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➤➵✐ ❤ä❝ ➤å♥❣ t❤➳♣ ❑❤♦❛ ❚♦➳♥ ◆❣✉②Ô♥ ❉➢➡♥❣ ❍♦➭♥❣ ❜➭✐ ❣✐➯♥❣ ♠ét sè ♣❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ t♦➳♥ ❤×♥❤ ❤ä❝ tr✉♥❣ ❤ä❝ ❝➡ së ➤å♥❣ t❤➳♣ ✲✷✵✵✾ ▼ô❝ ❧ô❝ ✶ ❝➳❝ ❞➵♥❣ t♦➳♥ t❤➢ê♥❣ ❣➷♣ ✹ ✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳ ✸✳ ❝➳❝ ❜➭✐ t♦➳♥ ✈Ị ➤❛ ❣✐➳❝ ✶✳✶✳ ❚❛♠ ❣✐➳❝ ✶✳✷✳ ❍×♥❤ t❤❛♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸✳ ❍×♥❤ ❜×♥❤ ❤➭♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹✳ ❍×♥❤ ❝❤÷ ♥❤❐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✺✳ ❍×♥❤ t❤♦✐ ✈➭ ❤×♥❤ ✈✉➠♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳ ➜Þ♥❤ ❧Ý ❚❛ ✲❧❡t tr♦♥❣ t❛♠ ❣✐➳❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳ ❚❛♠ ❣✐➳❝ ➤å♥❣ ❞➵♥❣ ✷✻ ❚❛♠ ❣✐➳❝ ➤å♥❣ ❞➵♥❣ ❍Ư t❤ø❝ ❧➢ỵ♥❣ tr♦♥❣ t❛♠ ❣✐➳❝ ✲ ❈➳❝ ❜➭✐ t♦➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✶✳ ❍Ư t❤ø❝ ❧➢ỵ♥❣ tr♦♥❣ t❛♠ ❣✐➳❝ ✈✉➠♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷✳ ❚Ø sè ❧➢ỵ♥❣ ❣✐➳❝ ❝đ❛ ❣ã❝ ♥❤ä♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✶✳ ❚Ý♥❤ ❞✐Ö♥ tÝ❝❤ ➤❛ ❣✐➳❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✷✳ ❚Ý♥❤ ❞✐Ö♥ tÝ❝❤ ➤❛ ❣✐➳❝ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ➤➵✐ sè ✳ ✳ ✳ ✸✻ ✹✳✸✳ ▼ét sè ➳♣ ❞ơ♥❣ ❝đ❛ tØ sè ❞✐Ư♥ tÝ❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✈Ị ❞✐Ư♥ tÝ❝❤ ✹✳ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❉✐Ö♥ tÝ❝❤ ➤❛ ❣✐➳❝ ✈➭ tØ sè ❞✐Ö♥ tí t ề ự trị ì ọ ✷✳ ✸✳ ✸✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✶✳✶✳ ❇➭✐ t♦➳♥ ❝ù❝ trÞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✶✳✷✳ ▼ét sè ❦✐Õ♥ t❤ø❝ t❤➢ê♥❣ ❞ï♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✶✳ ➜❛ ❣✐➳❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✷✳ ❚❛♠ ❣✐➳❝ ➤å♥❣ ❞➵♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✸✳ ➜➢ê♥❣ trß♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ P❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ ❜➭✐ t♦➳♥ ự trị t ề ự trị ì ọ P❤➢➡♥❣ ♣❤➳♣ ♣❤➯♥ ❝❤ø♥❣✱ ♥❣✉②➟♥ ❧Ý ❞✐✐rr✐❝❤❧❡ ✹✼ ✸✳✶✳ P❤➢➡♥❣ ♣❤➳♣ ♣❤➯♥ ❝❤ø♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✷✳ ◆❣✉②➟♥ ❧Ý ➜✐r✐❝❤❧❡ ✹✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ✸ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ✳✳✳ ❝➳❝ ➤✐Ó♠ t❤➻♥❣ ❤➭♥❣✱ ❝➳❝ ➤➢ê♥❣ t❤➻♥❣ ➤å♥❣ q✉②✱ ❝➳❝ ➤✐Ĩ♠ ♥➺♠ tr➟♥ ➤➢ê♥❣ trß♥ ✺✶ ✶✳ ✷✳ ✸✳ ✹✳ ✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ➤Þ♥❤ ❧Ý ✈Ị tØ sè ✈➭ ➳♣ ❞ơ♥❣ ✶✳✶✳ ❈➳❝ ➤Þ♥❤ ❧Ý ✶✳✷✳ ❈➳❝ ♣❤➢➡♥❣ ❞➵♥❣ t♦➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✷✳✶✳ ❈➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✷✳✷✳ ❈➳❝ ❞➵♥❣ t♦➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✳ ✳ ✳ ✳ ✺✹ ✸✳✶✳ ❈➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✷✳ ❈➳❝ ♣❤➢➡♥❣ ❞➵♥❣ t♦➳♥ ✺✹ ❈❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤✐Ó♠ t❤➻♥❣ ❤➭♥❣ ❈❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤➢ê♥❣ t❤➻♥❣ ➤å♥❣ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤ø♥❣ ♠✐♥❤ ➤➢ê♥❣ t❤➻♥❣ ➤✐ q✉❛ ➤✐Ĩ♠ ❝è ➤Þ♥❤ ✺✺ ✹✳✶✳ ❈➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✷✳ ❈➳❝ ♣❤➢➡♥❣ ❞➵♥❣ t♦➳♥ ✺✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤ø♥❣ ♠✐♥❤ ♥❤✐Ị✉ ➤✐Ĩ♠ ♥➺♠ tr➟♥ ➤➢ê♥❣ trß♥ ✺✻ ✺✳✶✳ ❈➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✺✳✷✳ ❈➳❝ ♣❤➢➡♥❣ ❞➵♥❣ t♦➳♥ ✺✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✫ ✸ ❈❤➢➡♥❣ ✶ ❝➳❝ ❞➵♥❣ t♦➳♥ t❤➢ê♥❣ ❣➷♣ ✶✳ ❝➳❝ ❜➭✐ t♦➳♥ ✈Ò ➤❛ ❣✐➳❝ ❚❛♠ ❣✐➳❝ ✶✳✶✳ ❦✐Õ♥ t❤ø❝ ❝➬♥ ♥❤í ✶✳ ❈➳❝ tr➢ê♥❣ ❤ỵ♣ ❜➺♥❣ ♥❤❛✉ ❝đ❛ t❛♠ ❣✐➳❝✿ ❍❛✐ t❛♠ ❣✐➳❝ ❆❇❈ ✈➭ ❆✬❇✬❈✬ ❜➺♥❣ ♥❤❛✉ ♥Õ✉ t❤á❛ ♠ét tr♦♥❣ ❝➳❝ tr➢ê♥❣ ❤ỵ♣ s❛✉ ➤➞②✿ ❛✮ AB = A B ; A = A ; AC = A C ❜✮ A = A ; AB = A B ; B = B ❝✮ AB = A B ; BC = B C ; CA = C A r t ã tr ì ❝đ❛ t❛♠ ❣✐➳❝ t❤× s♦♥❣ s♦♥❣ ✈í✐ ❝➵♥❤ t❤ø ❜❛ ✈➭ ❜➺♥❣ ♥ư❛ ❝➵♥❤ ✃②✳ • ❇❛ ➤➢ê♥❣ ❝❛♦ ➤å♥❣ q✉② t➵✐ ♠ét ➤✐Ĩ♠ ❣ä✐ ❧➭ trù❝ t➞♠ ❝đ❛ t❛♠ ❣✐➳❝✳ • ❇❛ ➤➢ê♥❣ tr✉♥❣ trù❝ ➤å♥❣ q✉② t➵✐ ➤✐Ĩ♠ ❖ ❣ä✐ ❧➭ t➞♠ ➤➢ê♥❣ trß♥ ♥❣♦➵✐ t✐Õ♣ t❛♠ ❣✐➳❝✳ • ❇❛ ➤➢ê♥❣ ♣❤➞♥ ❣✐➳❝ ➤å♥❣ q✉② t➵✐ ➤✐Ó♠ ■ ọ t trò ộ tế t ã ❇❛ ➤➢ê♥❣ tr✉♥❣ t✉②Õ♥ ➤å♥❣ q✉② t➵✐ ♠ét ➤✐Ó♠ ● ❣ä✐ ❧➭ trä♥❣ t➞♠ ❝đ❛ t❛♠ ❣✐➳❝ ✈➭ ❆●❂✷●▼ ✭✈í✐ ▼ ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❇❈✮✳ ❈➳❝ ❞➵♥❣ t♦➳♥ ➜Ị ✶✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈✳ ❉ù♥❣ r❛ ♣❤Ý❛ ♥❣♦➭✐ t❛♠ ❣✐➳❝ ❆❇❈ ❝➳❝ t❛♠ ❣✐➳❝ ✈✉➠♥❣ ❝➞♥ t➵✐ ❆ ❧➭ ❆❇❉ ✈➭ ❆❈❊✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ➤➢ê♥❣ ❝❛♦ ❆❍ ❝ñ❛ t❛♠ ❣✐➳❝ ❆❇❈ ➤✐ q✉❛ tr✉♥❣ ➤✐Ĩ♠ ■ ❝đ❛ ➤♦➵♥ ❉❊✳ ❍➢í♥❣ ❞➱♥ ✿ ✹ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ơ♥ ❉ ❍♦➭♥❣ ✳✳✳ ●ä✐ ■ ❧➭ ❣✐❛♦ ➤✐Ĩ♠ ❝đ❛ ❆❍ ✈➭ ❉❊✳ ◗✉❛ ❉ ❦❰ DD ⊥ AH ✳ ◗✉❛ ❊ ❦❰ EE ⊥ AH ✳ ❍❛✐ t❛♠ ❣✐➳❝ ✈✉➠♥❣ ❆❇❍ ✈➭ ❉❆❉✬ ❝ã ABH = DAD ✱ ❆❇❂❆❉✱ s✉② r❛ ❉❉✬❂❆❍✭✶✮✳ ❚➢➡♥❣ tù t❛♠ ❣✐➳❝ ❆❍❈❂t❛♠ ❣✐➳❝ ❊❊✬❆ s✉② r❛ ❊❊✬❂❆❍✭✷✮✳ ❚õ ✭✶✮ ✈➭ ✭✷✮ s✉② r❛ ❉❉✬❂❊❊✬✳ ❍❛✐ t❛♠ ❣✐➳❝ ✈✉➠♥❣ ❉❉✬■ ✈➭ ❊❊✬■ ✭❉❉✬❂❊❊✬ ✈➭ IDD = IEE ✭s♦ ❧❡ tr♦♥❣✮ ❜➺♥❣ ♥❤❛✉ s✉② r❛ ❉■❂❊■✱ tø❝ ❧➭ ■ ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❉❊✳ ➜Ị ✷✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈ ✈✉➠♥❣ t➵✐ ❇✱ tr➟♥ t✐❛ ➤è✐ ❝đ❛ t✐❛ ❇❆✱ ❧✃② ➤✐Ĩ♠ ❉ s❛♦ ❝❤♦ ❆❉❂✸❆❇✳ ➜➢ê♥❣ t❤➻♥❣ ✈✉➠♥❣ ❣ã❝ ✈í✐ ❈❉ t➵✐ ❉ ❝➽t ➤➢ê♥❣ t❤➻♥❣ ✈✉➠♥❣ ❣ã❝ ✈í✐ ❆❈ ë ❊✳ ❈❤ø♥❣ ♠✐♥❤ t❛♠ ❣✐➳❝ ❇❉❊ ❝➞♥ ❍➢í♥❣ ❞➱♥ ✿ ●ä✐ ◆✱ ■✱ ▼ ❧➬♥ ❧➢ỵt ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❈❊✱ ❆❉✱ ❇❉✳ ❚õ ❆❉ ❂✸❆❇ s✉② r❛ ■ ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❇▼✳ ❚❛ ❝ã ❆◆❂❉◆❂✶✴✷❈❊ ✭❞♦ t❛♠ ❣✐➳❝ ❆❈❊ ✈➭ ❉❈❊ ✈✉➠♥❣✮ ⇒ IN ⊥ AD✳ ●ä✐ ❏ ❧➭ ❣✐❛♦ ➤✐Ĩ♠ ❝đ❛ ◆■ ❦Ð♦ ❞➭✐ ✈í✐ ❇❊ t❤× ◆❏ ❧➭ ➤➢ê♥❣ tr✉♥❣ ❜×♥❤ ❝đ❛ ∆ ❇❈❊✳ ❙✉② r❛ ■❏ ❧➭ ➤➢ê♥❣ tr✉♥❣ ❜×♥❤ ❝ñ❛ ∆ ❇▼❊ ❞♦ ➤ã ▼❊ ✴✴ ■❏ ⇒ M E ⊥ BD ✳ ❚❛♠ ❣✐➳❝ ❇❊❉ ❝➞♥ t➵✐ ❊✳ ➜Ị ✸✳ ❳➳❝ ➤Þ♥❤ ❝➳❝ ❣ã❝ ❝đ❛ t❛♠ ❣✐➳❝ ❆❇❈✱ ❜✐Õt r➺♥❣ ➤➢ê♥❣ ❝❛♦ ❆❍ ✈➭ tr✉♥❣ t✉②Õ♥ ❆▼ ❝❤✐❛ ❣ã❝ ❍➢í♥❣ ❞➱♥ BAC t❤➭♥❤ ✸ ♣❤➬♥ ❜➺♥❣ ♥❤❛✉✳ ✿ ❑❰ M K ⊥ AC ✱ ❦❤✐ ➤ã ∆AHM = ∆AKM s✉② r❛ ▼❍❂▼❑ ✭✶✮✳ A1 = A2 s✉② r❛ t❛♠ ❣✐➳❝ ❆❇▼ ❝➞♥ t➵✐ ❆ ⇒ BH = M H ✭✷✮ ❚õ ✭✶✮ ✈➭ ✭✷✮ t❛ ❝ã ▼❑❂✶✴✷▼❈✱ s✉② r❛ t❛♠ ❣✐➳❝ ✈✉➠♥❣ ❑▼❈ ❧➭ ♥ư❛ t❛♠ ❣✐➳❝ ➤Ị✉ ❝➵♥❤ ▼❈✱ ➤➢ê♥❣ ❝❛♦ ❈❑✳ ✫ ✺ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ✳✳✳ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ C = 30o ✱ s✉② r❛ M3 = 60o ✳ ❚❛ ❝ã M1 = M2 ✭❞♦ ∆AHM = ∆AKM ✮ ♥➟♥ M1 = M2 = 1/2HKM = 60o ✱ ⇒ A1 = A2 = A3 = 30o ✳ ❱❐② ∆ABC ✈✉➠♥❣ t➵✐ ❆ ✈➭ ❝ã B = 60o ; C = 30o ❉♦ ➤ã ➜Ò ✹✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈✱ ❦❰ ➤➢ê♥❣ ❝❛♦ ❆❍✳ ●ä✐ ❈✬ ❧➭ ➤✐Ĩ♠ ➤è✐ ①ø♥❣ ❝đ❛ ❍ q✉❛ ❆❇❀ ❇✬ ❧➭ ➤✐Ĩ♠ ➤è✐ ①ø♥❣ ❝đ❛ ❍ q✉❛ ❆❈✳ ●ä✐ ❣✐❛♦ ➤✐Ĩ♠ ❝đ❛ ❇✬❈✬ ✈í✐ ❆❈ ✈➭ ❆❇ t❤❡♦ t❤ø tù ❧➭ ■ ✈➭ ❑✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ❇■✱ ❈❑ ❧➭ ❝➳❝ ➤➢ê♥❣ ❝❛♦ ❝đ❛ t❛♠ ❣✐➳❝ ❆❇❈✳ ❍➢í♥❣ ❞➱♥ ✿ ❚❛ ❝ã AC K = AHK ✭❞♦ tÝ♥❤ ➤è✐ ①ø♥❣✮❀ AHI = AB I ✳ ▼➭ AC K = AB I ✭t❛♠ ❣✐➳❝ ❈✬❆❇✬ ❝➞♥✮ s✉② r❛ AHK = AHI ⇒ ❆❍ ❧➭ ♣❤➞♥ ❣✐➳❝ ❝ñ❛ ❣ã❝ KHI ✳ ❱❐② ❇❍ ❧➭ ♣❤➞♥ ❣✐➳❝ ♥❣♦➭✐ ❝ñ❛ t❛♠ ❣✐➳❝ ❍❑■✳ ❚õ ➤Þ♥❤ ❧Ý ❜❛ ➤➢ê♥❣ ♣❤➞♥ ❣✐➳❝ s✉② r❛ ❇■ ❧➭ ♣❤➞♥ ❣✐➳❝ tr♦♥❣ ❝ñ❛ ❣ã❝ HIK ⇒ BI ⊥ IC ✭❤❛✐ ➤➢ê♥❣ ♣❤➞♥ ❣✐➳❝ tr♦♥❣ ✈➭ ♥❣♦➭✐ t❤× ✈✉➠♥❣ ❣ã❝✮✳ ❱❐② ❇■ ❧➭ ➤➢ê♥❣ ❝❛♦ ❝ñ❛ t❛♠ ❣✐➳❝ ❆❇❈✳ ❚➢➡♥❣ tù ❈❑ ❧➭ ➤➢ê♥❣ ❝❛♦ t❛♠ ❣✐➳❝ ❆❇❈ ➜Ò ✺✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈ ✈✉➠♥❣ t➵✐ ❆ ❝ã √ AB = c, AC = b ✈➭ ➤➢ê♥❣ ♣❤➞♥ 1 ❣✐➳❝ tr♦♥❣ AD = d✳ ❈❤ø♥❣ ♠✐♥❤✿ = + ✳ d b c ❍➢í♥❣ ❞➱♥ ✿ ❚❛ ❝ã AC K = AHK ✭❞♦ tÝ♥❤ ➤è✐ ①ø♥❣✮❀ AHI = AB I ✳ ▼➭ AC K = AB I ✭t❛♠ ❣✐➳❝ ❈✬❆❇✬ ❝➞♥✮ s✉② r❛ AHK = AHI ⇒ ❆❍ ❧➭ ♣❤➞♥ ❣✐➳❝ ❝ñ❛ ❣ã❝ KHI ✳ ❱❐② ❇❍ ❧➭ ♣❤➞♥ ❣✐➳❝ ♥❣♦➭✐ ❝ñ❛ t❛♠ ❣✐➳❝ ❍❑■✳ ✫ ✻ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ➜Ị ✳✳✳ ❚❙✳ ◆❣✉②Ơ♥ ❉ ❍♦➭♥❣ ✻✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈ ➤Ị✉ ✈➭ ➤✐Ĩ♠ ▼ ë tr♦♥❣ t ọ ợt ì ❝❤✐Õ✉ ✈✉➠♥❣ ❣ã❝ ❝ñ❛ ▼ ❧➟♥ ❝➳❝ ❝➵♥❤ ❇❈✱ ❈❆✱ ❆❇✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣✿ ❛✮ M A + M B + M C ❦❤➠♥❣ ➤æ✐✳ ❜✮ AC + BA + CB ❦❤➠♥❣ ➤ỉ✐✳ ❍➢í♥❣ ❞➱♥ ✿ ❛✮ ➜➷t ❆❇❂❇❈❂❈❆❂❛✳ ❑❰ ➤➢ê♥❣ ❝❛♦ ❆❍❂❤✳ ❚❛ ❝ã SABC = SM AB + 1 1 SM BC + SM CA ✳ ❙✉② r❛ a.h = a.M A + a.M B + a.M C ✳ ❍❛② 2 2 h = M A + M B + M C ❦❤➠♥❣ ➤æ✐✳ ❜✮ ◗✉❛ ▼ ❦❰ ❝➳❝ ➤♦➵♥ t❤➻♥❣ s♦♥❣ s♦♥❣ ✈í✐ ❝➳❝ ❝➵♥❤ ❝đ❛ t❛♠ ❣✐➳❝✳ ❉Ơ t❤✃② ❝➳❝ t❛♠ ❣✐➳❝ ▼■❏✱ ▼❑▲✱ ▼P◗ ➤Ò✉ ❝ã ❝➳❝ ➤➢ê♥❣ ❝❛♦ t➢➡♥❣ ø♥❣ ❧➭ ▼❆✬✱ ▼❇✬✱ ▼❈✬✳ ●ä✐ ①✱ ②✱ ③ ❧➬♥ ❧➢ỵt ❧➭ ❝➵♥❤ ❝đ❛ t❛♠ ❣✐➳❝ ➤Ị✉ ▼■❏✱ ▼❑▲✱ ▼P◗✱ t❛ ❝ã✿ ❛❂❆❇❂❆P✰P◗✰◗❇❂▼▲✰P◗✰▼■❂①✰②✰③✳ ❉♦ ➤ã t❛ ❝ã✿ ❆❈✬✰❇❆✬✰❈❇✬❂✭❆P✰P❈✬✮✰✭❇■✰■❆✬✮✰✭❈❑✰❑❇✬✮✳ z x y (y + ) + (z + ) + (x + ) 2 3a ❂ (x + y + z) = ✳ 2 ❂ ➜Ò ✼✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈✱ tr➟♥ t✐❛ ➤è✐ ❝ñ❛ t✐❛ ❇❆✱ ❈❆ ❧➬♥ ❧➢ỵt ❧✃② ➤✐Ĩ♠ P✱ ◗ s❛♦ ❝❤♦ ❇P❂❈◗✳ ●ä✐ ▼✱ ◆ ❧➬♥ ❧➢ỵt ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❝➳❝ ➤♦➵♥ ❇❈✱ P◗✳ ➜➢ê♥❣ t❤➻♥❣ ▼◆ ❝➽t ➤➢ê♥❣ t❤➻♥❣ ❆❇ ✈➭ ❆❈ t❤❡♦ t❤ø tù t➵✐ ❇✬✱ ❈✬✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ t❛♠ ❣✐➳❝ ❇✬❆❈✬ ❝➞♥✳ ❍➢í♥❣ ❞➱♥ ✿ ✫ ✼ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ✳✳✳ ●ä✐ ■ ❧➭ tr✉♥❣ ➤✐Ó♠ ➤♦➵♥ ❇◗✳ ❑❤✐ ➤ã ■▼ ❧➭ ➤➢ê♥❣ tr✉♥❣ ❜×♥❤ ❝đ❛ t❛♠ ❣✐➳❝ ❇❈◗✳ ❙✉② r❛ ■▼ ✴✴ ❈◗ ✈➭ IM = 1/2CQ ✭✶✮ ▼➷t ❦❤➳❝ ◆■ ❧➭ ➤➢ê♥❣ tr✉♥❣ ❜×♥❤ ❝đ❛ t❛♠ ❣✐➳❝ ❇P◗✱ s✉② r❛ ■◆ ✴✴ ❇❈ ✈➭ IN = 1/2BP ✭✷✮ ❚❤❡♦ ❣✐➯ t❤✐Õt ❇P❂❈◗ ♥➟♥ tõ ✭✶✮ ✈➭ ✭✷✮ s✉② r❛ ■▼❂■◆✳ ❉♦ ➤ã t❛♠ ❣✐➳❝ ■▼◆ ❝➞♥ ✈➭ IM N = IN M ✭✸✮ ❚❤❡♦ ✭✶✮ t❛ ❝ã ■▼ ✴✴ ❆◗✱ s✉② r❛ ❞♦ ➤ã IM N = C1 ✭✹✮ ❚❤❡♦ ✭✷✮ t❛ ❝ã ■◆ ✴✴ ❆P✱ s✉② r❛ ❚õ ✭✸✮✱ ✭✹✮✱ ✭✺✮ s✉② r❛ ✶✳✷✳ IM N = C2 ✭➤å♥❣ ✈Þ✮✳ ▼➭ C1 = C2 IN M = AB C ✭s♦ ❧❡ tr♦♥❣✮ ✭✺✮ C1 = AB C ✳ ❉♦ ➤ã t❛♠ ❣✐➳❝ ❇✬❆❈✬ ❝➞♥✳ ❍×♥❤ t❤❛♥❣ ❦✐Õ♥ t❤ø❝ ❝➬♥ ♥❤í • ❚ø ❣✐➳❝ ❆❇❈❉ ❧➭ ❤×♥❤ t❤❛♥❣ ⇔ AB//CD ∨ AD//BC • ➜♦➵♥ t❤➻♥❣ ♥è✐ tr✉♥❣ ➤✐Ĩ♠ ❤❛✐ ❝➵♥❤ ❜➟♥ ợ ọ tr ì ì t ã tr✉♥❣ ❜×♥❤ ❤×♥❤ t❤❛♥❣ s♦♥❣ s♦♥❣ ✈í✐ ❤❛✐ ❝➵♥❤ ➤➳② ✈➭ ❝ã ➤é ❞➭✐ ❜➺♥❣ ♥ư❛ tỉ♥❣ ➤é ❞➭✐ ❤❛✐ ã ì t ì t ó ó ã ì t ó é ì t ợ ❧➵✐✳ ❈➳❝ ❞➵♥❣ t♦➳♥ ➜Ò ✶✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ♥Õ✉ ➤♦➵♥ t❤➻♥❣ ♥è✐ ❝➳❝ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❝➷♣ ❝➵♥❤ ➤è✐ ❞✐Ư♥ ❝đ❛ ♠ét tø ❣✐➳❝ ❜➺♥❣ ♥÷❛ tỉ♥❣ ❤❛✐ ❝➵♥❤ ❦✐❛ t❤× tø ❣✐➳❝ ➤ã ❧➭ ❤×♥❤ t❤❛♥❣ ❍➢í♥❣ ❞➱♥ ✿ ✫ ✽ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ✳✳✳ ❈➳❝❤ ✶✿ ●✐➯ sư ▼✱ ◆ ❧➬♥ ❧➢ỵt ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❤❛✐ ❝➵♥❤ ➤è✐ ❆❉ ✈➭ ❇❈ ❝ñ❛ tø ❣✐➳❝ ❆❇❈❉ ✈➭ MN = AB + CD ✱ t❛ ❝❤ø♥❣ ♠✐♥❤ ❆❇ ✴✴ ❈❉✳ ●ä✐ ■ ❧➭ tr✉♥❣ ➤✐Ó♠ ❆❈✳ ❚❛ ❝ã ▼■ ❧➭ ➤➢ê♥❣ tr✉♥❣ ❜×♥❤ ❝đ❛ t❛♠ ❣✐➳❝ ❆❉❈ ♥➟♥ ▼■ ✴✴ ❈❉ ✈➭ CD ✭✶✮ MI = AB ✳ ❚õ ✭✶✮ ✈➭ ✭✷✮ s✉② r❛ M N + AB + CD AB + CD NI = ✳ ❚❤❡♦ ➤Ò ❜➭✐ M N = ♥➟♥ tõ ✭✸✮ s✉② r❛ M N = M I + N I ✳ ❉♦ ➤ã ▼✱ ◆✱ ■ t❤➻♥❣ ❤➭♥❣✳ ❚õ ➤ã s✉② r❛ ❆❇ ❚➢➡♥❣ tù ◆■ ✴✴ ❆❇ ✈➭ NI = ✴✴ ❈❉ ✴✴ ▼◆✳ ✿ ❈➳❝❤ ✷ AB + CD ✭✶✮✳ ❚r➟♥ t✐❛ ❆◆ ❧✃② ➤✐Ó♠ P s❛♦ ❝❤♦ ❆◆❂◆P✳ ❑❤✐ ➤ã ∆ ❆◆❇❂ ∆ P◆❈ ✭❝✱❣✱❝✮ ⇒ ❆❇✰❈P✳ ❚❛ ❧➵✐ ❝ã ▼◆ PD ❧➭ ➤➢ê♥❣ tr✉♥❣ ❜×♥❤ ❝đ❛ ∆ ❆❉P ♥➟♥ ▼◆ ✴✴ ❉P ✈➭ M N = ✭✷✮✳ ❚õ ✭✶✮ ✈➭ ✭✷✮ s✉② r❛ ❉P❂❆❇✰❈❉ ⇒ ❉P❂❉❈✰❈P ✭✈× ❆❇❂❈P✮ ⇒ ❉✱ ❈✱ P t❤➻♥❣ ❤➭♥❣✳ ❉♦ ➤ã ❆❇ ✴✴ ❈❉ ⇒ ❆❇❈❉ ❧➭ ❤×♥❤ t❤❛♥❣✳ ●✐➯ sư ➜Ị MN = ✷✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ tæ♥❣ ❝➳❝ ❦❤♦➯♥❣ ❝➳❝❤ tõ ❜❛ ➤Ø♥❤ ❝ñ❛ ♠ét t❛♠ ❣✐➳❝ ➤Õ♥ ♠ét ➤➢ê♥❣ t❤➻♥❣ ♥➺♠ ♥❣♦➭✐ t❛♠ ❣✐➳❝ ❜➺♥❣ ❜❛ ❧➬♥ ❦❤♦➯♥❣ ❝➳❝❤ tõ trä♥❣ t➞♠ t❛♠ ❣✐➳❝ ➤Õ♥ ➤➢ê♥❣ t❤➻♥❣ ➤ã✳ ✿ ●ä✐ ❊ ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❆❈❀ ● ❧➭ trä♥❣ t➞♠ ❝đ❛ ❍➢í♥❣ ❞➱♥ ∆ ❆❇❈❀ ▼ ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❇●✳ ●ä✐ ❆✬✱ ❇✬✱ ❈✬✱ ●✬✱ ▼✱ ❊✬ ❧➬♥ ❧➢ỵt ❧➭ ❤×♥❤ ❝❤✐Õ✉ ✈✉➠♥❣ ❣ã❝ ❝đ❛ ❆✱ ❇✱ ❈✱ ●✱ ▼✱ ❊ tr➟♥ ➤➢ê♥❣ t❤➻♥❣ ①②✳ ❑❤✐ ➤ã ✫ ✾ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ✳✳✳ t❛ ❝ã✿ ❆❆✬ ✴✴ ❇❇✬ ✴✴ ❈❈✬ ✴✴ ●●✬ ✴✴ ▼▼✬ ✴✴ ❊❊✬ ✭✈× ❝❤ó♥❣ ❝ï♥❣ ✈✉➠♥❣ ❣ã❝ ✈í✐ ➤➢ê♥❣ t❤➻♥❣ ①②✮✳ ❚❛ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ ❆❆✬✰❇❇✬✰❈❈✬❂✸●●✬✳ ❚❤❡♦ tÝ♥❤ ❝❤✃t trä♥❣ t➞♠ t❛ ❝ã ● ❧➭ tr✉♥❣ ➤✐Ó♠ ▼❊✳ ❚❛ ❝ã ợt tr ì ủ ❝➳❝ ❤×♥❤ t❤❛♥❣ ✈✉➠♥❣ ▼▼✬❊✬❊✱ ❆❆✬❈✬❈✱ ❇❇✬●✬●✳ ❉♦ ➤ã t❤❡♦ tÝ♥❤ ❝❤✃t ➤➢ê♥❣ tr✉♥❣ ❜×♥❤ ❝đ❛ ❤×♥❤ t❤❛♥❣ t❛ ❝ã✿ ✷●●✬❂▼▼✬✰❊❊✬ ✭✶✮ ✷▼▼✬❂❇❇✬✰●●✬✬ ✭✷✮ ✷❊❊✬❂❆❆✬✰❈❈✬ ✭✸✮ ❚❤❛② ✭✷✮ ✈➭ ✭✸✮ ✈➭♦ ✭✶✮ ❦❤✐ ➤❛❝ ♥❤➞♥ ❤❛✐ ✈Õ ✈í✐ ✷ t❤× ợ ề ì t ❝➞♥ ❆❇❈❉ ✭❇❈ ✴✴ ❆❉✮✳ ●ä✐ ▼✱ ◆ ❧➬♥ ❧➢ỵt ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❇❈✱ ❆❉✳ ❚r➟♥ t✐❛ ➤è✐ ❝đ❛ t✐❛ ❆❇ ❧✃② ➤✐Ĩ♠ P ❜✃t ❦×✳ P◆ ❝➽t ❇❉ t➵✐ ◗✳ ❈❤ø♥❣ ♠✐♥❤ ▼◆ ❧➭ ♣❤➞♥ ❣✐➳❝ ❝đ❛ ❣ã❝ ❍➢í♥❣ ❞➱♥ ✿ ❚❛ ❝ã P M Q✳ M N ⊥ AD ✈➭ M N ⊥ BC ✭✶✮✳ ❆❉ ❝➽t P▼ ✈➭ ▼◗ t➵✐ ■✱ ❑✳ P◗ ❝➽t ➤➢ê♥❣ t❤➻♥❣ ❇❈ t➵✐ ❘✳ ❱× ❇❈ ✴✴ ❆❉ ♥➟♥ IN AN KN ND = ✈➭ = ✳ MR BR MR BR IN KN ▼➭ ❆◆❂ ◆❉ ♥➟♥ = ⇒ IN = KN ✭✷✮ MR MR ✫ ✶✵ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ✳✳✳ ❑❰ t✐❛ ❖♠ ♥➺♠ ♥❣♦➭✐ ❣ã❝ ①❖② s❛♦ ❝❤♦ yOm = xOA✳ ❚r➟♥ t✐❛ ❖♠ ❧✃② ➤✐Ó♠ ❉ s❛♦ ❝❤♦ ❖❉❂❖❆✬✳ ❈➳❝ ➤✐Ó♠ ❆ ✈➭ ❉ ❝è ➤Þ♥❤✳ ∆DOC = ∆AOB ✭❝✱❣✱❝✮ ⇒ CD = AB ✳ ❉♦ ➤ã✿ AC + AB = AC + CD ✳ ❚❛ ❧➵✐ ❝ã AC + CD ≥ AD ✭➤é ❞➭✐ ❆❉ ❦❤➠♥❣ ➤æ✐ ❳➲② r❛ ➤➻♥❣ t❤ø❝ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❈ t❤✉é❝ ➤♦➵♥ ❍×♥❤ ✶ t❤➻♥❣ ❆❉✳ ❇ t❤✉é❝ t✐❛ ❖① s❛♦ ❝❤♦ ❖❇❂❖❈✳ ◆❤❐♥ ①Ðt ✈Ò ♣❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐✿ ❚r♦♥❣ ❝➳❝❤ ❣✐➯✐ tr➟♥✱ ➤♦➵♥ t❤➻♥❣ ❇❆ ➤➲ ➤➢ỵ❝ ✧➤ỉ✐ ♣❤Ý❛✧ ➤è✐ ✈í✐ ❖② t❤➭♥❤ ➤♦➵♥ ❈❉✳ ❚ỉ♥❣ ❆❈✰❆❇ ➤➢ỵ❝ ✈✐Õt ❞➢í✐ ❞➵♥❣ ❆❈✰❈❉ ✈í✐ ❆ ✈➭ ❉ ❝è ➤Þ♥❤✳ ❇✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ AC +CD ≥ AD ❣✐ó♣ t❛ ❣✐➯✐ ➤➢ỵ❝ ❜➭✐ t♦➳♥✳ ✸✳ ❙ư ❞ơ♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr♦♥❣ ➤➢ê♥❣ trß♥ ✳ ❈➳❝ ❜✃t tứ tr trò ợ tể ệ tr ➤Þ♥❤ ❧Ý✿ ✲ ➜➢ê♥❣ ❦Ý♥❤ ❧➭ ❞➞② ❧í♥ ♥❤✃t ❝đ❛ ➤➢ê♥❣ trß♥✳ ✲ ❚r♦♥❣ ❤❛✐ ❞➞② ❝đ❛ ♠ét ➤➢ê♥❣ trß♥✱ ❞➞② ❧í♥ ❤➡♥ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❦❤♦➯♥❣ ❝➳❝❤ ➤Õ♥ t➞♠ ♥❤á ❤➡♥✳ ✲ ❚r♦♥❣ ❤❛✐ ❝✉♥❣ ♥❤á ❝ñ❛ ♠ét ➤➢ê♥❣ trß♥✱ ❝✉♥❣ ❧í♥ ❤➡♥ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❣ã❝ ë t➞♠ ❧í♥ ❤➡♥✳ ✲ ❚r♦♥❣ ❤❛✐ ❝✉♥❣ ♥❤á ❝đ❛ ♠ét ➤➢ê♥❣ trß♥✱ ❝✉♥❣ ❧í♥ ❤➡♥ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❞➞② tr➢➡♥❣ ❝✉♥❣ ❧í♥ ❤➡♥✳ ❱Ý ❞ơ ✼ ✳ ❈❤♦ ❤❛✐ ➤➢ê♥❣ trß♥ ✭❖✮ ✈➭ ✭❖✬✮ ❝➽t ♥❤❛✉ t➵✐ ❆ ✈➭ ❇✳ ▼ét ❝➳t t✉②Õ♥ ❝❤✉♥❣ ❜✃t ❦× ❈❇❉ ✭❇ ữ t trò ✈➭ ✭❖✬✮ t➵✐ ❈ ✈➭ ❉✳ ❳➳❝ ➤Þ♥❤ ✈Þ trÝ ❝đ❛ ❝➳t t✉②Õ♥ ❈❇❉ ➤Ĩ t❛♠ ❣✐➳❝ ❆❈❉ ❝ã ❝❤✉ ✈✐ ❧í♥ ♥❤✃t✳ ●✐➯✐✿ ❚❛ t❤✃② sè ➤♦ ❝➳❝ ❣ã❝ ❝đ❛ t❛♠ ❣✐➳❝ ❆❈❉ ❦❤➠♥❣ ➤ỉ✐✳ ❚❤❐t ✈❐② sdC = 1/2sd AmB, sdD = 1/2sd AnB ❉♦ ♥❤❐♥ ①Ðt tr➟♥✱ ∆ACD ❝ã ❝❤✉ ✈✐ ❧í♥ ♥❤✃t ❦❤✐ ❝ã ♠ét ❝➵♥❤ ❝đ❛ ♥ã✱ ❝❤➻♥❣ ❤➵♥ ❆❈ ❧➭ ❧í♥ ♥❤✃t✳ ❈➵♥❤ ❆❈ ❧➭ ❞➞② ❝đ❛ ➤➢ê♥❣ trß♥ ✭❖✮✳ ❉♦ ➤ã ❆❈ ❧í♥ ♥❤✃t ❦❤✐ ❆❈ ❧➭ ➤➢ê♥❣ ❦Ý♥❤ ❝đ❛ ➤➢ê♥❣ trß♥ ✭❖✮✱ ❦❤✐ ➤ã ❆❉ ❧➭ ➤➢ê♥❣ ❦Ý♥❤ ❝đ❛ ➤➢ê♥❣ trß♥ ✭❖✬✮✱ t tế ị trí ì ①Ðt ✈Ị ♣❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐✿ ✈✉➠♥❣ ❣ã❝ ✈í✐ ❞➞② ❝❤✉♥❣ ❆❇✳ ✫ ✹✷ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ✳✳✳ ❛✮ ❚r♦♥❣ ❝➳❝❤ ❣✐➯✐ tr➟♥ t❛ ➤➲ sö ❞ô♥❣ ♥❤❐♥ ①Ðt✿ ◆Õ✉ ♠ét t❛♠ ❣✐➳❝ ❝ã sè ➤♦ ❝➳❝ ❣ã❝ ❦❤➠♥❣ ➤ỉ✐ t❤× ❝❤✉ ✈✐ ❝đ❛ ♥ã ❧í♥ ♥❤✃t ❦❤✐ ♠ét ❝➵♥❤ ❝đ❛ ♥ã ❧í♥ ♥❤✃t✳ ❜✮ ❈ị♥❣ ❝ã t❤Ó ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❝❤✉ ✈✐ ♥❤✃t ❜➺♥❣ ❝➳❝❤ s❛✉✿ ∆ACD ∼ ∆AOO (g.g) ⇒ ∆ACD ❧í♥ ♥❤✃t ❦❤✐ ❆❈ ❧í♥ cvACD AC = cvAOO AO ❉♦ ❝❤✉ ✈✐ ❆❖❖✬ ✈➭ ❆❖ ❦❤➠♥❣ ➤ỉ✐ ♥➟♥ ❝❤✉ ✈✐ ❆❈❉ ❧í♥ ♥❤✃t ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❆❈ ❧í♥ ♥❤✃t✳ ✹✳ ✳ ❈➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✈Ị ❧ị② ❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✈Ị ❧ị② t❤õ❛ ❜❐❝ ❤❛✐ t❤õ❛ ❜❐❝ ❤❛✐ ➤➢ỵ❝ sư ❞ơ♥❣ ❞➢í✐ ❞➵♥❣✿ A2 ≥ 0; −A2 ≤ ❉♦ ➤ã ✈í✐ ♠ ❧➭ ❤➺♥❣ sè✱ t❛ ❝ã✿ f = A2 + m ≥ m; minf = m ⇔ A = f = −A2 + m ≤ m; maxf = m ⇔ A = ❱Ý ❞ô ✽ ✳ ❈❤♦ ❤×♥❤ ✈✉➠♥❣ ❆❇❈❉ ❝ã ❝➵♥❤ ❜➺♥❣ ✹ ❝♠✳ ❚r➟♥ ❝➳❝ ❝➵♥❤ ❆❇✱ ❇❈✱ ❈❉✱ ❉❛✱ ❧✃② t❤❡♦ t❤ø tù ❝➳❝ ➤✐Ó♠ ❊✱ ❋✱ ●✱ ❍ s❛♦ ❝❤♦ ❆❊❂❇❋❂❈●❂❉❍✳ ❚Ý♥❤ ➤é ❞➭✐ ❆❊ s❛♦ ❝❤♦ tø ❣✐➳❝ ❊❋●❍ ❝ã ❝❤✉ ✈✐ ♥❤á ♥❤✃t✳ ❉Ô t❤✃② ❍❊❂❊❋❂❋●❂●❍ ✭tø ❣✐➳❝ ❊❋●❍ ò ì ỏ t HE = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❍❊ ♥❤á ♥❤✃t✳ ➜➷t ❆❊❂① t❤× ❍❆❂❊❇❂✹✲①✳ ❳Ðt t❛♠ ❣✐➳❝ ❍❆❊ ✈✉➠♥❣ t❤❡♦ ị í Pt ì AE + HA2 = x2 + (4 − x)2 = x2 + 16 − 8x + x2 = 2(x −√2)2 + √ 8≥8 √ HE = = 2 ⇔ x = 2✳ ❈❤✉ ✈✐ tø ❣✐➳❝ ❊❋●❍ ♥❤á ♥❤✃t ❜➺♥❣ ❝♠✱ ❦❤✐ ➤ã ❆❊❂✷❝♠✳ ✿ ❇➭✐ t♦➳♥ tr➟♥ ❝ã t❤Ó ❣✐➯✐ ❜➺♥❣ ❝➳❝❤ ❜✐Ĩ✉ ◆❤❐♥ ①Ðt ✈Ị ♣❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ tị ì t t➞♠ ❝❤✉♥❣ ❝đ❛ ❤❛✐ ❤×♥❤ ✈✉➠♥❣ ❊❋●❍ ✈➭ ❆❇❈❉✮ rå✐ ①➳❝ ➤Þ♥❤ ✈Þ trÝ ❝đ❛ ❊ ➤Ĩ ➤é ❞➭✐ ❖❊ ♥❤á ♥❤✃t✳ ✺✳ ❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ❈➠ s✐ ✳ ❇✃t ➤➻♥❣ t❤ø❝ ❈➠ s✐ ✈➭ ❝➳❝ ❤Ö q✉➯ ❝đ❛ ♥ã ➤➢ỵ❝ sư ❞ơ♥❣ tr♦♥❣ ❝➳❝ ❜➭✐ t♦➳♥ ❝ù❝ trị ì ọ ể tị ộ t❤❛② ❜ë✐ ❤❛✐ ❜✐Õ♥ ①✱ ②✳ ✫ ✹✸ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ✳✳✳ ❇✃t ➤➺♥❣ t❤ø❝ ❈➠ s✐ t❤Ĩ ❤✐Ư♥ q✉❛♥ ❤Ư ❣✐÷❛ tỉ♥❣ ❝đ❛ ❤❛✐ sè ❦❤➠♥❣ ➞♠ ✈➭ tÝ❝❤ ❝đ❛ ❝❤ó♥❣✿ ❚r✉♥❣ ❜×♥❤ ❝é♥❣ ❝đ❛ ❤❛✐ sè ❦❤➠♥❣ ➞♠ ❦❤➠♥❣ ♥❤á ❤➡♥ tr✉♥❣ ❜×♥❤ ♥❤➞♥ ❝đ❛ ❝❤ó♥❣✳ x+y √ ≥ xy ❤❛② (x + y)2 ≥ 4xy ❳➯② r❛ ❞✃✉ ❜➺♥❣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ①❂②✳ ❇✃t ➤➻♥❣ t❤ø❝ ❈➠ s✐ t❤➢ê♥❣ ➤➢ỵ❝ ❞ï♥❣ ❞➢í✐ ❞➵♥❣ s❛✉✿ (x + y)2 ❉➵♥❣ ✶✿ x + y ≥ ≥ 2xy ✳ ❉✃✉ ❜➺♥❣ ①➲② ❦❤✐ ①❂②✳ x2 + y xy ❉➵♥❣ ✷✿ ≥ 4; ≥ ✳ xy (x + y)2 x2 + y (x + y)2 ≥ ; ≥ 2✳ ❳➯② r❛ ❞✃✉ ❜➺♥❣ ❦❤✐ ①❂②✳ (x + y)2 x + y2 2 ✿ ◆Õ✉ tæ♥❣ ❝đ❛ ❤❛✐ sè ❦❤➠♥❣ ➤ỉ✐ t❤× tÝ❝❤ ❝đ❛ ❝❤ó♥❣ ❧í♥ ♥❤✃t ❦❤✐ ❉➵♥❣ ✸ ✈➭ ❝❤Ø ❦❤✐ ❤❛✐ sè ➤ã ❜➺♥❣ ♥❤❛✉✳ ❉➵♥❣ ✹ ✿ ◆Õ✉ ❤❛✐ sè ❞➢➡♥❣ ❝ã tÝ❝❤ ❦❤➠♥❣ ➤ỉ✐ t❤× tỉ♥❣ ❝đ❛ ❝❤ó♥❣ ♥❤á ♥❤✃t ❦❤✐ ❤❛✐ sè ✃② ❜➺♥❣ ♥❤❛✉✳ ❱Ý ❞ô ✿ ❈❤♦ ➤♦➵♥ t❤➻♥❣ ❆❇✱ ➤✐Ó♠ ▼ ❞✐ ❝❤✉②Ó♥ tr➟♥ ➤♦➵♥ t❤➻♥❣ ✃②✳ ❱Ï ❝➳❝ ➤➢ê♥❣ trß♥ ❝ã ➤➢ê♥❣ ❦Ý♥❤ ▼❆ ✈➭ ▼❇✳ ❳➳❝ ➤Þ♥❤ ✈Þ trÝ ❝đ❛ ➤✐Ĩ♠ ▼ ➤Ĩ tỉ♥❣ ❞✐Ư♥ tí ủ ì trò ó trị ỏ t ●✐➯✐✿ ➜➷t ▼❆❂①✱ ▼❇❂②✱ t❛ ❝ã ①✰②❂❆❇ ✭✮❁①✱②❁❆❇✮✳ ●ä✐ ❙ ✈➭ ❙✬ t❤❡♦ t❤ø tù ❧➭ ❞✐Ư♥ tÝ❝❤ ❝đ❛ ❤❛✐ ì trò ó í x y x2 + y S + S = π( )2 + π( )2 = π 2 (x + y) ❚❛ ❝ã ❜✃t ➤➻♥❣ t❤ø❝ x2 + y ≥ ♥➟♥✿ (x + y)2 AB S+S ≥π = π ✳ 8 ❚❛ ❝ã✿ ❳➯② r❛ ❞✃✉ ➤➻♥❣ t❤ø❝ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ①❂②✳ ❉♦ ➤ã ♠✐♥✭❙✰❙✬✮❂π ➤ã ▼ ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❆❇✳ AB ✱ ❦❤✐ ✿ ❚r♦♥❣ ❝➳❝❤ ❣✐➯✐ tr➟♥✱ t❛ ➤➲ sư ❞ơ♥❣ ❜✃t ◆❤❐♥ ①Ðt ✈Ị ♣❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ ➤➻♥❣ t❤ø❝ ❈➠ s✐ ❞➵♥❣ ✶✳ ✷✳ ✷✳✶✳ ❇➭✐ ❈➳❝ t ề ự trị ì ọ ❈❤♦ ➤➢ê♥❣ t❤➻♥❣ ❛ ✈➭ ♠ét ➤✐Ó♠ ❆ ❦❤➠♥❣ t❤✉é❝ ❛✳ ❍❛✐ ➤➢ê♥❣ t❤➻♥❣ t❤❛② ➤æ✐ ➤✐ q✉❛ ❆ ✈✉➠♥❣ ❣ã❝ ✈í✐ ♥❤❛✉✱ ❝➽t ➤➢ê♥❣ t❤➻♥❣ ❛ ë ❇ ✈➭ ✫ ✹✹ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ✳✳✳ ❈✳ ❳➳❝ ➤Þ♥❤ ✈Þ trÝ ❤❛✐ ➤➢ê♥❣ t❤➻♥❣ ✈✉➠♥❣ ❣ã❝ tr➟♥ s❛♦ ❝❤♦ ❦❤♦➯♥❣ ❝➳❝❤ ❇❈ ♥❤á ♥❤✃t✳ ❍➢í♥❣ ❞➱♥✿ ✿ ❳Ðt ❈➳❝❤ ✶ ∆ABC ✈✉➠♥❣ t➵✐ ❆✳ ●ä✐ ❆❍✱ ❆▼ t❤❡♦ t❤ø tù ❧➭ ➤➢ê♥❣ ❝❛♦✱ tr✉♥❣ t✉②Õ♥ t❛♠ ❣✐➳❝✳ ❱× ∆ABC ✈✉➠♥❣ ♥➟♥ BC = 2AM ≥ 2AH ✭q✉❛♥ ❤Ư ❣✐÷❛ ➤➢ê♥❣ ✈✉➠♥❣ ❣ã❝ ✈➭ ➤➢ê♥❣ ①✐➟♥✮✳ ì t ể ố ị ♥➟♥ ❆❍ ❝ã ➤é ❞➭✐ ❦❤➠♥❣ ➤æ✐✳ ▼✐♥ ❇❈❂❆❍ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ▼ tr✉♥❣ ✈í✐ ❍✱ ❦❤✐ ➤ã ∆ABC ✈✉➠♥❣ ❝➞♥✳ ❍×♥❤ ✶ ❈➳❝❤ ❞ù♥❣ ❝➳❝ ➤➢ê♥❣ t❤➻♥❣ ❆❇ ✈➭ ❆❈✿ ✲ ❉ù♥❣ AH ⊥ a✳ ✲ ❉ù♥❣ ❇✱ ❈ t❤✉é❝ ❛ s❛♦ ❝❤♦ ❍❇❂❍❈❂❍❆✳ ✷✿ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❦❤✐ ❈➳❝❤ ∆ABC ✈✉➠♥❣ t➵✐ ❆ t❤× ❇❈ ❝ã ➤é ❞➭✐ ♥❤á ♥❤✃t✳ ❚❤❐t ✈❐②✱ ①Ðt ❤❛✐ ➤➢ê♥❣ t❤➻♥❣ ❜✃t ❦× ✈✉➠♥❣ ❣ã❝ ✈í✐ ♥❤❛✉ t➵✐ ❆✱ ❝➽t ➤➢ê♥❣ t❤➻♥❣ ❛ t➵✐ ❉ ✈➭ ❊✱ ❣✐➯ sö ❆❉❃❆❊✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ❉❊❃❇❈✳ ❚r➟♥ t✐❛ ❆❉ ❧✃② ➤✐Ó♠ ▼ s❛♦ ❝❤♦ ❍×♥❤ ✶ ❆▼❂❆❊ ∆ABM = ∆ACE(c.g.c) ⇒ BM = CE ✭✶✮ ❉Ơ t❤✃② ▼ ♥➺♠ ❣✐÷❛ ❆ ✈➭ ❉ ♥➟♥✿ DM B > M BA = ACB = ABC > D ❚r♦♥❣ t❛♠ ❣✐➳❝ ▼❉❇ ❝ã DM B > D ♥➟♥ DB > M B ✭✷✮ ✭q✉❛♥ ❤Ư ❣✐÷❛ ❣ã❝ ✈➭ ❝➵♥❤ ➤è✐ ❞✐Ö♥ tr♦♥❣ t❛♠ ❣✐➳❝✮ ❚õ ✭✶✮ ✈➭ ✭✷✮ s✉② r❛ ❉❇❃❈❊✳ ✐t❡♠ ❈❤♦ t❛♠ ❣✐➳❝ ♥❤ä♥ ❆❇❈✳ ❉ù♥❣ ♠ét t❛♠ ❣✐➳❝ ❝ã ❝❤✉ ✈✐ ♥❤á ♥❤✃t ♥é✐ t✐Õ♣ t❛♠ ❣✐➳❝ ❆❇❈✱ tø❝ ❧➭ ❝ã ❜❛ ➤Ø♥❤ ♥➺♠ tr➟♥ ❜❛ ❝➵♥❤ ❝ñ❛ t❛♠ ❣✐➳❝ ✃②✳ ✫ ✹✺ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ✷✳✷✳ ❇➭✐ ✳✳✳ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ❚❛♠ ❣✐➳❝ ➤å♥❣ ❞➵♥❣ ✶✳ ❈❤♦ ➤✐Ó♠ ▼ ❞✐ ❝❤✉②Ĩ♥ tr➟♥ ➤♦➵♥ t❤➻♥❣ ❆❇✳ ❱Ï ❝➳❝ t❛♠ ❣✐➳❝ ➤Ị✉ ❆▼❈ ✈➭ ❇▼❉ ✈Ị ♠ét ♣❤Ý❛ ❝đ❛ ❆❇✳ ❳➳❝ ➤Þ♥❤ ✈Þ trÝ ❝đ❛ ▼ ➤Ĩ tỉ♥❣ ❞✐Ư♥ tÝ❝❤ ❤❛✐ t❛♠ ❣✐➳❝ ➤Ò✉ tr➟♥ ❧➭ ♥❤á ♥❤✃t✳ ✿ ●ä✐ ❑ ❧➭ ❣✐❛♦ ➤✐Ĩ♠ ❝đ❛ ❆❈ ✈➭ ❇❉✳ ❈➳❝ t❛♠ ❣✐➳❝ ❆▼❈✱ ❍➢í♥❣ ❞➱♥ ❇▼❉ ➤å♥❣ ❞➵♥❣ ✈í✐ t❛♠ ❣✐➳❝ ❆❑❇✳ ➜➷t ❆▼❂①✱ ▼❇❂②✱ ❆❇❂❛✱ SAM C = S1 , SBM D = S2 , SKAB = S ✳ ❚❛ ❝ã✿ x S2 y S1 = ( )2 ; = ( )2 ✳ ◆➟♥✿ S a S a 2 x +y (x + y)2 a2 S1 + S2 = ≥ = = S a2 2a2 2a2 ❳➯② r❛ ❞✃✉ ➤➻♥❣ t❤ø❝✿ ①❂②✳ ❉♦ ➤ã✿ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❆❇✳ ✷✳✸✳ ➜Ò min(S1 + S2 ) = S ⇔ M ❧➭ ➜➢ê♥❣ trß♥ ✶✳ ❚r♦♥❣ t❛♠ ❣✐➳❝ ❆❇❈ ❝ã ❣ã❝ ❆ ❦❤➠♥❣ ➤æ✐✱ ♥Õ✉ tæ♥❣ ❤❛✐ ❝➵♥❤ ❜➟♥ ❆❇✰❆❈ ❦❤➠♥❣ ➤ỉ✐ t❤× t❛♠ ❣✐➳❝ ❝➞♥ t➵✐ ❆ ❝ã ❝❤✉ ✈✐ ♥❤á ♥❤✃t✳ ➜Ị ✷✳ ❈❤♦ ❤❛✐ ➤✐Ĩ♠ ❆✱ ❇ ♥➺♠ tr♦♥❣ ♥ư❛ ♠➷t ♣❤➻♥❣ ✈í✐ ❜ê ❧➭ ➤➢ê♥❣ t❤➻♥❣ ❞✳ ❚×♠ tr➟♥ ❞ ➤✐Ĩ♠ P s❛♦ ❝❤♦ tỉ♥❣ ❆P✰❇P ♥❤á ♥❤✃t✳ ➜Ị ✸✳ ❈❤♦ ❣ã❝ ♥❤ä♥ xOy ✈➭ ➤✐Ĩ♠ ❆ ♥➺♠ tr♦♥❣ ❣ã❝ ➤ã✳ ❚×♠ ➤✐Ĩ♠ ❇ tr➟♥ ❖① ✈➭ ➤✐Ó♠ ❈ tr➟♥ ❖② s❛♦ ❝❤♦ ❝❤✉ ✈✐ t❛♠ ❣✐➳❝ ❆❇❈ ➤➵t ❣✐➳ trÞ ♥❤á ♥❤✃t✳ ➜Ị ✹✳ ❚❛ ♥ã✐ t❛♠ ❣✐➳❝ ❉▼◆ ♥é✐ t✐Õ♣ tr♦♥❣ t❛♠ ❣✐➳❝ ❆❇❈ ♥Õ✉ ❜❛ ➤Ø♥❤ ❝ñ❛ t❛♠ ❣✐➳❝ ❉▼◆ ♥➺♠ tr➟♥ ❜❛ ❝➵♥❤ ❝đ❛ t❛♠ ❣✐➳❝ ❆❇❈✳ ❈❤♦ tr➢í❝ ♠ét t❛♠ ❣✐➳❝ ♥❤ä♥ ❆❇❈✳ ❍➲② t×♠ t❛♠ ❣✐➳❝ ❝ã ❝❤✉ ✈✐ ♥❤á ♥❤✃t ♥é✐ t✐Õ♣ t❛♠ ❣✐➳❝ ♥➭②✳ ➜Ò ✺✳ ❈❤♦ t❛♠ ❣✐➳❝ ➤Ị✉ ❆❇❈ ♥é✐ t✐Õ♣ tr♦♥❣ ➤➢ê♥❣ trß♥ ✭❖✱❘✮✳ ❳➳❝ ➤Þ♥❤ ▼ tr➟♥ ❝✉♥❣ ♥❤á ❇❈ ➤Ĩ tỉ♥❣ ▼❆✰▼❇✰▼❈ ➤➵t trị t ề trò ✭❖✱❘✮ ➤➢ê♥❣ ❦Ý♥❤ ❆❇✳ ❱Ï t✐Õ♣ t✉②Õ♥ ❇① t➵✐ ❇ ❝đ❛ ✭❖✮✳ ●ä✐ ▼ ❧➭ ➤✐Ĩ♠ ❞✐ ➤é♥❣ t❤✉é❝ ♥ư❛ ➤➢ê♥❣ trß♥ ✭❖✮ ✈➭ ❆▼ ❝➽t ❇① t➵✐ ◆✳ ❳➳❝ ➤Þ♥❤ ✈Þ trÝ ❝đ❛ ▼ ➤Ĩ ✷❆▼✰❆◆ ➤➵t ❣✐➳ trÞ ♥❤á ♥❤✃t✳ ➜Ị ✼✳ ❈❤♦ ➤✐Ĩ♠ ▼ ♥➺♠ tr♦♥❣ t❛♠ ❣✐➳❝ ❆❇❈ ❝ã ❝➳❝ ❝➵♥❤ ❧➭ ❛✱ ❜✱ ❝✳ ●ä✐ ❝➳❝ ❦❤♦➯♥❣ ❝➳❝❤ tõ ➤✐Ó♠ ▼ ➤Õ♥ ❝➳❝ ❝➵♥❤ ❛✱ ❜✱ ❝ t➢➡♥❣ ø♥❣ ❧➭ ①✱ ②✱ ✫ ✹✻ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ✳✳✳ ③✳ ❍➲② ①➳❝ ➤Þ♥❤ ✈Þ trÝ ❝đ❛ ➤✐Ĩ♠ ▼ tr♦♥❣ t❛♠ ❣✐➳❝ s❛♦ ❝❤♦ ❜✐Ó✉ t❤ø❝ a b c + + ♥❤á ♥❤✃t✳ x y z P = ✸✳ P❤➢➡♥❣ ♣❤➳♣ ♣❤➯♥ ❝❤ø♥❣✱ ♥❣✉②➟♥ ❧Ý ❞✐✐rr✐❝❤❧❡ ✸✳✶✳ P❤➢➡♥❣ ♣❤➳♣ ♣❤➯♥ ❝❤ø♥❣ ❈❤ø♥❣ ♠✐♥❤ ♠ét ❜➭✐ t♦➳♥ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ♣❤➯♥ ❝❤ø♥❣ ❣å♠ ❜❛ ❜➢í❝✿ ✲ ❇➢í❝ ✶ ✿ ✭♣❤đ ➤Þ♥❤ ❦Õt ❧✉❐♥✮✳ ❚❛ ❣✐➯ sư r➺♥❣ ❦Õt ❧✉❐♥ ❝đ❛ ❜➭✐ t♦➳♥ ❧➭ ❦❤➠♥❣ ➤ó♥❣✳ ✲ ✿ ✭➤➢❛ ➤Õ♥ ♠➞✉ t❤✉➱♥✮✳ ❚õ ➤✐Ị✉ ❣✐➯ sư tr➟♥ ✈➭ tõ ❣✐➯ t❤✐Õt ❝đ❛ ❇➢í❝ ✷ ❜➭✐ t♦➳♥✱ t❛ s✉② r❛ ♠ét ➤✐Ị✉ ♠➞✉ t❤✉➱♥ ✈í✐ ❣✐➯ t❤✐Õt ❤♦➷❝ ♠➞✉ t❤✉➱♥ ✈í✐ ♠ét ❦✐Õ♥ t❤ø❝ ➤➲ ❤ä❝✳ ✲ ✿ ✭❦❤➻♥❣ ➤Þ♥❤ ❦Õt ❧✉❐♥✮✳ ❱❐② ❦Õt ❧✉❐♥ ❝đ❛ ❜➭✐ t♦➳♥ ❧➭ ➤ó♥❣✳ ❇➢í❝ ✸ ➛✉ ➤✐Ĩ♠ ❝đ❛ ♣❤➢➡♥❣ ♣❤➳♣ ♣❤➯♥ ❝❤ø♥❣ ❧➭ t➵♦ t❤➟♠ ➤➢ỵ❝ ♠ét ❣✐➯ t❤✐Õt ♠í✐ ✭❣✐➯ t❤✐Õt ♣❤➯♥ ❝❤ø♥❣✮ ✈➭♦ ❣✐➯ t❤✐Õt ❝đ❛ ❜➭✐ t♦➳♥✳ ❈❤➻♥❣ ❤➵♥ ➤Ó ❝❤ø♥❣ ♠✐♥❤ A ≥ B ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ♣❤➯♥ ❝❤ø♥❣✱ t❛ sư ❞ơ♥❣ ❣✐➯ t❤✐Õt ♣❤➯♥ ❝❤ø♥❣ A < B ➤Ĩ ❝❤Ø r❛ ➤✐Ị✉ ✈➠ ❧Ý✳ ❱Ý ❞ơ ✶✳ ❈❤♦ ❤❛✐ ➤✐Ĩ♠ ▼✱ ◆ ♥➺♠ ❜➟♥ tr♦♥❣ ♠ét ➤❛ ❣✐➳❝ ❧å✐✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ tå♥ t➵✐ ♠ét ➤Ø♥❤ ❝ñ❛ ➤❛ ❣✐➳❝ s❛♦ ❝❤♦ ❦❤♦➯♥❣ ❝➳❝❤ tõ ➤Ø♥❤ ➤ã ➤Õ♥ ▼ ❧í♥ ❤➡♥ ❦❤♦➯♥❣ ❝➳❝❤ tõ ➤Ø♥❤ ➤ã ➤Õ♥ ◆✳ ●✐➯✐✿ ✭❤✳✶✮✳ ●ä✐ ➤❛ ❣✐➳❝ ➤➲ ❝❤♦ ❧➭ A1 A2 · · · An ●✐➯ sö A1 M ≤ A1 N A2 M ≤ A2 N ········· An M ≤ An N ✫ ✹✼ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ✳✳✳ ●ä✐ ❞ ❧➭ ➤➢ê♥❣ tr✉♥❣ trù❝ ❝ñ❛ ▼◆✳ ➜➢ê♥❣ t❤➻♥❣ ❞ ❝❤✐❛ ♠➷t ♣❤➻♥❣ ❝đ❛ ➤❛ ❣✐➳❝ t❤➭♥❤ ❤❛✐ ♥ư❛ ♠➷t ♣❤➻♥❣✱ ♥ư❛ ♠➷t ♣❤➻♥❣ ❝❤ø❛ ▼ ❣ä✐ ❧➭ ✭■✮ ✱ ♥ö❛ ♠➷t ♣❤➻♥❣ ❝❤ø❛ ◆ ❣ä✐ ❧➭ ✭■■✮✳ ❉♦ A1 M ≤ A1 N ♥➟♥ (A1 ) ∈ (I)✱ t➢➡♥❣ tù A2 , A3 , · · · , An ❝ò♥❣ t❤✉é❝ ✭■✮✳ ▼ä✐ ➤Ø♥❤ ❝đ❛ ➤❛ ❣✐➳❝ t❤✉é❝ ♥ư❛ ♠➷t ♣❤➻♥❣ ✭■✮✱ ❝ß♥ ◆ t❤✉é❝ ♥ư❛ ♠➷t ♣❤➻♥❣ ✭■■✮ ♥➟♥ ◆ ♥➺♠ ❜➟♥ ♥❣♦➭✐ ➤❛ ❣✐➳❝✱ trÝ❛ ❣✐➯ t❤✐Õt✳ ❱❐② tå♥ t➵✐ ♠ét ➤Ø♥❤ ✸✳✷✳ Ak ❝ñ❛ ➤❛ ❣✐➳❝ s❛♦ ❝❤♦ Ak M > Ak N ◆❣✉②➟♥ ❧Ý ➜✐r✐❝❤❧❡ ❉➵♥❣ ➤➡♥ ❣✐➯♥✿ ◆Õ✉ ♥❤èt ♥✰✶ t❤á ✈➭♦ ♥ ❧å♥❣ t❤× tå♥ t➵✐ ♠ét ❧å♥❣ ❝ã Ýt ♥❤✃t ✷ t❤á✳ ❚æ♥❣ q✉➳t✿ ♥Õ✉ ♥❤èt ♥ t❤á ✈➭♦ ♠ ❧å♥❣ ♠➭ ♣❤Ð♣ ❝❤✐❛ ó t ò tì tå♥ t➵✐ ♠ét ❧å♥❣ ❝❤ø❛ Ýt ♥❤✃t ❦✰✶ ❝♦♥ t❤á✳ ❱Ý ❞ơ ✷✿ ❈❤♦ ♠ét ❤×♥❤ ✈✉➠♥❣ ✈➭ ✶✸ ➤➢ê♥❣ t ỗ ề ì t tứ ❣✐➳❝ ❝ã tØ sè ❞✐Ö♥ tÝ❝❤ ❜➺♥❣ ✷✿✸✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ tr♦♥❣ ✶✸ ➤➢ê♥❣ t❤➻♥❣ ➤ã✱ ❝ã Ýt ♥❤✃t ❜è♥ ➤➢ê♥❣ t❤➻♥❣ ❝ï♥❣ ➤✐ q✉❛ ♠ét ➤✐Ó♠✳ ●✐➯✐✿ ●ä✐ ❞ ❧➭ ➤➢ê♥❣ t❤➻♥❣ ❝❤✐❛ ❤×♥❤ ✈✉➠♥❣ ❆❇❈❉ t❤➭♥❤ ❤❛✐ tø ❣✐➳❝ ❝ã tØ sè ❞✐Ö♥ tÝ❝❤ ❜➺♥❣ ✷✿✸✳ ➜➢ê♥❣ t❤➻♥❣ ❞ ❦❤➠♥❣ t❤Ĩ ❝➽t ❤❛✐ ❝➵♥❤ ❦Ị ♥❤❛✉ ❝đ❛ ❤×♥❤ ✈✉➠♥❣ ✈× ❦❤✐ ➤ã ❦❤➠♥❣ t➵♦ t❤➭♥❤ ❤❛✐ tø ❣✐➳❝✳ ●✐➯ sö ❞ ❝➽t ❤❛✐ ❝➵♥❤ ❆❇ ✈➭ ❈❉ t➵✐ ▼ ✈➭ ◆✱ ❦❤✐ ➤ã ♥ã ❝➽t ➤➢ê♥❣ tr✉♥❣ ❜×♥❤ ❊❋ t➵✐ ■✳ ●✐➯ sö 2 SAM N D = SBM N C t❤× EI + IF 3 ỗ t tr ❜×♥❤ ❝đ❛ ❤×♥❤ ✈✉➠♥❣ t❤❡♦ tØ sè ✷✿✸✳ ❈ã ❜è♥ ➤✐Ĩ♠ ❝❤✐❛ ➤➢ê♥❣ tr✉♥❣ ❜×♥❤ ❝đ❛ ❤×♥❤ ✈✉➠♥❣ t❤❡♦ tØ sè ✷✿✸ ✭❧➭ ❝➳❝ ➤✐Ó♠ ■✱ ❑✱ ●✱ ❍ tr➟♥ ❤×♥❤ ✷✳ ✫ ✹✽ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ơ♥ ó tỗ t q✉❛ ♠ét tr♦♥❣ ❜è♥ ➤✐Ó♠✳ P❤Ð♣ ❝❤✐❛ ✶✸ ❝❤♦ ✹ ❝ã t❤➢➡♥❣ ❧➭ ✸ ✈➭ ❝ß♥ ❞➢ ♥➟♥ tå♥ t➵✐ ♠ét ➤✐Ó♠ ❝ã Ýt ♥❤✃t ❜è♥ ✭✸✰✶✮ ➤➢ê♥❣ t❤➻♥❣ ➤✐ q✉❛✳ ➜Ị ✶✳ ❈❤♦ ✷✵✵✺ ➤✐Ĩ♠ tr➟♥ ♠➷t ♣❤➻♥❣✱ ❜✐Õt r ỗ ó ể t ì ủ ể ❜❛♦ ❣✐ê ❝ị♥❣ ❝❤ä♥ r❛ ➤➢ỵ❝ ✷ ➤✐Ĩ♠ ❝ã ❦❤♦➯♥❣ ❝➳❝❤ ❜Ð ❤♦✇♥✶✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ tr♦♥❣ ❝➳❝ ➤✐Ó♠ tr➟♥ ❝ã Ýt ♥❤✃t ✶✵✵✸ ➤✐Ĩ♠ ♥➺♠ tr♦♥❣ ♠ét ➤➢ê♥❣ trß♥ ❝ã ❜➳♥ ❦Ý♥❤ ❜➺♥❣ ✶✳ ➜Ị ✷✳ ❚r♦♥❣ ❤×♥❤ ✈✉➠♥❣ ộ ỗ trớ ➤✐Ó♠✱ tr♦♥❣ ➤ã ❦❤➠♥❣ ❝ã ✸ ➤✐Ó♠ ♥➭♦ t❤➻♥❣ ❤➭♥❣✳ ◆❣➢ê✐ t❛ ✈Ï ❝➳❝ ➤➢ê♥❣ trß♥ ❝ã ❜➳♥ ❦Ý♥❤ ➤Ị✉ ❜➺♥❣ √ ❝ã t➞♠ ❧➭ ❝➳❝ ➤✐Ó♠ ➤➲ ❝❤♦✳ ❍á✐ ❝ã ❤❛② ❦❤➠♥❣ ✸ ➤✐Ó♠ tr♦♥❣ sè ❝➳❝ ➤✐Ó♠ ♥ã✐ tr➟♥ ♠➭ ❝❤ó♥❣ ➤Ị✉ t❤✉é❝ ♣❤➬♥ ❝❤✉♥❣ ❝đ❛ ✸ ì trò ó t ũ í ể ➤ã❄ ➜Ị ✸✳ ❈❤♦ ♠ét ❤×♥❤ ✈✉➠♥❣ ✈➭ ✾ ➤➢ê♥❣ t tr ó ứ ỗ t ề ì ✈✉➠♥❣ t❤➭♥❤ ❤❛✐ tø ❣✐➳❝ ❝ã tØ sè ❞✐Ö♥ tÝ❝❤ ❧➭ ✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ tr♦♥❣ sè ✾ ➤➢ê♥❣ t❤➻♥❣ ➤ã ❝ã Ýt ♥❤✃t ✸ ➤➢ê♥❣ t❤➻♥❣ ➤å♥❣ q✉② t➵✐ ♠ét ➤✐Ĩ♠✳ ➜Ị ✹✳ ▼ét ♥➢í❝ ❝ã ✽✵ s➞♥ ❜❛②✱ ♠➭ ❦❤♦➯♥❣ ❝➳❝❤ ❣✐÷❛ ❤❛✐ s➞♥ ❜➞② ♥➭♦ ũ ỗ t từ ột s➞♥ ❜❛② ✈➭ ❜❛② ➤Õ♥ s➞♥ ❜❛② ♥➭♦ ❣➬♥ ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ tr➟♥ ❜✃t ❦× s➞♥ ❜❛② ♥➭♦ ❝ị♥❣ ❦❤➠♥❣ t❤Ĩ ❝ã q✉➳ ✺ ♠➳② ❜❛② ➤Õ♥✳ ➜Ị ✺✳ ❚r♦♥❣ ❤×♥❤ ✈✉➠♥❣ ❝ã ❝➳❝ ❝➵♥❤ ❜➺♥❣ ✶ ❝ã ✶✵✶ ➤✐Ó♠ tï② ý✱ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❝ã Ýt ♥❤✃t ✺ ể tr ì trò í ✫ ✹✾ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ➜Ị ❚❙✳ ◆❣✉②Ơ♥ ❉ ❍♦➭♥❣ ✳✳✳ ✻✳ ❈❤♦ ♠ét ❤×♥❤ ✈✉➠♥❣ ✈➭ ✶✸ t ỗ t ề ì t ❤❛✐ tø ❣✐➳❝ ❝ã tØ sè ❞✐Ö♥ tÝ❝❤ ❜➺♥❣ ✷✿ ✸✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ tr♦♥❣ ✶✸ ➤➢ê♥❣ t❤➻♥❣ ➤ã✱ ❝ã Ýt ♥❤✃t ❜è♥ ➤➢ê♥❣ t❤➻♥❣ ❝ï♥❣ ➤✐ q✉❛ ♠ét ➤✐Ó♠✳ ➜Ò ✼✳ ●✐➯✐ ❜➭✐ t♦➳♥ ❜➺♥❣ q✉② ♥➵♣✿ ✧❈❤♦ ♥ ➤➢ê♥❣ trß♥ tr♦♥❣ ♠➷t ♣❤➻♥❣✱ ❤❛✐ ➤➢ê♥❣ trß♥ ♥➭♦ ❝ị♥❣ ❝➽t ♥❤❛✉ t➵✐ ❤❛✐ ➤✐Ĩ♠✱ ❦❤➠♥❣ ❝ã ❜❛ ➤➢ê♥❣ trß♥ ♥➭♦ ❝ï♥❣ ➤✐ q✉❛ ♠ét ➤✐Ó♠✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ♥ ➤➢ê♥❣ trß♥ ➤ã ❝❤✐❛ ♠➷t ♣❤➻♥❣ t❤➭♥❤ ✷✰♥✭♥✲✶✮ ♠✐Ị♥✧✳ ➜Ị ✽✳ ●✐➯✐ ❜➭✐ t♦➳♥ ❜➺♥❣ q✉② ♥➵♣✿ ✧❈❤ø♥❣ ♠✐♥❤ r➺♥❣ tỉ♥❣ ❝➳❝ ❣ã❝ tr♦♥❣ ❝đ❛ ➤❛ ❣✐➳❝ ❧å✐ ♥ ❝➵♥❤ (n ≥ 3) ❜➺♥❣ Sn = (n − 2).180o ✳ ✫ ✺✵ ❈❤➢➡♥❣ ✸ ❝➳❝ ➤✐Ó♠ t❤➻♥❣ ❤➭♥❣✱ ❝➳❝ ➤➢ê♥❣ t❤➻♥❣ ➤å♥❣ q✉②✱ ❝➳❝ ➤✐Ĩ♠ ♥➺♠ tr➟♥ ➤➢ê♥❣ trß♥ ✶✳ ➤Þ♥❤ ❧Ý ✈Ị tØ sè ✈➭ ➳♣ ❞ơ♥❣ ✶✳✶✳ ❈➳❝ ➤Þ♥❤ ❧Ý ➜Þ♥❤ ❧Ý ❈❡✈❛✿ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈✱ ❣✐➯ sư ❆✬✱ ❇✬✱ ❈✬ ❧➬♥ ❧➢ỵt ❧➭ ❝➳❝ ➤✐Ĩ♠ ♥➺♠ tr➟♥ ❜❛ ➤➢ê♥❣ t❤➻♥❣ ❇❈✱ ❈❆✱ ❆❇✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ❆❆✬✱ ❇❇✬✱ ❈❈✬ ➤å♥❣ q✉② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐✿ AB BC C A =1 AC BA C B ❍➢í♥❣ ❞➱♥ ✿ ❛✮ ●✐➯ sư ❆❆✬✱ ❇❇✬✱ ❈❈✬ ➤å♥❣ q✉② ë ■✳ ❑❰ ❆❉ ✈➭ ❈❊ ❝ï♥❣ s♦♥❣ s♦♥❣ ❇❇✬✱ ❝➽t ❈❈✬ ë ❉ ✈➭ ❆❆✬ ë ❊✳ AB BI = ✭✶✮✳ AC CE IC CE BC ∆ ❈■❇✬ ➤å♥❣ ❞➵♥❣ ∆ ❈❉❆ ♥➟♥ = = ✭✷✮✳ BA ID DA CA DA ∆ ❈✬❉❆ ➤å♥❣ ❞➵♥❣ ∆ ❈✬■❇ ♥➟♥ = ✭✸✮✳ CB BI ∆ ❆✬❇■ ➤å♥❣ ❞➵♥❣ ∆ ❆✬❈❊ từ ế tì ợ AB BC C A BI CE DA = = ✭➤❝♠✮✳ AC BA C B CE DA BI ✺✶ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❚❙✳ ◆❣✉②Ô♥ ❉ ❍♦➭♥❣ ✳✳✳ AB BC C A = ✈➭ ❆❆✬ ❝➽t ❇❇✬ t➵✐ ■✳ ❚✐❛ ❈■ ❝➽t ❆❇ t➵✐ AC BA C B AB BC C A C”A CA ❈✧✳ ❚❤❡♦ ❝➞✉ ❛✮ t❤× = 1✳ ❚õ ➤ã s✉② r❛ = ❤❛② AC BA C B C”B CB C” ≡ C ✳ ❜✮ ●✐➯ sö ❱❐② ❆❆✬✱ ❇❇✬✱ ❈❈✬ ➤å♥❣ q✉② t➵✐ ■ ✭➤❝♠✮✳ ➜Þ♥❤ ❧Ý ▼❡♥❡❧❛✉s✿ ❈❤♦ ▼✱ ◆✱ P ❧➬♥ ❧➢ỵt ♥➺♠ tr➟♥ ❝➳❝ ➤➢ê♥❣ t❤➻♥❣ ❝❤ø❛ ❝➳❝ ❝➵♥❤ ❆❇✱ ❇❈✱ ❈❆ ❝ñ❛ t❛♠ s ỗ ể trù ỉ t❛♠ ❣✐➳❝✱ ✈➭ ❝ã ❦❤➠♥❣ q✉➳ ✷ ➤✐Ó♠ t❤✉é❝ ❤❛✐ ❝➵♥❤ ❝đ❛ t❛♠ ❣✐➳❝ ✭tø❝ ❝ã Ýt ♥❤✃t ♠ét ➤✐Ĩ♠ ♥➺♠ tr➟♥ ♣❤➬♥ ❦Ð♦ ❞➭✐ ❝ñ❛ ♠ét ❝➵♥❤✮✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ▼✱ ◆✱ P t❤➻♥❣ ❤➭♥❣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ MA NB P C =1 MB N C P A ❍➢í♥❣ ❞➱♥ ✿ ❛✮ ●✐➯ sư ▼✱ ◆✱ P t❤➻♥❣ ❤➭♥❣✳ ❑❰ ❆◗ ✴✴ ❇❈ ❝➽t ▼◆ ë ◗✳ ❧Ý ❚❛ ❧❡t tr♦♥❣ ∆ ▼❇◆✿ ✭✷✮ ➳♣ ❞ô♥❣ ➤Þ♥❤ MA AQ PC NC = ✭✶✮✱ ✈➭ tr♦♥❣ ∆ P◆❈✿ = MB NB PA AQ ❚õ ✭✶✮ ✈➭ ✭✷✮ s✉② r❛✿ MA NB P C AQ N B N C = = ✭➤❝♠✮✳ MB NC P A N B N C AQ ✫ ✺✷ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ❜✮ ●✐➯ sư ✳✳✳ ❚❙✳ ◆❣✉②Ơ♥ ❉ ❍♦➭♥❣ MA NB P C = 1✳ ◆è✐ ▼P ❦Ð♦ ❞➭✐ ❝➽t ❇❈ ë ◆✬✱ t❤❡♦ ❝❤ø♥❣ MB NC P A ♠✐♥❤ tr➟♥ t❤×✿ MA N B P C == MB N C P A NB NB ❚õ ➤ã s✉② r❛ = ✳ NC NC ì ù ữ ♥➟♥ ◆≡ ◆✬✱ tø❝ ▼✱ ◆✱ P t❤➻♥❣ ❤➭♥❣✳ ✶✳✷✳ ➜Ò ❈➳❝ ♣❤➢➡♥❣ ❞➵♥❣ t♦➳♥ ✶✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈✳ ●ä✐ ❉ ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❇❈✱ ❊✱ ❋ ❧➬♥ ❧➢ỵt ❧➭ ❤❛✐ ➤✐Ĩ♠ tr➟♥ ❆❇ ✈➭ ❆❈✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ♥Õ✉ ❆❉✱ ❇❋✱ ❈❊ ➤å♥❣ q✉② t❤× ❊❋ s♦♥❣ s♦♥❣ ✈í✐ ❇❈✳ ✿ ❍➢í♥❣ ❞➱♥ ➳♣ ❞ơ♥❣ ➤Þ♥❤ ❧Ý ❈❡✈❛ tr♦♥❣ t❛♠ ❣✐➳❝ ❆❇❈ ✈í✐ ❜❛ ➤➢ê♥❣ t❤➻♥❣ ➤å♥❣ q✉② ❆❉✱ ❇❋✱ ❈❊ t❛ ❝ã✿ EA DB F C = 1✳ EB DC F A EA F C = 1✳ EB F A EA FA ❙✉② r❛✿ = ✳ ❚❤❡♦ ➤Þ♥❤ ❧Ý ❚❛✲❧Ðt t❛ ❝ã ❊❋ ✴✴ ❇❈ ✭➤❝♠✮✳ EB FC ➜Ò ✷✳ ❈❤♦ ❣ã❝ xOy ✱ tr➟♥ t✐❛ ❖① ❧✃② ❤❛✐ ➤✐Ó♠ ❈ ✈➭ ❆✱ tr➟♥ t✐❛ ❖② ❧✃② ❤❛✐ ❱× ❉❇❂❉❈ ♥➟♥ ➤✐Ĩ♠ ❉ ✈➭ ❇ s❛♦ ❝❤♦ ❆❉ ❝➽t ❇❈ t➵✐ ❊✱ ❝➳❝ ➤➢ê♥❣ t❤➻♥❣ ❆❇ ✈➭ ❈❉ ❝➽t ♥❤❛✉ t➵✐ ❑❀ t✐❛ ❖❊ ❝➽t ❆❇ t➵✐ ■✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣✿ IA KA = IB KB ✫ ✺✸ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ✷✳ ✳✳✳ ❈❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤✐Ó♠ t❤➻♥❣ ❤➭♥❣ ✷✳✶✳ ❈➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ ✷✳✷✳ ❈➳❝ ❞➵♥❣ t♦➳♥ ➜Ị ❚❙✳ ◆❣✉②Ơ♥ ❉ ❍♦➭♥❣ ✶✳ ❚r➟♥ ❝➳❝ ❝➵♥❤ ❜➟♥ ❆❇ ✈➭ ❇❈ ❝đ❛ t❛♠ ❣✐➳❝ ❝➞♥ ❆❈❇ ❧✃② ➤✐Ĩ♠ ▼✱ ◆ s❛♦ ❝❤♦ ❆▼❂❈◆✳ ●ä✐ ❉✱ ❊✱ ❋ ❧➬♥ ❧➢ỵt ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❆❈✱ ❇❈✱ ▼◆✳ ❈❤ø♥❣ ♠✐♥❤ ❉✱ ❊✱ ❋ t ề ì ữ t ọ P ❧➭ ➤✐Ó♠ tï② ý tr➟♥ ➤➢ê♥❣ ❝❤Ð♦ ❇❉✳ ●ä✐ ▼ ❧➭ ➤✐Ĩ♠ ➤è✐ ①ø♥❣ ❝đ❛ ❈ q✉❛ P❀ ❊✱ ợt ì ế ủ tr ✈➭ ❆❇✳ ❈❤ø♥❣ ♠✐♥❤ ❊✱ ❋✱ P t❤➻♥❣ ❤➭♥❣✳ ➜Ò ✸✳ ❚õ ♠ét ➤Ø♥❤ ❝ñ❛ ♠ét t❛♠ ❣✐➳❝ ✈Ï ❝➳❝ ➤➢ê♥❣ ✈✉➠♥❣ ❣ã❝ ①✉è♥❣ ❜è♥ ➤➢ê♥❣ ♣❤➞♥ ❣✐➳❝ ❝➳❝ ❣ã❝ tr♦♥❣ ✈➭ ♥❣♦➭✐ ❝đ❛ t❛♠ ❣✐➳❝ ë ❤❛✐ ➤Ø♥❤ ❝ß♥ ❧➵✐✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ❜è♥ ❝❤➞♥ ➤➢ê♥❣ ✈✉➠♥❣ ❣ã❝ ➤ã ♥➺♠ tr➟♥ ❝ï♥❣ ♠ét ➤➢ê♥❣ t❤➻♥❣✳ ➜Ò ✹✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈ ❝ã ❜❛ ❣ã❝ ♥❤ä♥ ♥é✐ t✐Õ♣ tr♦♥❣ ➤➢ê♥❣ trò ọ ợt trự t ✈➭ trä♥❣ t➞♠ ❝ñ❛ t❛♠ ❣✐➳❝ ❆❇❈✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ❍✱ ●✱ ❖ t❤➻♥❣ ❤➭♥❣✳ ➜Ị ✺✳ ❚õ ❣✐❛♦ ➤✐Ĩ♠ ❖ ❝ñ❛ ❝➳❝ ➤➢ê♥❣ ♣❤➞♥ ❣✐➳❝ tr♦♥❣ ❝ñ❛ t❛♠ ❣✐➳❝ ❆❇❈✱ ❦❰ ❝➳❝ ➤➢ê♥❣ ✈✉➠♥❣ ❣ã❝ ✈í✐ ❝➳❝ ❝➵♥❤ ❇❈✱ ❈❆✱ ❆❇ ❧➬♥ ❧➢ỵt t➵✐ ❉✱ ❊✱ ❋✳ ❑❰ BB1 ⊥ AO✱ ❦❰ AA1 ⊥ BO✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ A1 , B1 , D, E ❝ï♥❣ ♥➺♠ tr➟♥ ♠ét ➤➢ê♥❣ t❤➻♥❣✳ ✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤➢ê♥❣ t❤➻♥❣ ➤å♥❣ q✉② ✸✳✶✳ ❈➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ ✸✳✷✳ ❈➳❝ ♣❤➢➡♥❣ ❞➵♥❣ t♦➳♥ ➜Ị ✶✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈ ✈➭ ➤✐Ĩ♠ ❖ ë tr♦♥❣ t❛♠ ❣✐➳❝ ✃②✳ ●ä✐ ▼✱ ◆✱ P ❧➬♥ ❧➢ỵt ❧➭ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❆❇✱ ❆❈✱ ❇❈✳ ❈➳❝ ➤✐Ĩ♠ ❆✬✱ ❇✬✱ ❈✬ ❧➬♥ ❧➢ỵt ❧➭ ❝➳❝ ➤✐Ĩ♠ ➤è✐ ①ø♥❣ ❝đ❛ ❖ q✉❛ P✱ ◆✱ ▼✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ❝➳❝ ➤➢ê♥❣ t❤➻♥❣ ❆❆✬✱ ❇❇✬✱ ❈❈✬ ➤å♥❣ q✉② t➵✐ ♠ét ➤✐Ĩ♠✳ ➜Ị ✷✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈✳ ❱Ï r❛ ♥❣♦➭✐ t❛♠ ❣✐➳❝ ❆❇❈ ❝➳❝ t❛♠ ❣✐➳❝ ✈✉➠♥❣ ❝➞♥ ❆❇❉✱ ❇❈❊✱ ❈❆❋ ❝ã ❝➳❝ ❝➵♥❤ ❤✉②Ị♥ ❧➬♥ ❧➢ỵt ❧➭ ❆❇✱ ❇❈✱ ❈❆✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ❝➳❝ ➤➢ê♥❣ t❤➻♥❣ ❈❉✱ ❇❋✱ ❆❊ ➤å♥❣ q✉②✳ ✫ ✺✹ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ➜Ị ❚❙✳ ◆❣✉②Ơ♥ ❉ ❍♦➭♥❣ ✳✳✳ ✸✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈ ♥é✐ t✐Õ♣ tr♦♥❣ ➤➢ê♥❣ trß♥ ❖✳ ❍ ❧➭ trù❝ t➞♠ t❛♠ ❣✐➳❝ ✳ ●ä✐ A1 , B1 , C1 ❧➬♥ ❧➢ỵt ❧➭ ❝➳❝ ➤✐Ĩ♠ ➤è✐ ①ø♥❣ ✈í✐ ❍ q✉❛ ❇❈✱ ❈❆✱ ❆❇❀ ❞ ❧➭ ♠ét ➤➢ê♥❣ t❤➻♥❣ q✉❛ ❍ ✈➭ ❦❤➠♥❣ s♦♥❣ s♦♥❣ ✈í✐ ❝➵♥❤ ♥➭♦ ❝đ❛ t❛♠ ❣✐➳❝ ❆❇❈✳ d1 , d2 , d3 ❧➬♥ ❧➢ỵt ❧➭ ➤➢ê♥❣ t❤➻♥❣ ➤è✐ ①ø♥❣ ❝đ❛ ❞ q✉❛ ❇❈✱ ❈❆✱ ❆❇✳ ❈❤ø♥❣ ♠✐♥❤ d1 , d2 , d3 ➤å♥❣ q✉② t➵✐ ♠ét ➤✐Ĩ♠ tr➟♥ ➤➢ê♥❣ trß♥ ❖✮✳ ➜Ị ✹✳ ❈❤♦ ❧ơ❝ ❣✐➳❝ ❧å✐ ❆❇❈❉❊❋✱ ❜✐Õt r➺♥❣ ỗ é ó t ♣❤➬♥ ❝ã ❞✐Ö♥ tÝ❝❤ ❜➺♥❣ ♥❤❛✉✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ❆❉✱ ❇❊✱ ❈❋ ➤å♥❣ q✉②✳ ✹✳ ❈❤ø♥❣ ♠✐♥❤ ➤➢ê♥❣ t❤➻♥❣ ➤✐ q✉❛ ➤✐Ĩ♠ ❝è ➤Þ♥❤ ✹✳✶✳ ❈➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ ✹✳✷✳ ❈➳❝ ♣❤➢➡♥❣ ❞➵♥❣ t♦➳♥ ➜Ò ✶✳ ❈❤♦ ❣ã❝ ✈✉➠♥❣ ột ì ữ t ó ➤ỉ✐✱ ❝ß♥ ❝➳❝ ❝➵♥❤ ❖❆✱ ❖❈ t❤❛② ➤ỉ✐ ♥❤➢♥❣ ❧✉➠♥ ♥➺♠ tr➟♥ ❝➳❝ t✐❛ ❖①✱ ❖②✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ➤➢ê♥❣ t❤➻♥❣ ➤✐ q✉❛ ❇ ✈➭ ✈✉➠♥❣ ❣ã❝ ✈í✐ ❆❈ ❧✉➠♥ ➤✐ q✉❛ ♠ét ➤✐Ĩ♠ ❝è ➤Þ♥❤✳ ➜Ị ✷✳ ❈❤♦ tø ❣✐➳❝ ❆❇❈❉ t❤❛② ➤ỉ✐ ♥é✐ t✐Õ♣ tr♦♥❣ ➤➢ê♥❣ trß♥ ❖✳ ❱Ï t✐❛ ❆① ✈✉➠♥❣ ❣ã❝ ❆❉ ✈➭ ❝➽t ❝➵♥❤ ❇❈ t➵✐ ❊ ✭❊ ♥➺♠ tr➟♥ ❝➵♥❤ ❇❈✮✳ ❱Ï t✐❛ ❆② ✈✉➠♥❣ ❣ã❝ ✈í✐ ❆❇✱ ❝➽t ❈❉ t➵✐ ❋✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ❊❋ ❧✉➠♥ ➤✐ q✉❛ ♠ét ➤✐Ĩ♠ ❝è ➤Þ♥❤✳ ➜Ị ✸✳ ❈❤♦ ➤➢ê♥❣ trß♥ t➞♠ ❖✱ ❞➞② ❆❇✳ ▼ ❞✐ ❝❤✉②Ĩ♥ tr➟♥ ❝✉♥❣ ❧í♥ ❆❇✳ ❈➳❝ ➤➢ê♥❣ ❝❛♦ ❆❊✱ ❇❋ ❝ñ❛ t❛♠ ❣✐➳❝ ❆❇▼ ❝➝t ♥❤❛✉ t➵✐ ❍✳ ❛✮ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ OM ⊥ EF ✳ ❜✮ ➜➢ê♥❣ trß♥ t➞♠ ❍ ❜➳♥ ❦Ý♥❤ ❍▼ ❝➽t ▼❆✱ ▼❇ ë ❈ ✈➭ ❉✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ➤➢ê♥❣ t❤➻♥❣ ❦❰ tõ ▼ ✈✉➠♥❣ ❣ã❝ ✈í✐ ❈❉ ❧✉➠♥ ➤✐ q✉❛ ♠ét ➤✐Ĩ♠ ❝è ➤Þ♥❤✳ ❝✮ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ➤➢ê♥❣ t❤➻♥❣ ❦❰ tõ ❍ ✈✉➠♥❣ ❣ã❝ ✈í✐ ❈❉ ❝ị♥❣ ➤✐ q✉❛ ♠ét ➤✐Ĩ♠ ❝è ➤Þ♥❤✳ ề trò ố ị ▼ tr➟♥ ❝✉♥❣ ♥❤á ❆❇✳ ●ä✐ ❑ ❧➭ tr✉♥❣ ➤✐Ó♠ ➤♦➵♥ ▼❇✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ❦❤✐ ▼ ❝❤➵② tr➟♥ ❝✉♥❣ ❆❇ t❤× ➤➢ê♥❣ t❤➻♥❣ q✉❛ ❑ ✈✉➠♥❣ ❣ã❝ ✈í✐ ❆▼ ❧✉➠♥ ➤✐ q✉❛ ♠ét ➤✐Ĩ♠ ❝è ➤Þ♥❤✳ ✫ ✺✺ P❤➢➡♥❣ ♣❤➳♣ ❞➵② ❤ä❝ ✺✳ ✳✳✳ ❈❤ø♥❣ ♠✐♥❤ ♥❤✐Ị✉ ➤✐Ĩ♠ ♥➺♠ tr➟♥ ➤➢ê♥❣ trß♥ ✺✳✶✳ ❈➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ ✺✳✷✳ ❈➳❝ ♣❤➢➡♥❣ ❞➵♥❣ t♦➳♥ ➜Ị ❚❙✳ ◆❣✉②Ơ♥ ❉ ❍♦➭♥❣ ✶✳ ❈❤♦ tø ❣✐➳❝ ❆❇❈❉ ♥é✐ t✐Õ♣ ➤➢ê♥❣ trß♥ ❖✳ ●ä✐ H1 , H2 , H3 , H4 ❧➬♥ ❧➢ỵt ❧➭ trù❝ t➞♠ ❝➳❝ t❛♠ ❣✐➳❝ ❇❈❉✱ ❆❈❉✱ ❆❇❉✱ ❆❈✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ✹ ➤✐Ĩ♠ ➜Ị H1 , H2 , H3 , H4 ❝ï♥❣ ♥➺♠ tr➟♥ ♠ét ➤➢ê♥❣ trß♥✳ ✷✳ ❈❤♦ t❛♠ ❣✐➳❝ ❆❇❈ ❝ã trù❝ t➞♠ ❍ ✈➭ ❆✬✱ ❇✬✱ ❈✬ ❧➬♥ ❧➢ỵt ❧➭ ❝➳❝ tr✉♥❣ ➤✐Ĩ♠ ❝đ❛ ❇❈✱ ❈❆✱ ❆❇✳ ❱Ï ❜❛ ➤➢ê♥❣ trß♥ ❜➺♥❣ ♥❤❛✉ ❝ã t➞♠ ❧➭ ❆✱ ❇✱ ❈✱ ➤➢ê♥❣ trß♥ t➞♠ ❆ ❝➽t ➤➢ê♥❣ t❤➻♥❣ ❇✬❈✬ t➵✐ ❉ ✈➭ ❉✬❀ ➤➢ê♥❣ trß♥ t➞♠ ❇ ❝➽t ➤➢ê♥❣ t❤➻♥❣ ❆✬❈✬ t➵✐ ❊ ✈➭ ❊✬❀ ➤➢ê♥❣ trß♥ ❝ã t➞♠ ❈ ❝➽t ➤➢ê♥❣ t❤➻♥❣ ❆✬❇✬ t➵✐ ❑ ✈➭ ❑✬✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ✻ ➤✐Ó♠ ❉✱ ❉✬✱ ❊✱ ❊✬✱ ❑✱ ù tộ ột trò ề ì ✈✉➠♥❣ ❆❇❈❉ t➞♠ ❖✱ ➤➢ê♥❣ t❤➻♥❣ ❞ q✉❛② q✉❛♥❤ ❖ ❝➽t ❤❛✐ ❝➵♥❤ ❆❉ ✈➭ ❇❈ ❧➬♥ ❧➢ỵt t➵✐ ❊ ✈➭ ❋ ✭❊✱❋ ❦❤➠♥❣ trï♥❣ ✈í✐ ❝➳❝ ➤Ø♥❤ ❤×♥❤ ✈✉➠♥❣✮✳ ❚õ ❊ ✈➭ ❋ ❧➬♥ ❧➢ỵt ✈Ï ❝➳❝ ➤➢ê♥❣ t❤➻♥❣ s♦♥❣ s♦♥❣ ✈í✐ ❉❇✱ ❆❈✱ ❝❤ó♥❣ ❝➽t ♥❤❛✉ t➵✐ ■✳ ❛✮ ❈❤ø♥❣ ♠✐♥❤ ❆✱ ■✱ ❇ t❤➻♥❣ ❤➭♥❣✳ ❜✮ ❚õ ■ ✈Ï ➤➢ê♥❣ ✈✉➠♥❣ ❣ã❝ ✈í✐ ❊❋ t➵✐ ❍✱ ❝❤ø♥❣ ♠✐♥❤ ❍ t❤✉é❝ ♠ét ➤➢ê♥❣ t❤➻♥❣ ❝è ➤Þ♥❤ ✈➭ ➤➢ê♥❣ t❤➻♥❣ ■❍ ➤✐ q✉❛ ♠ét ➤✐Ĩ♠ ❝è ➤Þ♥❤✳ ➜Ị ✹✳ trò ố ị tr ❝✉♥❣ ♥❤á ❆❇✳ ●ä✐ ❑ ❧➭ tr✉♥❣ ➤✐Ó♠ ➤♦➵♥ ▼❇✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ❦❤✐ ▼ ❝❤➵② tr➟♥ ❝✉♥❣ ❆❇ t❤× ➤➢ê♥❣ t❤➻♥❣ q✉❛ ❑ ✈✉➠♥❣ ❣ã❝ ✈í✐ ❆▼ ❧✉➠♥ ➤✐ q✉❛ ♠ét ➤✐Ĩ♠ ❝è ➤Þ♥❤✳ ✫ ✺✻