ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC - NGƠ TRỌNG THÀNH ĐƯỜNG TRỊN SODDY VÀ CÁC VẤN ĐỀ LIÊN QUAN LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2019 ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC - NGÔ TRỌNG THÀNH ĐƯỜNG TRÒN SODDY VÀ CÁC VẤN ĐỀ LIÊN QUAN Chuyên ngành: Phương pháp Toán sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC PGS.TS Nguyễn Việt Hải THÁI NGUN - 2019 ✐ ▼ư❝ ❧ư❝ ❉❛♥❤ ♠ư❝ ❤➻♥❤ ▲í✐ ❝↔♠ ì♥ ▼ð ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❜ê s✉♥❣ ✶✳✶ ✶✳✷ ✐✐✐ ✐✈ ✶ ✸ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ tr♦♥❣ ♠➦t ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✷ ❈æ♥❣ t❤ù❝ ❦❤♦↔♥❣ ❝→❝❤✱ t➼♥❤ ❝❤➜t ❜↔♦ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✷ ▼ët sè ❦➳t q✉↔ tr♦♥❣ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ❚å❛ ✤ë ❜❛r②❝❡♥tr✐❝ t❤✉➛♥ ♥❤➜t ✷ ❈→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ✷✵ ✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝❤ ❞ü♥❣ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❇→♥ ❦➼♥❤ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ✷✸ ✷✳✸ ✷✳✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ❇→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ❙♦❞❞② ♥ë✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷✳✷ ❇→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ❙♦❞❞② ♥❣♦↕✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✣÷í♥❣ trá♥ ❙♦❞❞② tr♦♥❣ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸✳✶ ❈→❝ ✤✐➸♠ ❙♦❞❞② ✈➔ ✤÷í♥❣ t❤➥♥❣ ❙♦❞❞② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸✳✷ P❤÷ì♥❣ tr➻♥❤ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ❚❛♠ ❣✐→❝ ❙♦❞❞② ✈➔ t❛♠ ❣✐→❝ ❊✉❧❡r✲●❡r❣♦♥♥❡✲❙♦❞❞② ✳ ✳ ✳ ✳ ✸ ▼ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✸✳✶ ❚❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ✷✾ ✸✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✐✐ ✸✳✷ ✸✳✸ ✸✳✶✳✶ ▼ët sè ❤➺ t❤ù❝ ❤➻♥❤ ❤å❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✳✷ ❚❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✸✳✶✳✸ ❚❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ❝↕♥❤ ♥❣✉②➯♥ ✸✳✶✳✹ ❉ü♥❣ t❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ❜✐➳t ♠ët ❝↕♥❤ κ = ta + tb + tc ✳ ✳ ✳ ✸✳✷✳✶ ❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ κ = ✸✳✷✳✷ ❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ κ = ❈→❝ t❛♠ ❣✐→❝ ❧ỵ♣ = tb + tc ✳ ✳ ✳ ✳ ✳ ✸✳✸✳✶ ❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ =1 ✸✳✸✳✷ ❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ =2 ❈→❝ t❛♠ ❣✐→❝ ❧ỵ♣ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✺✼ ✺✽ ✐✐✐ ❉❛♥❤ ♠ö❝ ❤➻♥❤ ✶✳✶ ❷♥❤ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ✤✐➸♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❛✮ ❷♥❤ ✤÷í♥❣ t❤➥♥❣ ❦❤ỉ♥❣ q✉❛ ❝ü❝❀ ❜✮ ❷♥❤ ✤÷í♥❣ trá♥ ❝â t➙♠ ❧➔ ❝ü❝ ✶✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❷♥❤ ❝õ❛ ✤÷í♥❣ trá♥ ❦❤ỉ♥❣ q✉❛ ❝ü❝ ♥❣❤à❝❤ ✤↔♦ ✹ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ AB = R · AB OA.OB ✶✳✹ ❑❤♦↔♥❣ ❝→❝❤ ✶✳✺ ❚➼♥❤ ❝❤➜t ❜↔♦ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✻ ❱➼ ❞ư ✈➲ ❝ỉ♥❣ t❤ù❝ ❈♦♥✇❛② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶ ✣÷í♥❣ trá♥ ❙♦❞❞② ♥ë✐ ✈➔ ✤÷í♥❣ trá♥ ❙♦❞❞② ♥❣♦↕✐ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❈→❝❤ ❞ü♥❣ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ❚å❛ ✤ë ❜❛r②❝❡♥tr✐❝ ❝õ❛ ❝→❝ ✤✐➸♠ ❙♦❞❞② ✈➔ ✤÷í♥❣ t❤➥♥❣ ❙♦❞❞② ✷✻ ✷✳✹ ❚➙♠ ❙♦❞❞② ♥ë✐✱ ♥❣♦↕✐ ✈➔ ✤✐➸♠ ❊♣♣st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✺ ❈→❝ ✤÷í♥❣ t❤➥♥❣ ❊✉❧❡r ✈➔ ●❡r❣♦♥♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✻ ❚❛♠ ❣✐→❝ ❊✉❧❡r✲●❡r❣♦♥♥❡✲❙♦❞❞② ✈✉æ♥❣ t↕✐ S ✳ ✳ ✸✷ ✷✳✼ ▼ët sè ✤✐➸♠ tr➯♥ ❝↕♥❤ t❛♠ ❣✐→❝ ❊✉❧❡r✲●❡r❣♦♥♥❡✲❙♦❞❞② ✳ ✳ ✸✸ ✸✳✶ AD✲❝❡✈✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❡✈✐❛♥ t✐➳♣ t✉②➳♥ ✤➾♥❤ ❆ ✸✳✸ ❈→❝ ❤➺ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ✸✳✹ P Q ⊥ AD ✸✳✺ ❚❛♠ ❣✐→❝ ❦✐➸✉ ữớ t r s s ợ t➼❝❤ ✤✐➸♠ ✸✳✽ ❉ü♥❣ t❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ❜✐➳t ♠ët ❝↕♥❤ ✸✳✾ ❚❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ A t✐➳♣ t✉②➳♥ ✤➾♥❤ E = X481 Fl = G ∩ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ C θ ABC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ AD ✹✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✶✵ ❚❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ =1✳ =2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✐✈ ▲í✐ ❝↔♠ ì♥ ✣➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♠ët ❝→❝❤ tổ ổ ữủ sỹ ữợ ❣✐ó♣ ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ P●❙✳❚❙✳ ◆❣✉②➵♥ ❱✐➺t ❍↔✐✱ ●✐↔♥❣ ✈✐➯♥ ❝❛♦ ❝➜♣ ❚r÷í♥❣ ✤↕✐ ❤å❝ ❍↔✐ P❤á♥❣✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛② ✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥ ♥❤➜t ❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉ t❤➛② ✤➣ ❞➔♥❤ ❝❤♦ tỉ✐✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ♣❤á♥❣ ✣➔♦ t↕♦✱ ❑❤♦❛ ❚♦→♥ qỵ t ổ ợ ✭✷✵✶✼ ✲ ✷✵✶✾✮ ❚r÷í♥❣ ✤↕✐ ❤å❝ ❦❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ♥❤ú♥❣ ❦✐➳♥ tự qỵ ụ ữ t tổ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ t t tợ ỳ ữớ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❤é trđ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦ ❍↔✐ P❤á♥❣✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✾ ◆❣÷í✐ ✈✐➳t ▲✉➟♥ ✈➠♥ ◆❣ỉ ❚rå♥❣ ❚❤➔♥❤ ✶ ▼ð ✤➛✉ ✶✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❈→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ❝õ❛ t❛♠ ❣✐→❝ ABC ❝â ♥❤ú♥❣ t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t✱ ❜➔✐ t♦→♥ ❞ü♥❣ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ❧➔ tr÷í♥❣ ❤đ♣ r✐➯♥❣ q✉❛♥ trå♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❆♣♦❧✐❧♦♥✐✉s✳ ❈❤❛ ✤➫ ❝õ❛ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ ✤✐➸♠ ❙♦❞❞②✱ ✤÷í♥❣ t❤➥♥❣ ❙♦❞❞②✱ t❛♠ ❣✐→❝ ❙♦❞❞②✱✳✳ ❧➔ ❋r❡❞❡r✐❝❦ ❙♦❞❞②✱ ♥❣÷í✐ ✤➣ ❞➔♥❤ ✤÷đ❝ ❣✐↔✐ t❤÷ð♥❣ ◆♦❜❡❧ ✈➲ ❍â❛ ❤å❝✳ P❤→t tr✐➸♥ ❝→❝ ❦❤→✐ ♥✐➺♠ ♥➔② tr♦♥❣ ♥❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙②✱ ♥❤✐➲✉ t→❝ ❣✐↔ ✭◆✳ ❉❡r❣✐❛❞❡s ♥➠♠ ✷✵✵✼✱ ▼✳ ❏❛❝❦s♦♥ ♥➠♠ ✷✵✶✸✱ ▼✳ ❏❛❝❦s♦♥ ✈➔ ❚❛❦❤❛❡✈ ♥➠♠ ✷✵✶✺✱ ✷✵✶✻ ✮ ✤➣ ❝æ♥❣ ❜è ❝→❝ ♣❤→t ❤✐➺♥ ❤➻♥❤ ❤å❝ s➙✉ s➢❝ s✐♥❤ r❛ tø ✤÷í♥❣ trá♥ ❙♦❞❞②✳ ❇➔✐ t♦→♥ ✤➦t r❛ ❧➔ ❧➔♠ t❤➳ ♥➔♦ ❞ü♥❣ ✤÷đ❝ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ ①→❝ ✤à♥❤ ❝→❝ ❜→♥ ❦➼♥❤ ❝õ❛ ❝❤ó♥❣ t❤❡♦ ❝→❝ tố t trữợ ữớ trỏ ữớ t õ q ợ ❝→❝ ✤÷í♥❣ trá♥ ✈➔ ✤÷í♥❣ t❤➥♥❣ ✤➣ ❜✐➳t ❦❤→❝❄ ❚r➻♥❤ qt t tr ỵ ❞♦ ✤➸ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ✧✣÷í♥❣ trá♥ ❙♦❞❞② ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥✧✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔✿ ✲ ❚r➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ ❝→❝❤ ①→❝ ✤à♥❤ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ t➼♥❤ ✤÷đ❝ ❝→❝ ❜→♥ ❦➼♥❤✱ t➻♠ ✤÷đ❝ t t ợ ữớ trỏ ✤÷í♥❣ trá♥ ❙♦❞❞② ♥❣♦↕✐✳ ❚ø ✤â ✤÷❛ r❛ ❝→❝❤ ❞ü♥❣ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❝→❝ ✤÷í♥❣ trá♥✱ ✤÷í♥❣ t❤➥♥❣ ❙♦❞❞② tr♦♥❣ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝✳ ✲ ❳→❝ ✤à♥❤ ♠è✐ q✉❛♥ ❤➺ ❝õ❛ t ợ ữớ t t P ữủ t ợ κ = ta + tb + tc ❦❤↔♦ s→t ❝→❝ trữớ ủ t ợ õ ợ = tb + tc ✱ ✷ ✷✳ ◆ë✐ ❞✉♥❣ ✤➲ t➔✐✱ ♥❤ú♥❣ ✈➜♥ ✤➲ ❝➛♥ ❣✐↔✐ q✉②➳t ◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ✸ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❜ê s✉♥❣ ◆❤➢❝ ❧↕✐ ✈➔ ❜ê s✉♥❣ ❤❛✐ ❝❤õ ✤➲ ❝ì ❜↔♥ ✤÷đ❝ sû ❞ư♥❣ ❧➔♠ ❝ỉ♥❣ ❝ư ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ ✤➦t r❛✿ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ✈➔ tå❛ ✤ë rtr ữỡ ỗ P tr♦♥❣ ♠➦t ♣❤➥♥❣ ✶✳✷✳ ❚å❛ ✤ë ❜❛r②❝❡♥tr✐❝ t❤✉➛♥ ♥❤➜t ❈❤÷ì♥❣ ✷✳ ❈→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ✤➲ ❝➟♣ ✤➳♥ sü ①→❝ ✤à♥❤ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ❝ị♥❣ ❝→❝ ❜ë ♣❤➟♥ ❝õ❛ ♥â ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤➻♥❤ ❤å❝ ❝➜♣ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë✳ ✣➙② ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ trå♥❣ t➙♠ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❈❤÷ì♥❣ ♥➔② ❜❛♦ ỗ s tờ ủ s tứ ❜➔✐ ❜→♦ ❬✶❪✱ ❬✸❪✱ ❬✼❪✮✿ ✷✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝❤ ❞ü♥❣ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ✷✳✷✳ ❇→♥ ❦➼♥❤ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ✷✳✸✳ ✣÷í♥❣ trá♥ ❙♦❞❞② tr♦♥❣ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝ ✷✳✹✳ ❚❛♠ ❣✐→❝ ❙♦❞❞② ✈➔ t❛♠ ❣✐→❝ ❊✉❧❡r✲●❡r❣♦♥♥❡✲❙♦❞❞② ❈❤÷ì♥❣ ✸✳ ▼ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ❈❤÷ì♥❣ ✸ ①➨t ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ t❛♠ ❣✐→❝ ❙♦❞❞②✱ t❤ü❝ ❝❤➜t ❧➔ ❝→❝ tr÷í♥❣ ❤đ♣ r✐➯♥❣ q✉❛♥ trå♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ❦❤→✐ ♥✐➺♠ ❦❤→❝ tr♦♥❣ ❤➻♥❤ ❤å❝✱ ❝❤➥♥❣ ❤↕♥ t❛♠ ❣✐→❝ ❍❡r♦♥✳ ❈❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ ✈➔ tê♥❣ ❤ñ♣ t❤❡♦ ❝→❝ ❜➔✐ ❜→♦ ❬✹❪✱ ❬✺❪✳ ◆ë✐ ỗ t ❣✐→❝ ❧ỵ♣ ✸✳✸✳ ❈→❝ t❛♠ ❣✐→❝ ❧ỵ♣ κ = ta + tb + tc = tb + tc ✳ ✸ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❜ê s✉♥❣ ❚❛ ♥❤➢❝ ❧↕✐ ✈➔ ❜ê s✉♥❣ ❤❛✐ ♥ë✐ ❞✉♥❣ ❝➛♥ ❝❤♦ ❝→❝ ❝❤÷ì♥❣ s❛✉✿ ❚❤ù ♥❤➜t✱ ✤✐➸♠ q✉❛ ✈➲ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ●✐→♦ tr➻♥❤ ❤➻♥❤ ❤å❝ ❝➜♣❀ ❚❤ù ❤❛✐✱ ❜ê s✉♥❣ t❤➯♠ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝ ✭❞↕♥❣ ❤➻♥❤ ❤å❝ ❣✐↔✐ t➼❝❤✮✱ ♣❤→t tr✐➸♥ tø ❦❤→✐ ♥✐➺♠ t➙♠ t✛ ❝ü q✉❡♥ t❤✉ë❝✳ ✶✳✶ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ tr♦♥❣ ♠➦t ♣❤➥♥❣ ❚❛ ♥❤➢❝ ❧↕✐ ♠ët sè ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ q✉❛ ✤÷í♥❣ trá♥ ❤❛② ❝á♥ ❣å✐ ❧➔ ♣❤➨♣ ✤è✐ ①ù♥❣ q✉❛ ✤÷í♥❣ trá♥ tr➯♥ ♠➦t ♣❤➥♥❣ ❊✉❝❧✐❞❡✳ ❈→❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝â t❤➸ t➻♠ t❤➜② tr♦♥❣ ❝→❝ ❣✐→♦ tr➻♥❤ ❍➻♥❤ ❤å❝ ❝➜♣ ❤✐➺♥ ❤➔♥❤✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❚r➯♥ ♠➦t ♣❤➥♥❣ ❝❤♦ ✤÷í♥❣ trá♥ t➙♠ O✱ ❜→♥ ❦➼♥❤ R✳ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝ü❝ O✱ ♣❤÷ì♥❣ t➼❝❤ k = R2 ❧➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ tr➯♥ ♠➦t ♣❤➥♥❣✱ ❜✐➳♥ P → P s❛♦ ❝❤♦ ♥➳✉ P = O t❤➻ OP.OP = R2 ❀ ♥➳✉ P ≡ O t❤➻ P ←→ ∞✳ ỵ õ ữớ trỏ ♥❣❤à❝❤ ✤↔♦✳ fRO2 ✱ ✤÷í♥❣ trá♥ (O, R) ✤÷đ❝ ❣å✐ ❧➔ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ♥➔② ❝ô♥❣ ❣å✐ ❧➔ ♣❤➨♣ ✤è✐ ①ù♥❣ q✉❛ ✤÷í♥❣ trá♥✳ ❉➵ t❤➜② ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝â t➼♥❤ ❝❤➜t ✤è✐ ❤ñ♣✱ tù❝ ❧➔ fRO2 = Id✳ ❚ø ✹ ❍➻♥❤ ✶✳✶✿ ❷♥❤ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ✤✐➸♠ ✤à♥❤ ♥❣❤➽❛ t❛ s✉② r❛ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦✿ ❍➻♥❤ ✶✳✷✿ ❛✮ ❷♥❤ ✤÷í♥❣ t❤➥♥❣ ❦❤ỉ♥❣ q✉❛ ❝ü❝❀ ❜✮ ❷♥❤ ✤÷í♥❣ trá♥ ❝â t➙♠ ❧➔ ❝ü❝ ❛✮ ◗✉❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ fRO2 ✱ ✤÷í♥❣ trá♥ ♥❣❤à❝❤ ✤↔♦ (O, R) ❜✐➳♥ t❤➔♥❤ ❝❤➼♥❤ ♥â✱ ♥â✐ ❝→❝❤ ❦❤→❝✱ ✤÷í♥❣ trá♥ ♥❣❤à❝❤ ✤↔♦ ❧➔ ❤➻♥❤ ❦➨♣ t✉②➺t ✤è✐ ✭t÷ì♥❣ tü trö❝ ✤è✐ ①ù♥❣ tr♦♥❣ ♣❤➨♣ ✤è✐ ①ù♥❣✮✳ ▼å✐ ✤✐➸♠ ð tr♦♥❣ ❜✐➳♥ t❤➔♥❤ ✤✐➸♠ ð ♥❣♦➔✐ ✈➔ ♥❣÷đ❝ ❧↕✐✳ (O, R) ✹✹ s−a s−b s−c s = = = m2 n2 m2 (m + n)2 n2 (m + n)2 (m2 + mn + n2 )2 ⇐⇒ (s − a) m2 + mn + n2 ⇐⇒ s · = s.m2 n2 m2 + n2 (m + n)2 = a · m2 + mn + n2 2 s m2 (m + n)2 + n2 = b m2 + mn + n2 ✈➔ s·[n2 (m+n)2 + m2 ] = c m2 + mn + n2 ✳ ❱➻ ✈➟② t❛ ❝â t❤➸ ❧➜② a = (m + n)2 m2 + n2 ❀ ❚÷ì♥❣ tü✱ b = m2 (m + n)2 + n2 ; c = n2 (m + n)2 + m2 ✳ ⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❍❡r♦♥ ✈➔ ❝→❝ ❝ỉ♥❣ t❤ù❝ ✤➣ ❜✐➳t t❛ ❝â✿ S = m2 n2 (m + n)2 m2 + mn + n2 ; s = m2 + mn + n2 m2 + n2 (m + n)2 + m2 (m + n)2 + n2 m2 n2 (m + n)2 R= ;r = (m2 + mn + n2 ) m + mn + n2 ❙❛✉ ✤➙② ❧➔ ♠ët sè t❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ❝↕♥❤ ♥❣✉②➯♥ ù♥❣ ✈ỵ✐ ❝→❝ ❝➦♣ m n c b a s S r R ✶ ✶ ✺ ✺ ✽ ✾ ✶✷ ✷ ✶ ✶✸ ✹✵ ✹✺ ✹✾ ✷✺✷ ✸ ✶ ✷✺ ✶✺✸ ✶✻✵ ✶✻✾ ✶✽✼✷ ✹ ✶ ✹✶ ✹✶✻ ✹✷✺ ✹✹✶ ✽✹✵✵ ✸ ✷ ✶✸✻ ✷✻✶ ✸✷✺ ✸✻✶ ✶✼✶✵✵ ✺ ✶ ✻✶ ✾✷✺ ✾✸✻ ✾✻✶ ✷✼✾✵✵ ✻ ✶ ✽✺ ✶✽✵✵ ✶✽✶✸ ✶✽✹✾ ✼✻✽✺✷ ✺ ✷ ✷✾✻ ✶✸✷✺ ✶✹✷✶ ✶✺✷✶ ✶✾✶✶✵✵ ✹ ✸ ✺✽✺ ✾✷✽ ✶✷✷✺ ✶✸✻✾ ✷✻✶✵✼✷ ✼ ✶ ✶✶✸ ✸✶✽✺ ✸✷✵✵ ✸✷✹✾ ✶✼✽✼✺✷ ✺ ✸ ✽✵✶ ✶✽✷✺ ✷✶✼✻ ✷✹✵✶ ✼✵✺✻✵✵ ✽ ✶ ✶✹✺ ✺✷✹✽ ✺✷✻✺ ✺✸✷✾ ✸✼✽✹✸✷ ✼ ✷ ✺✷✵ ✹✶✻✺ ✹✷✾✸ ✹✽✾ ✶✵✻✸✻✾✷ ✺ ✹ ✶✻✾✻ ✷✹✷✺ ✸✸✷✶ ✸✼✷✶ ✶✾✼✻✹✵✵ ✾ ✶ ✶✽✶ ✽✶✽✶ ✽✷✵✵ ✽✷✽✶ ✼✸✼✶✵✵ ✼ ✸ ✶✸✹✶ ✺✸✹✶ ✺✽✵✵ ✻✷✹✶ ✸✹✽✸✾✵✵ 36 144 13 400 21 900 31 900 31 1764 43 4000 39 7056 37 3136 57 14400 49 5184 73 15876 67 32400 61 8100 91 44100 79 25 325 14 2125 26 9061 42 6409 38 29341 62 78625 86 56869 78 47125 74 183625 114 110449 98 386125 146 292825 134 210781 122 749521 182 470980 158 (m, n)✿ ✹✺ ✸✳✶✳✹ ❉ü♥❣ t❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ❜✐➳t ♠ët ❝↕♥❤ ❳➨t t❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ❝â ✤→② ❧➔ AB A(0, 0), B(c, 0)✳ ●✐↔ sû tå❛ ✤ë C ợ m, n số tỹ ữỡ t❛ ❝â ❣â❝ ✈ỵ✐ ✸✳✶✳✸ tr♦♥❣ ❤➺ tå❛ ✤ë ❉❡s❝❛rt❡s ✈✉ỉ♥❣ ❧➔ (x, y)✳ ❉ị♥❣ ❜✐➸✉ ❞✐➵♥ tr♦♥❣ m3 m2 + mn + n2 x b2 + c2 − a2 , = = c 2c2 (m + n) (m2 + n2 )2 2m2 n2 m2 + mn + n2 y 2S = = c c (m + n)2 (m2 + n2 )2 ❱✐➳t m = tn t❛ t❤✉ ✤÷đ❝ q✉ÿ t➼❝❤ ❝õ❛ (x, y) = c C ♣❤ö t❤✉ë❝ t❤❛♠ sè t ♥❤÷ s❛✉ 2t2 + t + 2t2 + t + 2t2 , (1 + t) (1 + t2 )2 (1 + t)2 (1 + t2 )2 ❍➻♥❤ ✸✳✼✿ ◗✉ÿ t➼❝❤ ✤✐➸♠ C ❚r➯♥ t❤ü❝ t ợ s a, s b trữợ tữợ t ỹ ữủ t ❦✐➸✉ ❙♦❞❞② ♠ët ❝→❝❤ ✤ì♥ ❣✐↔♥ ♥❤÷ s❛✉✿ ❈→❝❤ ❞ü♥❣✳ AB ✈➔ ✤✐➸♠ Z ∈ AB s❛♦ ❝❤♦ AZ = s − a, BZ = s − b✳ ◆❤÷ ✈➟② AB = AZ + BZ = s − a + s − b = c✳ B1 ❉ü♥❣ ⊥ AB t↕✐ Z ✱ ❝➢t ♥û❛ ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ AB ð P ❀ B2 ▲➜② ❆✬✱❇✬ ♥➡♠ ✈➲ ❝ò♥❣ ♠ët ♣❤➼❛ ❝õ❛ ❆❇ ♠➔ AA , BB ⊥ AB ✈➔ AA = AZ, BB = BZ B3 ◆è✐ P A , P B ✱ ❝❤ó♥❣ ❝➢t AB t÷ì♥❣ ù♥❣ t↕✐ X, Y ❀ B4 ❉ü♥❣ ✤÷í♥❣ trá♥ q✉❛ P, X, Y ✱ ❝➢t ❧↕✐ P Z t↕✐ Q❀ B5 ❉ü♥❣ X ∈ AZ, Y ∈ ZB s❛♦ ❝❤♦ X Z = ZY = ZQ❀ ❈❤♦ ✤♦↕♥ t❤➥♥❣ ✹✻ B6 ❉ü♥❣ ❝→❝ ✤÷í♥❣ trá♥ t➙♠ A ✈➔ B ✱ t÷ì♥❣ ù♥❣ q✉❛ Y , X ✳ ❍❛✐ ✤÷í♥❣ C ✳ ∆ABC ❧➔ t ỹ ợ ữớ ú AB ð Z ✱ ❤➻♥❤ ✸✳✽✳ trá♥ ♥➔② ❣➦♣ ♥❤❛✉ ð trá♥ ♥ë✐ t✐➳♣ t✐➳♣ ❍➻♥❤ ✸✳✽✿ ❉ü♥❣ t❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ❜✐➳t ♠ët ❝↕♥❤ ❈❤ù♥❣ ♠✐♥❤✳ AZ = u, BZ = v ✳ ❚ø ❝æ♥❣ t❤ù❝ t➼♥❤ ❜→♥ ❦➼♥❤ √ trá♥ ❙♦❞❞② t❛ ❝â ZP = uv ✈➔ tø ✭✸✳✽✮ s✉② r❛ √ √ ZP uv u v √ =√ √ ZX = ZA · =u· ZP + AA u + uv u+ v √ v u √ ✳ ỵ ữỡ t ữỡ tỹ ZY = u+ v ✣➦t ZQ = ❚❛ s✉② r❛ = ZQ √ ZX.XY uv √ = √ ZP ( u + v)2 √ u+ v 1 1 √ =√ +√ =√ +√ ✳ uv u v AZ ZB ✤÷í♥❣ ✹✼ ❉♦ ✤â t❛♠ ❣✐→❝ ABC t❤ä❛ ♠➣♥✿ BC = BX = BZ + ZX = BZ + ZQ, AC = AY = AZ + ZY = AZ + ZQ, AB = AZ + ZB, 1 1 1 = √ = √ +√ = √ +√ ✳ ✣â ZQ s−c s−a s−b AZ ZB ❣✐→❝ ợ ữớ trỏ t t ú AB t↕✐ Z ✳ √ ✈ỵ✐ t❛♠ ❝❤➼♥❤ ❧➔ ✸✳✷ ❈→❝ t❛♠ ❣✐→❝ ❧ỵ♣ κ = ta + tb + tc ❚❛ ①→❝ ✤à♥❤ ♠ët ❧ỵ♣ t❛♠ ❣✐→❝ ✤➦❝ ❜✐➺t✱ ❞ü❛ t r t ABC ỵ ✣à♥❤ ♥❣❤➽❛ ✸✳✸✳ A B C , tb = tan , tc = tan ✳ 2 ❚❛ ♥â✐ ❆❇❈ ❧➔ t❛♠ ❣✐→❝ ❧ỵ♣ κ ♥➳✉ ta + tb + tc = κ✳ ta = tan ❚ø ❤❛✐ ❝æ♥❣ t❤ù❝ ✭✸✳✻✮✱ ✭✸✳✼✮✱ ♥❤➙♥ ❝↔ ✷ ✈➳ ✈ỵ✐ r✱ t❛ ✤÷đ❝ r ξ r ta + tb + tc − = ξ ta + tb + tc + = ❚❛ s✉② r❛✿ ξ κ+2 = ξ κ−2 ✭✸✳✶✵✮ ❈æ♥❣ t❤ù❝ ✭✸✳✶✵✮ t❤➸ ❤✐➺♥ q✉❛♥ ❤➺ ❣✐ú❛ ❝→❝ ❜→♥ ❦➼♥❤ ❙♦❞❞② ✈➔ ✤➦❝ sè κ✳ κ✳ Ð ✤➙② t❛ ❝❤➾ ①➨t ❜➔✐ t♦→♥ t➻♠ t➜t ❝↔ ❝→❝ t❛♠ ợ số ữỡ ởt t❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ t❤✉ë❝ ❧ỵ♣ κ ❧➔ ♠ët t❛♠ ❣✐→❝ ❍❡r♦♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝→❝ t❛♥❣ ❝õ❛ ♠é✐ ♥û❛ ❣â❝ ❝õ❛ ♥â ❧➔ sè ❤ú✉ t✛✳ ❳➨t t❛♠ ❣✐→❝ ABC ✱ ❣å✐ θ = ADB ✈ỵ✐ AD ❧➔ ❝❡✈✐❛♥ t✐➳♣ t✉②➳♥ ✤➾♥❤ A✱ t❛ ❝â tb − tc = (tb + tc ) cos ỵ ✤➳♥ sè κ ✈➔ ❤➺ t❤ù❝ ❝ì ❜↔♥ ✭✸✳✸✮ t❛ ữủ ữỡ tr ợ ta , tb , tc ✿ = (tb + tc ) cos θ tb − tc ta + tb + tc =κ t t +t t +t t =1 a b b c c a ❉➵ t❤➜② ❝â ✈ỉ sè ❝→❝ t❛♠ ❣✐→❝ t❤✉ë❝ ❧ỵ♣ ✹✽ ●✐↔✐ ❤➺ t❤✉ ✤÷đ❝ √ κ + cos2 θ + 2c κ2 − − cos2 θ ta = + cos2 θ √ (1 + cos θ) κ − κ2 − − cos2 θ tb = + cos2 θ √ (1 − cos θ) κ − κ2 − − cos2 θ tc = + cos2 θ ✭✸✳✶✶✮ ✭✸✳✶✷✮ ✭✸✳✶✸✮ = ±1✳ ❘ã r➔♥❣ ta , tb , tc ❧➔ sè ❤ú✉ t✛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ κ2 − − cos2 θ = v 2 ✈ỵ✐ v ❤ú✉ t✛✳ ❚ù❝ ❧➔ κ − ❧➔ tê♥❣ ❤❛✐ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ ❤❛✐ sè ❤ú✉ t✛✳ ▼ët ❝→❝❤ t÷ì♥❣ ✤÷ì♥❣✱ κ − ❧➔ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ✷ sè ♥❣✉②➯♥ t❤❡♦ ❜ê ✤➲ s❛✉ ợ ởt số tờ ữỡ ❝→❝ sè ❤ú✉ t✛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔ tê♥❣ ❝õ❛ ❜➻♥❤ ♣❤÷ì♥❣ ❤❛✐ sè ♥❣✉②➯♥✳ ✭✤÷đ❝ ❋✳▼✳ ❏❛❝❦s♦♥ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❬✸❪✮✿ ▼➺♥❤ ✤➲ ✸✳✷✳ ●✐↔ sû κ > số ữỡ r ợ tỗ t ❧➔ tê♥❣ ❝õ❛ ✷ ❜➻♥❤ ♣❤÷ì♥❣ ❝→❝ sè ♥❣✉②➯♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ s✉② r❛ tø ❜ê ✤➲ tr➯♥✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ κ2 − ❧➔ tê♥❣ ❝õ❛ ✷ ❜➻♥❤ ♣❤÷ì♥❣ ❝→❝ sè ♥❣✉②➯♥ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ✤â ❜➡♥❣ ❝→❝❤ ①➙② ❞ü♥❣ t❤➔♥❤ ❝æ♥❣ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ κ✳ ✸✳✷✳✶ ❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ κ = ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ♥➔②✱ ❝→❝ t❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ❦❤↔♦ s→t ð ♠ö❝ ✸✳✶ ❧➔ t❛♠ ❣✐→❝ t❤✉ë❝ ợ t q t ữủ ự tọ ú ❧➔ ♥❤ú♥❣ t❛♠ ❣✐→❝ s✐♥❤ r❛ tø ✤÷í♥❣ trá♥ ❙♦❞❞②✳ ◆❣❛② s❛✉ ✤➙②✱ t❛ s➩ ①➙② ❞ü♥❣ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ ✹✳ P❤➨♣ ①➙② ❞ü♥❣ rã r➔♥❣ →♣ ❞ư♥❣ ✈➔♦ ❧ỵ♣ κ ✈ỵ✐ κ2 − ❧➔ tê♥❣ ❝õ❛ ❤❛✐ ❜➻♥❤ ♣❤÷ì♥❣ ❝→❝ sè ♥❣✉②➯♥✳ ✸✳✷✳✷ ❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ κ = ❚✛ sè ❝→❝ ❜→♥ ❦➼♥❤ ✷ ✤÷í♥❣ trá♥ ❙♦❞❞② ❣✐→ trà ♥❣✉②➯♥ κ✱ t❛ t❤➜② ♥❣❛② t✛ sè κ+2 κ−2 κ+2 ξ = = 1+ ✳ ξ κ−2 κ−2 ❧➔ sè ♥❣✉②➯♥ ❝❤➾ ❦❤✐ ❱ỵ✐ κ = 3, 4, ✹✾ ✈➔ ❜↔♥ t❤➙♥ ❣✐→ trà t✛ sè ♥➔② t÷ì♥❣ ù♥❣ ❜➡♥❣ ❦❤ỉ♥❣ ❝â t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ κ = 3, 6✳ 5, 3, 2✳ ❚❤❡♦ ♠➺♥❤ ✤➲ ✸✳✷ Ð ✤➙② t❛ s➩ ①➙② ❞ü♥❣ t❛♠ ❣✐→❝ κ = 4✳ ❑❤æ♥❣ ♠➜t t➼♥❤ ❝❤➜t tê♥❣ q✉→t✱ ❣✐↔ sû a ≥ b ≥ c✳ ❈→❝ t❤❛♠ sè ta , tb , tc ✤÷đ❝ ❝❤♦ ❜ð✐ ✭✸✳✶✶✮✱ ✭✸✳✶✷✮✱ ✭✸✳✶✸✮ ✈ỵ✐ κ = 4✳ ❚❛ ❝â √ κ2 − = 13 ✈➔ ✤á✐ ❤ä✐ cos θ ✈➔ v = 13 − cos2 θ ❧➔ ❝→❝ sè ❤ú✉ t✛✳ ❱➻ 13 = 32 + 22 ♥➯♥ t❛ ✈✐➳t v = 13 − cos2 θ ❧➔ v − = − cos2 θ ⇐⇒ (3 − cos θ)(3 + cos θ) = (v − 2)(v + 2)✳ ❱➻ t➜t ❝↔ ❝→❝ ♥❤➙♥ tû ❧➔ sè ❤ú✉ t✛ ♥➯♥ t❛ ❝â t❤➸ ❣✐↔ sû − cos θ = w(v + 2) ✈ỵ✐ w ❧➔ sè ❤ú✉ t✛ ♥➔♦ ✤â✳ ❚❛ s✉② r❛ w(3 + cos θ) = v − 2✳ ❚❛ ♥❤➟♥ ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❍❡r♦♥ ✈ỵ✐ − cos θ = w(v + 2) w(3 + cos θ) = v − = w(v + 2) 3w = v − 2 ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ v ❦❤ỉ♥❣ t❤➸ ❤ú✉ t✛ ❞♦ v = 13✳ ❇ð✐ ✈➟② t❛ ❝â t❤➸ ❝♦✐ b > c✱ ❦❤✐ ✤â θ ❧➔ ❣â❝ ♥❤å♥ ✈➔ < cos θ < 1✳ ●✐↔✐ ❤➺ ✤â ✈ỵ✐ ➞♥ cos v t ữủ ú ỵ r➡♥❣ ♥➳✉ tb = tc t❤➻ cos θ = 0✳ ❍➺ trð t❤➔♥❤ − 4w − 3w2 + 6w − 2w2 , v= ✭✸✳✶✹✮ cos θ = + w2 + w2 √ √ √ 3−1 3−2 ✈➔ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ♥➔② ❧✉æ♥ ❧✉æ♥ ♥❤å♥ ❱➻ ❝→❝ ❣✐→ trà ✺✺ > ta > ✈➔ tb > > tc > ✈➔ ✈➻ tb + tc = ♥➯♥ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ < ta + tb + tc < ✳ r ♥➯♥ ξ = s − a > ✈➔ ✤÷í♥❣ ▲↕✐ tø ✭✸✳✻✮ t❛ ❝â✿ ta + tb + tc − = ξ ✈ỵ✐ b ❧➔ ❝↕♥❤ ợ t ú ỵ r tb > trỏ s ổ t ú ợ ữớ trỏ t ú ứ t = ữợ ✹ t❤✉ ✤÷đ❝✿ a = 2m2 n b = (m + n) m2 − 2mn + 2n2 c = −(m n) m2 + 2mn + 2n2 ú ỵ r➡♥❣ ✈➻ < cos θ < ♥➯♥ ❦❤æ♥❣ t❤➸ ❝â t❛♠ ❣✐→❝ ❍❡r♦♥ ❝➙♥ t❤✉ë❝ ❧ỵ♣ ♥➔②✳ ❚ø ✭✸✳✷✸✮ t❛ ❝â ❤➺ t❤ù❝ ❣✐ú❛ ❝→❝ ❝↕♥❤ a3 = s(b − c) ❱ỵ✐ ❝→❝ sè ♥❣✉②➯♥ ♥❣✉②➯♥ t❤õ② ❧ỵ♣ S= ❤♦➦❝ a(s − a) ≤ m < n ≤ t❛ ♥❤➟♥ ✤÷đ❝ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ = a, b, c ữợ ợ t ú ữ t tr ữợ m n a b c s S r R ✶ ✷ ✹ ✶✺ ✶✸ ✶✻ ✷✹ ✶ ✸ ✸ ✷✻ ✷✺ ✷✼ ✸✻ ✷ ✸ ✶✷ ✷✺ ✶✼ ✷✼ ✾✵ ✶ ✹ ✽ ✶✷✺ ✶✷✸ ✶✷✽ ✹✽✵ ✸ ✹ ✼✷ ✶✶✾ ✻✺ ✶✷✽ ✷✵✶✻ ✶ ✺ ✺ ✶✷✸ ✶✷✷ ✶✷✺ ✸✵✵ ✷ ✺ ✷✵ ✶✶✾ ✶✶✶ ✶✷✺ ✶✵✺✵ ✸ ✺ ✹✺ ✶✶✻ ✽✾ ✶✷✺ ✶✽✵✵ ✹ ✺ ✽✵ ✶✶✼ ✺✸ ✶✷✺ ✶✽✵✵ ✶ ✻ ✶✷ ✹✷✼ ✹✷✺ ✹✸✷ ✷✺✷✵ ✺ ✻ ✸✵✵ ✹✵✼ ✶✺✼ ✹✸✷ ✶✾✽✵✵ 10 15 63 12 42 72 72 35 275 65 325 24 85 1025 16 1105 16 2501 40 629 10 2581 40 689 10 5185 24 5809 24 ❳➨t ✈à tr➼ t➙♠ ❙♦❞❞② ♥❣♦↕✐ ❝õ❛ ❧ỵ♣ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ♥➔② ❝ò♥❣ ❜→♥ ❦➼♥❤ ❝õ❛ ♥â ♥❤÷ tr➯♥ ❤➻♥❤ ✸✳✶✵✳ ✺✻ ❍➻♥❤ ✸✳✶✵✿ ❚❛♠ ❣✐→❝ ❍❡r♦♥ ợ r ợ ố ỵ ✶✳✹✱ ❬✺❪✮ =2 ✳ ❚➙♠ ❝õ❛ ✤÷í♥❣ trá♥ ❙♦❞❞② ❝õ❛ t❛♠ ❣✐→❝ = ❧➔ ✤è✐ ①ù♥❣ ❝õ❛ ♠ët tr♦♥❣ ❝→❝ ✤➾♥❤ ❝õ❛ t❛♠ ❣✐→❝ q✉❛ ❝↕♥❤ ❈❤ù♥❣ ♠✐♥❤✳ = tb + tc = 2✳ ❚ø r ✭✸✳✼✮ t❛ ❝â ta + tb + tc − = ♥➯♥ ❜→♥ ❦➼♥❤ ❝õ❛ ✤÷í♥❣ trá♥ ❙♦❞❞② ♥❣♦↕✐ ξ ❜➡♥❣ ξ = s − a > ♥❤÷ ❧➔ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝ ❦➳t ❤đ♣ ✈ỵ✐ ta ♠➔ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✳ ợ > ữớ trỏ ổ t ú ợ ữớ trỏ t ú õ BF = s − b + ξ = s − b + s − a = c ✈➔ t❛♠ ❣✐→❝ ABF ❧➔ t❛♠ ❣✐→❝ ❝➙♥✳ ❚❛ ❝ơ♥❣ ♥❤➟♥ ✤÷đ❝ F ❧➔ ✤è✐ ①ù♥❣ ❝õ❛ A q✉❛ BC ✳ ❑❤æ♥❣ ♠➜t t➼♥❤ ❝❤➜t tê♥❣ q✉→t t❛ ❧➜② ❈❤÷ì♥❣ ✸ ❝❤õ ②➳✉ ❦❤↔♦ s→t ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ t❤✉ë❝ ❧ỵ♣ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ t❤✉ë❝ ❧ỵ♣ = 1, 2✳ κ = 2, ✈➔ ❇➡♥❣ ❝→❝ s✉② ❧✉➟♥ sè ❤å❝ ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ✤÷đ❝ ❦➳t q✉↔ tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ t tữỡ ự ợ số κ ❦❤→❝ ✈➔ ✳ ❍♦➔♥ t♦➔♥ t÷ì♥❣ tü ❝â t❤➸ ❦❤↔♦ s→t ✤è✐ ✈ỵ✐ ❝→❝ ❣✐→ trà ✺✼ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➲ ❝➟♣ ✤➳♥ ❝→❝ ❦➳t q✉↔ s❛✉✿ ✶✳ ❚r➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦➳t q✉↔ ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ tr➯♥ ♠➦t ♣❤➥♥❣ ✈➔ ❧÷đ❝ ✈➲ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝ ❞ị♥❣ ❝❤♦ ❝→❝ ❝❤÷ì♥❣ ✷✱ ✸✳ ✷✳ ❚rå♥❣ t➙♠ ợ t ữớ trỏ ❙♦❞❞②✿ ✣à♥❤ ♥❣❤➽❛✱ ❝→❝❤ ❞ü♥❣✱ ❜→♥ ❦➼♥❤ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ✈➔ ❜✐➸✉ ❞✐➵♥ ❝❤ó♥❣ tr♦♥❣ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝✳ ✸✳ Ù♥❣ ❞ư♥❣ ✈➔♦ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ✈ỵ✐ ✤➦❝ sè κ = 2, ✈➔ ✤➦❝ sè = 1, 2✳ ú tổ t õ ữợ ự t t❤❡♦✿ ✲ ❚➻♠ t❤➯♠ ✈➲ ❝→❝ ❜➔✐ t♦→♥ ù♥❣ ❞ö♥❣ t q ỵ tö❝ ❦❤❛✐ t❤→❝ ❝→❝ ②➳✉ tè ❤➻♥❤ ❤å❝ ❝õ❛ t❛♠ ❣✐→❝ ❊✉❧❡r✲●❡r❣♦♥♥❡✲ ❙♦❞❞② ✈➔ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ✤➦❝ sè κ ✈➔ ❦❤→❝✳ ▼➦❝ ❞ị ✤➣ r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳✱ ❦❤✐➳♠ ❦❤✉②➳t✳ rt sỹ õ ỵ s t ổ ỗ ❧➔♠ ❝❤♦ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❤♦➔♥ ❝❤➾♥❤ ✈➔ ❝â ➼❝❤ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ✺✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❆♥❤ ❬✶❪ ❉❡r❣✐❛❞❡s✱ ◆✳ ✭✷✵✵✼✮✱ ✏❚❤❡ s♦❞❞② ❝✐r❝❧❡s✑✱ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✱ ✼ ✱ ♣♣✳ ✶✾✶✕✶✾✼✳ ❆♥ ❆❜str❛❝t ♦❢ ❛ ❚❤❡s✐s ❙✉❜♠✐tt❡❞ ✐♥ P❛rt✐❛❧ ❋✉❧✲ ❢✐❧❧♠❡♥t ♦❢ t❤❡ ❘❡q✉✐r❡♠❡♥t ❢♦r t❤❡ ❉❡❣r❡❡ ▼❛st❡r ♦❢ ❆rts✱ ❯♥✐✈❡rs✐t② ❬✷❪ ●✐s❝❤✱ ❉✳❏✳ ✭✷✵✵✻✮✱ ♦❢ ◆♦rt❤❡r♥ ■♦✇❛✱ ✶✲✾✳ ❬✸❪ ❏❛❝❦s♦♥✱ ❋✳▼✳ ✭✷✵✶✸✮✱ ✏❙♦❞❞②✐❛♥ tr✐❛♥❣❧❡s✑✱ ✶✸ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✱ ✱ ♣♣✳ ✶✕✻✳ ❬✹❪ ❏❛❝❦s♦♥✱ ❋✳▼✳✱ ❚❛❦❤❛❡✈✱ ❙✳ ✭✷✵✶✺✮✱ ✏❍❡r♦♥✐❛♥ tr✐❛♥❣❧❡s ♦❢ ❝❧❛ss ❈♦♥❣r✉❡♥t ✐♥❝✐r❝❧❡s ❝❡✈✐❛♥ ♣❡rs♣❡❝t✐✈❡✑✱ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✱ K✿ ✶✺ ✱ ♣♣✳ ✺✕✶✷✳ ❬✺❪ ❏❛❝❦s♦♥✱ ❋✳▼✳✱ ❚❛❦❤❛❡✈✱ ❙✳ ✭✷✵✶✻✮✱ ✏❍❡r♦♥✐❛♥ tr✐❛♥❣❧❡s ♦❢ ❝❧❛ss ❏✿ ❈♦♥✲ ❣r✉❡♥t ✐♥❝✐r❝❧❡s ❝❡✈✐❛♥ ♣❡rs♣❡❝t✐✈❡✑✱ ♣✉t❡r ❉✐s❝♦✈❡r❡❞ ▼❛t❤❡♠❛t✐❝s✱ ✶ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❈♦♠✲ ✭✸✮✱ ♣♣✳ ✶✕✽✳ ❬✻❪ ❑✐♠❜❡r❧✐♥❣✱ ❈✳ ✭✷✵✵✵✮✱ ✏❊♥❝②❝❧♦♣❡❞✐❛ ♦❢ tr✐❛♥❣❧❡ ❝❡♥t❡rs✑✱ ❆✈❛✐❧❛❜❧❡ ❛t ❤tt♣✿✴✴❢❛❝✉❧t②✳❡✈❛♥s✈✐❧❧❡✳❡❞✉✴❝❦✻✴❡♥❝②❝❧♦♣❡❞✐❛✴❊❚❈✳❤t♠❧✳ ❬✼❪ ❨✐✉✱ P✳ ✭✷✵✵✶✮✱ ■♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ●❡♦♠❡tr② ♦❢ t❤❡ ❚r✐❛♥❣❧❡✱ ❆t❧❛t✐❝ ❯♥✐✈❡rs✐t② ▲❡❝t✉r❡ ◆♦t❡s✳ ❋❧♦r✐❞❛ ...ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC - NGƠ TRỌNG THÀNH ĐƯỜNG TRỊN SODDY VÀ CÁC VẤN ĐỀ LIÊN QUAN Chuyên ngành: Phương pháp Toán sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN