦s ✉♠ t❡①t♦ s♦❜r❡ ❚❛❧❡s ❞❡ ▼✐❧❡t♦✿
◆ã♦ s❡ s❛❜❡ ❡♠ q✉❛✐s s✐t✉❛çõ❡s ❚❛❧❡s ❞❡ ▼✐❧❡t♦ ✐♥t❡r❡ss♦✉✲s❡ ♣❡❧❛ ●❡♦♠❡tr✐❛✳ ❚r❛❞✐❝✐♦♥❛❧♠❡♥t❡✱ ❡❧❡ ✈✐s✐t❛✈❛ ♦ ❊❣✐t♦ ❡♠ s✉❛s ✈✐❛❣❡♥s ❝♦♠❡r❝✐❛✐s ❡ ❝✉❧t✉r❛✐s✳ ❊♠ ❝♦♠♣❛♥❤✐❛ ❞♦ ❢❛r❛ó ❆♠❛s✐s ❡ ❝♦♥t❡♠✲ ♣❧❛♥❞♦ ❛ ♣✐râ♠✐❞❡ ❞❡ ◗✉é♦♣s✱ ♠❡❞✐✉ ❛s s♦♠❜r❛s ❞❛ ♣✐râ♠✐❞❡ ❡ ❞❡ ✉♠ ❜❛stã♦ q✉❡ ❝♦❧♦❝❛r❛ ✈❡rt✐❝❛❧♠❡♥t❡ ♥❛ ❛r❡✐❛✱ ❛ ♠❡t❛❞❡ ❞❛ ♠❡❞✐❞❛ ❞❛ ❜❛s❡ ❞❛ ♣✐râ♠✐❞❡ ❡ ❛ ❛❧t✉r❛ ❞♦ ❜❛stã♦✱ ❝❛❧❝✉❧❛♥❞♦ ❛ ❛❧t✉r❛ ❞♦ ♠♦✲ ♥✉♠❡♥t♦ ❛ ♣❛rt✐r ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ s❡♥❞♦ r❡s♣♦♥sá✈❡❧ ♣♦r ✉♠ ❞♦s ❛❝♦♥t❡❝✐♠❡♥t♦s ♠❛✐s ✐♥t❡r❡ss❛♥t❡s ❞❛ ❍✐stór✐❛ ❞❛ ●❡♦♠❡tr✐❛✳ ❉❡✜♥✐çã♦ ✳✺✵✳ ❉♦✐s tr✐â♥❣✉❧♦s s❡rã♦ s❡♠❡❧❤❛♥t❡s q✉❛♥❞♦ ❡①✐st✐r ✉♠❛ ❝♦rr❡s♣♦♥✲ ❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ s❡✉s ✈ért✐❝❡s✱ t❛❧ q✉❡ ♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❝♦rr❡s♣♦♥❞❡♥t❡s s❡❥❛♠ ✐❣✉❛✐s ❡ ❛ r❛③ã♦ ❡♥tr❡ ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✭❤♦♠ó❧♦❣♦s✮ s❡❥❛ ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧ k >0✱ ♦♥❞❡ k é ❝❤❛♠❛❞♦ ❛ r❛③ã♦ ❞❡ s❡♠❡❧❤❛♥ç❛✳
❋✐❣✉r❛ ✸✸✿ ❉❡✜♥✐çã♦ ❞❡ tr✐â♥❣✉❧♦s s❡♠❡❧❤❛♥t❡s
P❡❧❛ ❞❡✜♥✐çã♦✱ Aˆ = ˆA′✱ ˆ
B = ˆB′ ❡ ˆ
C = ˆC′ ✭♣❡❧❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦s ✈ért✐❝❡s ✮ ❡ AB
A′B′ =
BC B′C′ =
AC
A′C′ = k✳ ❙í♠❜♦❧♦ ♣❛r❛ tr✐â♥❣✉❧♦s s❡♠❡❧❤❛♥t❡s✿
ABC ∼ A′
B′
C′
✳ ❖❜s❡r✈❛♠♦s q✉❡ q✉❛♥❞♦ k = 1 ✱ t❡r❡♠♦s ✉♠ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞❡
s❡♠❡❧❤❛♥ç❛ ❡♥tr❡ tr✐â♥❣✉❧♦s✱ ♦✉ s❡❥❛✱ ♦s tr✐â♥❣✉❧♦s s❡rã♦ ❝♦♥❣r✉❡♥t❡s✳
❚❡♦r❡♠❛ ✳✺✶✳ ✭▲▲▲✮ ❙❡❥❛♠ABC ❡A′
B′
C′ tr✐â♥❣✉❧♦s ♥♦ ♣❧❛♥♦✱ t❛✐s q✉❡ AB A′B′ =
BC B′C′ =
AC
A′C′✳ ❊♥tã♦✱ ∆ABC ∼∆A
′
B′
C′✳
❋✐❣✉r❛ ✸✹✿ tr✐â♥❣✉❧♦s s❡♠❡❧❤❛♥t❡s✿ ❝❛s♦ ▲▲▲
❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❛♥❞♦ K ❛ r❛③ã♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✸✹✱ t❡✲
♠♦s✿ AB = k.A′
B′
✱ BC = k.B′
C′
❡ AC = k.A′
C′
✳ ◆❛ ✜❣✉r❛ ✸✺✱ t❡♠♦s q✉❡
B′′
∈AB ❡C′′
∈AC ✱ t❛❧ q✉❡AB′′
=A′
B′ ❡
AC′′
=A′
B′✳ ❖s ♣♦♥t♦s B′′ ❡
C′′ ❢♦r❛♠ ♠❛r❝❛❞♦s t❛❧ q✉❡
B′′
C′′
//BC ❡ C′′
D//AB✳ ❙❡❣✉❡
❞♦ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ s♦❜r❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ q✉❡
❋✐❣✉r❛ ✸✺✿ ❉❡♠♦♥str❛çã♦ ❞♦ ❝❛s♦ ▲▲▲
AC AC′′ =
AB AB′′ =
k.A′
B′
A′B′ =k⇐⇒AC =kAC ′′
=k.A′
C′ ❡
BC B′′C′′ =
BC
BD =
AC AC′′ =
k.A′
C′
A′C′ =k ⇐⇒BC =k.B ′′
C′′
=k.B′
C′
❊♥tã♦✱AB′′
=A′
B′
, AC′′
=A′
C′ ❡B′′
C′′
=B′
C′
✱ ✐st♦ é✱ ♦s tr✐â♥❣✉❧♦sAB′′
C′′ ❡A′
B′
C′ sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦ ▲▲▲✳ P♦rt❛♥t♦✱ t❡♠♦s Bˆ = ABCˆ =ABˆ′′
C′′
=A′ ˆ
B′
C′
= ˆB′✱ ❡✱ ❛♥❛❧♦❣❛♠❡♥t❡✱ ˆ
A = ˆA′ ❡Cˆ = ˆC′✳
❚❡♦r❡♠❛ ✳✺✷✳ ✭❆❆✮
❙❡❥❛♠ ABC ❡ DEF tr✐â♥❣✉❧♦s ♥♦ ♣❧❛♥♦✱ t❛✐s q✉❡ Aˆ= ˆD ❡ Bˆ = ˆE✳ ❊♥tã♦✱
△ABC ∼ △DEF✳
❉❡♠♦♥str❛çã♦✳
❋✐❣✉r❛ ✸✻✿ ❞❡♠♦♥str❛çã♦ ❞♦ ❝❛s♦ ❆❆
❱❛♠♦s ♠♦str❛r q✉❡ AB
DE =
AC
DF = k ❈♦♥s✐❞❡r❡ E
′
∈ AB ❡ F′
∈ AC✱ t❛✐s q✉❡ AE′
= DE ❡ AF′
= DF✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✸✻✳ P♦r ❆▲❆✱ △AE′
F′ ∼
= △DEF✳
❈♦♠♦✱ ♣♦r ❤✐♣ót❡s❡✱Bˆ = ˆE ❡Eˆ′
= ˆE ✭♣❡❧❛ ❝♦♥❣r✉ê♥❝✐❛ ❛♥t❡r✐♦r✮✱ ❡♥tã♦Eˆ′
= ˆB✳
❚❡♠♦s ❞♦✐s ❝❛s♦s ❛ ❝♦♥s✐❞❡r❛r✿
1.o ❖s ♣♦♥t♦s E′
= B✿ ♦s tr✐â♥❣✉❧♦s AE′
F′
❡ ABC sã♦ ♦ ♠❡s♠♦ tr✐â♥❣✉❧♦
✭△AE′
F′ ∼
=△ABC✮❡✱ ♣♦rt❛♥t♦ AB
DE =
AC DF = 1 ❀ 2.o ❙✉♣♦♥❤❛♠♦s
E′
6
= B✳ P❡❧♦ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ s♦❜r❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✱
t❡♠♦s✿ AB AE′ =
AC
AF′ ⇐⇒
AB
DE =
AC DF =k✳
P♦rt❛♥t♦✱ △ABC ∼ △DEF✳
❚❡♦r❡♠❛ ✳✺✸✳ ✭▲❆▲✮ ❙❡❥❛♠ ABC ❡ DEF✱ tr✐â♥❣✉❧♦s ♥♦ ♣❧❛♥♦ t❛✐s q✉❡ AB
DE =
AC
DF =k ❡ Aˆ= ˆD✳ ❊♥tã♦✱ △ABC ∼ △DEF✳
❋✐❣✉r❛ ✸✼✿ ❉❡♠♦♥str❛çã♦ ❞♦ ❝❛s♦ ▲❆▲
❉❡♠♦♥str❛çã♦✳
❙❡❥❛♠E′
∈AB ❡F′
∈AC✱ t❛✐s q✉❡AE′
=DE ❡AF′
=DF ✭✜❣✉r❛✿ ✸✼✮✳❈♦♠♦✱
♣♦r ❤✐♣ót❡s❡✱Aˆ= ˆD✱ ❡♥tã♦✱ ♣♦r ▲❆▲✱△AE′
F′ ∼
=△DEF✳ ▲♦❣♦✱Eˆ′ = ˆE,Fˆ′ = ˆF ❡Aˆ= ˆD❀ AE′
=DE, AF′
=DF ❡ E′
F′
=EF✳ P❡❧♦ ❝♦❧♦rár✐♦ ✳✹✾✱ t❡♠♦s✿ AB
AE′ =
AC
AF′ =k✱ ❡ ❝♦♠♦ E
′
F′
||BC✱ ♣♦rt❛♥t♦ Bˆ = ˆE′
= ˆE ✱Cˆ = ˆF′
= ˆF✳
P♦r ❆❆✱ t❡♠♦s q✉❡△ABC ∼ △DEF✳
✷✳✸ ➪r❡❛s
❉❡✜♥✐çã♦ ✳✺✹✳ ➪r❡❛ é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ q✉❡ ❡stá ❛ss♦❝✐❛❞♦ à ✉♠❛ s✉♣❡r✲ ❢í❝✐❡✱ q✉❡ q✉❛♥t✐✜❝❛ ♦ ❡s♣❛ç♦ ♦❝✉♣❛❞♦ ♣♦r ❡st❛ s✉♣❡r❢í❝✐❡✳
P♦st✉❧❛❞♦s
✶✳ P♦❧í❣♦♥♦s ❝♦♥❣r✉❡♥t❡s t❡♠ ár❡❛s ✐❣✉❛✐s✳
✷✳ ❙❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ é ♣❛rt✐❝✐♦♥❛❞♦ ❡♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♦✉tr♦s ♣♦❧í❣♦♥♦s ❝♦♥✈❡①♦s✱ ❡♥tã♦ ❛ ár❡❛ ❞♦ ♣♦❧í❣♦♥♦ ♠❛✐♦r é ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s ♣♦❧í❣♦♥♦s ♠❡♥♦r❡s✳
✸✳ ❙❡ ✉♠ ♣♦❧í❣♦♥♦ ♠❛✐♦r ❝♦♥tê♠ ♦✉tr♦ ♠❡♥♦r ❡♠ s❡✉ ✐♥t❡r✐♦r✱ ❡♥tã♦ ❛ ár❡❛ ❞♦ ♣♦❧í❣♦♥♦ ♠❛✐♦r é ♠❛✐♦r q✉❡ ❛ ár❡❛ ❞♦ ♣♦❧í❣♦♥♦ ♠❡♥♦r✳
✹✳ ❆ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ✶ ❝♠ é 1cm2 ✳
Pr♦♣♦s✐çã♦ ✳✺✺✳ ❯♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ l t❡♠ ár❡❛ l2 ✳
❉❡♠♦♥str❛çã♦✳ ❉✐s❝✉t✐r❡♠♦s ❛ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦l✱l ∈N✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ P♦st✉❧❛❞♦ ✷ ❝✐t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ l é ♣❛rt✐❝✐♦♥❛❞♦
❡♠ l2 q✉❛❞r❛❞♦s ❞❡ ❧❛❞♦ ✉♥✐tár✐♦✱ ♦✉ s❡❥❛✱
l2 é ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s q✉❛❞r❛❞♦s
❞❡ ❧❛❞♦s ✉♥✐tár✐♦s ✲ Al = l2✱ ♦♥❞❡
Al r❡♣r❡s❡♥t❛ ❛ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ l✳ P♦r ❡①❡♠♣❧♦✱ s❡ l = 3 cm✱ ❡♥tã♦ A3 = 3
2
= 9 cm2✳ ❖❜s❡r✈❡ ❛ ✜❣✉r❛ ✸✽✳
❋✐❣✉r❛ ✸✽✿ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ 3 cm
❊ s❡ ♦ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦ ❢♦r ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❞♦ t✐♣♦ m
n✱ ❝♦♠ m, n∈N ❄
❈♦♥s✐❞❡r❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ l = 3
4 cm✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ ❝♦♥str✉í♠♦s 4 2
❝ó♣✐❛s ❞❡ss❡ q✉❛❞r❛❞♦ ❞❡ ár❡❛A3
4 ❡✱ ❡♠ s❡❣✉✐❞❛✱ ♦ s❡❣✉✐♥t❡ ❛rr❛♥❥♦✱ ❞❡ ♠♦❞♦ q✉❡ t❡♥❤❛♠♦s ✉♠ q✉❛❞r❛❞♦ ♠❛✐♦r ❞❡ ❧❛❞♦ 3
4.4 = 3 cm✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✸✾✳
❋✐❣✉r❛ ✸✾✿ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ 3 4 cm ✳
P❡❧♦ P♦st✉❧❛❞♦s ✸✱ ❛ ár❡❛ ❞❡ss❡ q✉❛❞r❛❞♦ ♠❛✐♦r é 32
= 42
.A3
4✳ ❊♥tã♦✱
A3 4 =
32
42 =
3 4
2
cm2✳
●❡♥❡r❛❧✐③❛♥❞♦✱ ❝♦♥s✐❞❡r❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ m
n✱ ❝♦♠ m, n ∈ N✳ P♦r ❝♦♥✲
s❡❣✉✐♥t❡✱ ❝♦♥str✉í♠♦s n2 ❝ó♣✐❛s ❞❡ss❡ q✉❛❞r❛❞♦ ❞❡ ár❡❛
Am
n ❡✱ ❡♠ s❡❣✉✐❞❛✱ ♦ s❡❣✉✐♥t❡ ❛rr❛♥❥♦✱ ❞❡ ♠♦❞♦ q✉❡ t❡♥❤❛♠♦s ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ♠❛✐♦r m
n.n =m ✳
P❡❧♦ P♦st✉❧❛❞♦ ✸✱ ❛ ár❡❛ ❞❡ss❡ q✉❛❞r❛❞♦ ♠❛✐♦r é m2
= n2
.Am
n✳ ❊♥tã♦✱ A m n =
m2
n2 =
m n
2
✳
❋✐❣✉r❛ ✹✵✿ ❞❡♠♦♥str❛çã♦ ❞❛ ár❡❛ ❞❡ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ m n
❱❛♠♦s ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ l ∈ R+ ❡ ♠♦str❛r❡♠♦s q✉❡
Al=l2✱ ♦♥❞❡Al é ❛ ár❡❛ ❞❡ss❡ q✉❛❞r❛❞♦✳
❋✐❣✉r❛ ✹✶✿ ❆ ❞❡♥s✐❞❛❞❡ ❞❡ Q ❡♠ R
P❛r❛ ❝❛❞❛ K ∈ N∗✱ ❝♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ✐♥t❡r✈❛❧♦
J = l− 1 2k, l+
1 2k
✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✹✶✳ ❈♦♠♦ Q é ❞❡♥s♦ ❡♠ R✱ ❡①✐st❡♠ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s xk ❡ yk
t❛✐s q✉❡✿
(i) l− 1
2k < xk < l
(ii)l < yk < l+ 1 2k.
❙✉❜tr❛✐♥❞♦ ✭✐✐✮ ❞❡ ✭✐✮✱ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦✱ ♦❜t❡♠♦s✿
1
2k < yk−xk < 1
2k ⇐⇒yk−xk= 1 2k
❝♦♠♦ 1 2k <
1
k✱ t❡♠♦s q✉❡
yk−xk< 1 k.
❊♠ s❡❣✉✐❞❛✱ ❝♦♥str✉í♠♦s q✉❛❞r❛❞♦s ❞❡ ♠❡❞✐❞❛sxk ❡ yk✱ ❝♦♥❢♦r♠❡ s❡❣✉❡✳
❋✐❣✉r❛ ✹✷✿ ◗✉❛❞r❛❞♦s ❞❡ ❧❛❞♦sxk✱ l ❡ yk
❉❛ ✜❣✉r❛ ✹✷ ♣♦❞❡♠♦s ❡s❝r❡✈❡rxk< l < yk✱ ❝✉❥❛s ár❡❛s sã♦x2k< l
2
< y2
k✳ P❡❧♦
P♦st✉❧❛❞♦ ✸✱ ❡♥tã♦x2
k < Al< y
2
k✳ ❈♦♥❝❧✉✐✲s❡ q✉❡ ♦s ♥ú♠❡r♦s l
2
, Al∈(x2
k, y2
k)✳
❊♥tã♦✱
Al−l2
< y2
k−x
2
k
Al−l2
<(yk+xk)(yk−xk)
❝♦♠♦yk−xk< 1k ❡ xk < l ✿
Al−l2
<(yk+xk)1 k
<(yk−xk+ 2xk)1 k
<(1 k + 2l)
1 k =
1 k2 +
2l k.
❖❜s❡r✈❛♠♦s q✉❡ lim k→∞
1 k2 +
2l
k = 0✱ ∀k ∈N✱l ∈R+ ✜①♦✳ P♦rt❛♥t♦✱
Al−l2
= 0 ⇐⇒Al =l2
.
Pr♦♣♦s✐çã♦ ✳✺✻✳ ❯♠ r❡tâ♥❣✉❧♦s ❞❡ ❧❛❞♦s ❛ ❡ ❜ t❡♠ ár❡❛ ❛❜✳
❉❡♠♦♥str❛çã♦✳
❋✐❣✉r❛ ✹✸✿ ❘❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s ♥❛t✉r❛✐s
❉✐s❝✉t✐r❡♠♦s ♦ ❝❛s♦ ❞♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s ♥❛t✉r❛✐s✳ ❙✉♣♦♥❤❛♠♦s ✉♠ r❡tâ♥✲ ❣✉❧♦ ❞❡ ❧❛❞♦sa= 3 cm❡b= 2 cm✳ ❉♦ P♦st✉❧❛❞♦ ✷✱ ♣❛rt✐❝✐♦♥❛♠♦s ❡ss❡ r❡tâ♥❣✉❧♦
❡♠ ✻ q✉❛❞r❛❞♦s ❞❡ ❧❛❞♦ ✉♥✐tár✐♦ ✭P♦st✉❧❛❞♦ ✹✮✳ ▲♦❣♦✱ ❛ ár❡❛ é ❛ s♦♠❛ ❞❛s ár❡❛s ❞❡ss❡s q✉❛❞r❛❞♦s✳ ❊♥tã♦✱ ➚r❡❛ ❂3.2 = 6 cm2 ✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✹✸✳
❋✐❣✉r❛ ✹✹✿ ❘❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s r❛❝✐♦♥❛✐s
❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s a = 2
3 cm ❡ b =
7
5 cm✳ ❈♦♥str✉í♠♦s
3.5 = 15❝ó♣✐❛s ❞❡ss❡ r❡tâ♥❣✉❧♦✱ ❞❡ t❛❧ ♠♦❞♦ q✉❡ q✉❛♥❞♦ ♠♦♥t❛r♠♦s ✉♠ r❡tâ♥❣✉❧♦
♠❛✐♦r t❡♥❤❛♠♦s ✉♠ ❧❛❞♦ ♠❡❞✐♥❞♦ 2
3.3 = 2 cm ❡ ♦✉tr♦ 7
5.5 = 7 cm ✭✜❣✉r❛ ✹✹✮✳
❊♠ s❡❣✉✐❞❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r2.7 = 15.A✱ ♦♥❞❡ Aé ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s
2 3 ❡
7
5 ✳ ❊♥tã♦✱ A= 14
15 cm
2 ✳
●❡♥❡r❛❧✐③❛♥❞♦✱ t♦♠❡♠♦s ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s m1
n1 ❡
m2
n2✱ ❝♦♠m1, m2, n1, n2 ∈ N✱ ❝✉❥❛ ár❡❛ é A✳ ❈♦♥str✉í♠♦s n1.n2 ❝ó♣✐❛s ❞❡ss❡ r❡tâ♥❣✉❧♦✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱
♠♦♥t❛♠♦s ✉♠ r❡tâ♥❣✉❧♦ ♠❛✐♦r ❞❡ ❧❛❞♦s m1
n1.n1 = m1 ❡
m2
n2.n2 = m2✳ ❊♥tã♦✱
m1.m2 =n1.n2.A⇐⇒A=
m1
n1.
m2
n2 =
m1.m2
n1.n2 ✳
P♦r ❝♦♥s❡❣✉✐♥t❡✱ t♦♠❡♠♦s ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s a, b∈ R+ ❡ ∀k ∈ N ✱ ♥ú♠❡r♦s
r❛❝✐♦♥❛✐s xk, yk, wk, zk✱ t❛✐s q✉❡ xk < a < yk✱ wk < b < zk ✱ yk −xk < 1
k ❡ zk − wk < 1
k ✳ ❙❡♥❞♦ ❆ ❛ ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s a ❡ b✱ t❡♠♦s q✉❡✱
♠✉❧t✐♣❧✐❝❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦✱ ❛s ❞✉❛s ♣r✐♠❡✐r❛s ❞❡s✐❣✉❛❧❞❛❞❡s✿
xk.wk< ab < yk.zk
✱ ♦✉ s❡❥❛✱
xk.wk < A < yk.zk.
❈♦♥❝❧✉✐✲s❡ q✉❡ A, ab ∈ (xk.wk, yk.zk) ❯s❛♥❞♦ ✉♠ ❛r❣✉♠❡♥t♦ ❛♥á❧♦❣♦ ❢❡✐t♦ ♣❛r❛
♦s q✉❛❞r❛❞♦s✱ t❡♠♦s✿
0<|A−ab|< yk.zk−xk.wk = (zk−wk)yk+wk(yk−xk) = 1 kyk+
1 kwk,
♣♦✐syk−xk < 1
k ❡ zk−wk< 1 k
|A−ab|< 1
k(yk+wk)< 1
k((yk−xk) + 2xk+ (zk−wk) + 2wk)
< 1 k(
1
k + 2a+ 1
k + 2b),
♣♦✐sxk < a ❡wk< b
<( 2 k2 +
2a k +
2b k ).
❖❜s❡r✈❛♠♦s q✉❡ lim k→∞(
2 k2 +
2a k +
2b
k ) = 0✱ ∀k∈N
∗
❡ a, b∈R+ ✜①♦s✳ P♦rt❛♥t♦✱
|A−ab|= 0 ⇐⇒A=ab.
Pr♦♣♦s✐çã♦ ✳✺✼✳ ❆ ár❡❛ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❞❡ ❜❛s❡ ❜ ❡ ❛❧t✉r❛ ❤ é ❛❤✳
❋✐❣✉r❛ ✹✺✿ ➪r❡❛ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ♣♦♥t♦s ❊ ❡ ❋✱ ♣r♦❥❡çõ❡s ♦rt♦❣♦♥❛✐s ❞♦s ♣♦♥t♦s ❉ ❡ ❈ à r❡t❛ AB−→✳ ❈♦♥s✐❞❡r❡ ❛ ♥♦t❛çã♦ ♣❛r❛ ár❡❛✱ ♣♦r ❡①❡♠♣❧♦✱ A(ABCD)✱ é ❛ ár❡❛ ❞❡ ✉♠ q✉❛❞r✐❧át❡r♦ ❞❡ ✈ért✐❝❡s ❆✱ ❇✱ ❈ ❡ ❉✳ ❖❜s❡r✈❛♥❞♦
❛ ✜❣✉r❛ ✹✺✱ ♦s tr✐â♥❣✉❧♦s ADE ❡ BCF sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦ ❤✐♣♦t❡♥✉s❛
❝❛t❡t♦✱ ❞❡ ♠♦❞♦ q✉❡
AE =BF ❡A(ADE) =A(BCF)✱ ❡♥tã♦
A(ABCD) =A(ADE) +A(BCDE)
=A(BCF) +A(BCDE) A(ABCD) = A(CDEF).
▼❛s✱ ❈❉❊❋ é ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❛❧t✉r❛ ❤ ❡ ❜❛s❡✿ EF =EB+BF =EB+AE = AB =a✳
P♦rt❛♥t♦✱ A(ABCD) = A(CDEF) = ah
Pr♦♣♦s✐çã♦ ✳✺✽✳ ❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦ ❞❡ ❧❛❞♦s BC = a✱ AC = b ❡ AB = c ❡ ❛❧t✉r❛s ha, hb ❡ hc✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ r❡❧❛t✐✈❛s ❛♦s ❧❛❞♦s a, b ❡ c✳ ❊♥tã♦✱ A(ABC) = aha
2 =
bhb
2 =
chc 2
❋✐❣✉r❛ ✹✻✿ ❞❡♠♦♥str❛çã♦ ❞❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦
❉❡♠♦♥str❛çã♦✳
❙❡❥❛ S = A(ABC) ❡ ❉ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛s ♣❛r❛❧❡❧❛s ❛
←→
BC ♣♦r ❆
❡ AB←→ ♣♦r C ✭✜❣✉r❛✿ ✹✻✮✳ ▲♦❣♦✱ ♦ q✉❛❞r✐❧át❡r♦ ABCD é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❞❡
ár❡❛ 2S ✭✉♠❛ ✈❡③ q✉❡ A(ABC) = A(ACD)✱ ♣♦✐s △ABC ∼= △ACD ♣❡❧♦ ❝❛s♦ LLL✮✳ P♦rt❛♥t♦✱ 2A(ABC) = 2S =aha −→S =
aha
2 ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ t❡♠♦s q✉❡
2S =bhb =chc
Pr♦♣♦s✐çã♦ ✳✺✾✳ ❆ ár❡❛ ❞❡ ✉♠ ❧♦s❛♥❣♦ ❞❡ ❞✐❛❣♦♥❛✐s d1 ❡ d2 é
d1·d2
2 ✳
❖ ❧♦s❛♥❣♦ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❞❡ q✉❛tr♦ ❧❛❞♦s ❝♦♥❣r✉❡♥t❡s✳ P♦❞❡♠♦s ❞❡❝♦♠♣ô✲ ❧♦ ❡♠ q✉❛tr♦ tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s ❡♥tr❡ s✐✱ ♦♥❞❡ ❞❡❞✉③✐r❡♠♦s ✉♠❛ r❡❧❛çã♦✳
❈♦♠ ❡ss❛ ❞❡❝♦♠♣♦s✐çã♦✱ ♣♦❞❡♠♦s ♠♦♥t❛r ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❜❛s❡ d1 ❡ ❛❧t✉r❛
d2
2✱
❝♦♥❢♦r♠❡ s❡❣✉❡✳ ❉❡♠♦♥str❛çã♦✳
❋✐❣✉r❛ ✹✼✿ ➪r❡❛ ❞❡ ✉♠ ❧♦s❛♥❣♦
❇❛st❛ ♦❜s❡r✈❛r ❞❛ ✜❣✉r❛ ✹✼ q✉❡ ❛ ár❡❛ ❞♦ ❧♦s❛♥❣♦ ABCD é ✐❣✉❛❧ ❛ ár❡❛ ❞♦
r❡tâ♥❣✉❧♦ ACD′
D′′
✱ ❡♥tã♦ A(ABCD) =d1·
d2
2 =
d1·d2
2 ✳
Pr♦♣♦s✐çã♦ ✳✻✵✳ ❆ ár❡❛ ❞❡ ✉♠ tr❛♣é③✐♦ ❞❡ ❛❧t✉r❛h✱ ❜❛s❡ ♠❛✐♦r B ❡ ❜❛s❡ ♠❡♥♦r b é (B +b)·h
2 ✳
❖ tr❛♣é③✐♦ é ✉♠ q✉❛❞r✐❧át❡r♦ q✉❡ ♣♦ss✉✐ ✉♠ ♣❛r ❞❡ ❧❛❞♦s ♣❛r❛❧❡❧♦s ❡ ✉♠ ♣❛r ❞❡ ❧❛❞♦s ♥ã♦ ♣❛r❛❧❡❧♦s✳ P♦❞❡♠♦s ❞❡❝♦♠♣ô✲❧♦ ❡♠ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡ ✉♠ tr✐â♥❣✉❧♦✱ ♦ q✉❡ ♠♦str❛r❡♠♦s ❛ s❡❣✉✐r✳
❉❡♠♦♥str❛çã♦✳
❋✐❣✉r❛ ✹✽✿ ➪r❡❛ ❞❡ ✉♠ tr❛♣é③✐♦
❈♦♥s✐❞❡r❡ ✉♠ tr❛♣é③✐♦ ABCD ✭✜❣✉r❛✿ ✹✽✮ ❞❡ ❛❧t✉r❛ h ❡ ❜❛s❡s BC = B ❡ AD=b✳ ❚r❛ç❛♠♦s ❛ r❡t❛ r ♣❡❧♦ ♣♦♥t♦ D✱ t❛❧ q✉❡ r||
←→
AB✳