(LUẬN văn THẠC sĩ) một số bài toán về đường tròn tiếp xúc

ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - VŨ THỊ NĂM MỘT SỐ BÀI TOÁN VỀ ĐƯỜNG TRỊN TIẾP XÚC LUẬN VĂN THẠC SĨ TỐN HỌC THÁI NGUYÊN - 2018 download by : skknchat@gmail.com ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - VŨ THỊ NĂM MỘT SỐ BÀI TOÁN VỀ ĐƯỜNG TRỊN TIẾP XÚC Chun ngành: Phương pháp Tốn sơ cấp Mã số: 8460113 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC PGS.TS Nguyễn Việt Hải THÁI NGUYÊN - 2018 download by : skknchat@gmail.com ✐ ❉❛♥❤ ♠ö❝ ❤➻♥❤ ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✶✳✺ ✶✳✻ ✶✳✼ ✶✳✽ ✶✳✾ ✶✳✶✵ ✶✳✶✶ ✶✳✶✷ ❇➔✐ t♦→♥ ❋❡✉❡r❜❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❉ü♥❣ ✤÷í♥❣ trá♥ ❚❤❡❜❛✉❧t ✳ ✳ ✳ ✳ ✳ ỵ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇ê s✉♥❣ t➼♥❤ ❝❤➜t ❝õ❛ t➙♠ I ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❛✮ P Q ✤✐ q✉❛ I ❀ ❜✮ P Q ✤✐ q✉❛ IC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❝✮ P Q ✤✐ q✉❛ IA ❀ ❞✮ P Q ✤✐ q✉❛ IB ✳ ✳ ✳ ✳ trữớ ủ ỵ t ✳ ✳ ✳ ✳ ✳ ❚ø ❜➔✐ t♦→♥ ❚❤❡❜❛✉❧t ✤➳♥ ỵ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ r ố ợ ữớ trỏ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ✷✳✺ ✷✳✻ ✷✳✼ ✷✳✽ ✷✳✾ ✷✳✶✵ ❇➔✐ t♦→♥ ▼❛❧❢❛tt✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚ê♥❣ ❞✐➺♥ t➼❝❤ ❝→❝ ❤➻♥❤ trá♥ ▼❛❧❢❛tt✐ ❦❤æ♥❣ ◆❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐ ❣è❝ ✳ ✳ ✳ ✳ ✳ ▲í✐ ❣✐↔✐ ✤↕✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▲í✐ ❣✐↔✐ ✤↕✐ sè✲❤➻♥❤ ❤å❝ ❝õ❛ ❙❝❤❡❧❧❜❛❝❤ ✳ ✳ ✳ ❑❤✐ R = 21 ❀ a = sin α, b = sin β, c = sin γ ✳ P❤➨♣ ❞ü♥❣ ♣❤ö ✶ ✈➔ ♣❤➨♣ ❞ü♥❣ ♣❤ö ✷ ✳ ✳ ✳ P❤➨♣ ❞ü♥❣ ❜➡♥❣ ♣❤➛♥ ♠➲♠ ●❡♦●❡❜r❛✱ ♥➠♠ ❇➔✐ t♦→♥ ❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✻ ✶✼ ✶✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣❤↔✐ ❧➔ ❧ỵ♥ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✶ ✷✹ ✷✺ ✷✻ ✷✽ ✷✾ ✸✵ ✸✷ ✸✸ ✸✳✶ ❛r❜❡❧♦s ✲ ❤➻♥❤ ✧❝♦♥ ❞❛♦ t❤ñ ✤è♥❣ ❣✐➛②✧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷ ✣÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❛r❜❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t❤ù ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ download by : skknchat@gmail.com ✐✐ ✸✳✹ ✸✳✺ ✸✳✻ ✸✳✼ ✸✳✽ ✸✳✾ ✸✳✶✵ ✸✳✶✶ ✸✳✶✷ ❇❛ ❝→❝❤ ❞ü♥❣ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❛r❜❡❧♦s ❆❇❈ ✳ ❈→❝❤ ❞ü♥❣ t❤ù t÷ ❝õ❛ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ✳ ✳ ✳ ❈➦♣ ✤÷í♥❣ trá♥ rs tự t tự ỵ tự ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ❜❛ ✈➔ t❤ù t÷ ✳ ❈➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ♥➠♠ ✈➔ t❤ù s→✉ ❈➦♣ ✤÷í♥❣ trá♥ t❤ù ❜↔②✱ t❤ù t→♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈➦♣ ✤÷í♥❣ trá♥ t❤ù ❝❤➼♥ ✈➔ ❝➦♣ t❤ù ♠÷í✐ ✳ ✳ ✳ ❈➦♣ t❤ù ♠÷í✐ ♠ët ✈➔ ❝➦♣ t❤ù ♠÷í✐ ❤❛✐ ✳ ✳ ✳ ✳ ✳ download by : skknchat@gmail.com ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✺ ✹✻ ✹✼ ✹✽ ✹✾ ✺✵ ✺✶ ✺✷ ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ▼ð ✤➛✉ ✶ ❚ø ❜➔✐ t♦→♥ ❚❤❡❜❛✉❧t ✤➳♥ ❜➔✐ t♦→♥ ❋❡✉❡r❜❛❝❤ ✶✳✶ ●✐ỵ✐ t❤✐➺✉ ✈➲ ❤❛✐ ❜➔✐ t♦→♥✿ ❜➔✐ t♦→♥ ❚❤❡❜❛✉❧t ✈➔ ❜➔✐ t♦→♥ ❋❡✉❡r❜❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✶ ❇➔✐ t♦→♥ ❋❡✉❡r❜❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷ ❇➔✐ t♦→♥ ❚❤❡❜❛✉❧t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❇➔✐ t♦→♥ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ⑩♣ ❞ư♥❣ ❜➔✐ t♦→♥ ❝ì ❜↔♥ ự ỵ t ứ ỵ t ỵ r ⑩♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❇➔✐ t♦→♥ ▼❛❧❢❛tt✐ ✷✳✶ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ▲í✐ ❣✐↔✐ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐ ❣è❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ▲í✐ ❣✐↔✐ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✶ ❈→❝❤ ❞ü♥❣ ✤↕✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✷ ❈→❝❤ ❞ü♥❣ ✤↕✐ sè✲❤➻♥❤ ❤å❝ ❝õ❛ ❙❝❤❡❧❧❜❛❝❤ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ▼ët sè ❜➔✐ t♦→♥ ❦✐➸✉ ▼❛❧❢❛tt✐ ❣è❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶ ❍❛✐ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐ ✤è✐ ♥❣➝✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✷ ❇➔✐ t♦→♥ ▼❛❧❢❛tt✐ ❝❤♦ t❛♠ ❣✐→❝ ✤➲✉ ✈➔ ❤➻♥❤ ✈✉æ♥❣ ✷✳✹✳✸ ❇➔✐ t♦→♥ ▼❛❧❢❛tt✐ ❝❤♦ ✤÷í♥❣ trá♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✣÷í♥❣ trá♥ t✐➳♣ ①ó❝ tr♦♥❣ ❤➻♥❤ ❤å❝ ❛r❜❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✈✐ ✶ ✹ ✹ ✹ ✼ ✾ ✶✷ ✶✹ ✶✼ ✷✵ ✷✵ ✷✷ ✷✹ ✷✹ ✷✻ ✸✶ ✸✶ ✸✹ ✸✼ ✸✽ ✸✳✶ ▼ët sè ❜➔✐ t♦→♥ ✤ì♥ ❣✐↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ download by : skknchat@gmail.com ✐✈ ✸✳✷ ✣÷í♥❣ trá♥ ♥ë✐ t✐➳♣ tr♦♥❣ ❛r❜❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷✳✶ ❚➼♥❤ ❝❤➜t ❝õ❛ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ tr♦♥❣ ❆r❜❡❧♦s ✳ ✳ ✸✳✷✳✷ ❈→❝❤ ❞ü♥❣ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❛r❜❡❧♦s ❆❇❈ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ❈→❝ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s tr♦♥❣ ❛r❜❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸✳✶ ❈➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ♥❤➜t ✈➔ t❤ù ❤❛✐ ✳ ✳ ✸✳✸✳✷ ❈➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ❜❛ ✈➔ t❤ù t÷ ✳ ✳ ✳ ✳ ✸✳✸✳✸ ❈→❝ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ♥➠♠ ✈➔ t❤ù s→✉ ✸✳✸✳✹ ❈→❝ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ❜↔② ✈➔ t❤ù t→♠ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ download by : skknchat@gmail.com ✹✵ ✹✵ ✹✹ ỵ stt ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ỵ O9 a (ABC) (O) (OA , rA ) [ABC] (P Q) O(r) ab t= a+b (Wk ), (Wk ) (CXY ) ỵ ✤÷í♥❣ trá♥ ❊✉❧❡r ✺ ❇→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ tr♦♥❣ A ✻ ✣÷í♥❣ trá♥ ✤✐ q✉❛ ✸ ✤✐➸♠ A, B, C ✶✶ ✣÷í♥❣ trá♥ t➙♠ O ✶✼ ❚➙♠✱ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ▼❛❧❢❛tt✐ tr♦♥❣ A ✷✹ ❍➻♥❤ ❛r❜❡❧♦s ✷✷ ◆û❛ ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ P Q ✸✽ ✣÷í♥❣ trá♥ t➙♠ O✱ ❜→♥ ❦➼♥❤ r ✸✽ ❇→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s ✹✺ ❈➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s ✣÷í♥❣ trá♥ ❇❛♥❦♦❢❢ tr♦♥❣ ❤➻♥❤ ❛r❜❡❧♦s ✹✻ ✹✼ download by : skknchat@gmail.com ✈✐ ▲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♥❣ ❦✐➳♥ t❤ù❝ 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 ▲✉➟♥ ✈➠♥ ❱ơ ❚❤à ◆➠♠ download by : skknchat@gmail.com ✶ ▼ð ✤➛✉ ✶✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❈→❝ ❜➔✐ t♦→♥ ✈➲ ✤÷í♥❣ trá♥ ❧✉ỉ♥ ❧➔ ♥❤ú♥❣ ❜➔✐ t♦→♥ ✤÷đ❝ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠✳ ◆❤✐➲✉ ❜➔✐ t♦→♥ ✈➲ sü t✐➳♣ ú ữớ trỏ ợ t t✉ê✐ ❝õ❛ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ♥❤÷ ❜➔✐ t♦→♥ ❚❤❡❜❛✉❧t✱ ❜➔✐ t♦→♥ ❋❡✉❡r❜❛❝❤✱ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐✱ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ✤÷í♥❣ trá♥ tr♦♥❣ ❤➻♥❤ ❤å❝ ❛r❜❡❧♦s ✭✧❤➻♥❤ ❝♦♥ ❞❛♦ ❝õ❛ t❤ñ ✤â♥❣ ❣✐➛②✧✮ ✳✳✳❙ü ❞➝♥ ❞➢t tø ❜➔✐ t♦→♥ ♥➔② s❛♥❣ ❜➔✐ t♦→♥ ❦❤→❝ ❝ị♥❣ ❝→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ❝❤ó♥❣ ✤➣ ♠❛♥❣ ❧↕✐ ♥❤✐➲✉ ❦➳t q✉↔ t✉②➺t ✈í✐ ❝õ❛ ❤➻♥❤ ❤å❝ ❊✉❝❧✐❞❡✳ ✣➸ ❤✐➸✉ ❜✐➳t t❤➯♠ ✈➲ ❝→❝ ❝→❝ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝✱ ❦❤❛✐ t❤→❝ ❝→❝ t➼♥❤ ❝❤➜t✱ ❝→❝❤ ①→❝ ✤à♥❤ ❝❤ó♥❣✱ →♣ ❞ư♥❣ ✤÷đ❝ ✈➔♦ ❝→❝ ❜➔✐ t♦→♥ ❦❤→❝✱ tæ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐ ✧▼ët sè ❜➔✐ t♦→♥ ✈➲ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝✧✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔✿ ✲❚➻♠ ❤✐➸✉ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝✿ ❜➔✐ t♦→♥ ❚❤❡❜❛✉❧t✱ ❜➔✐ t♦→♥ ❋❡✉❡r❜❛❝❤✱ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐✱ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝ tr♦♥❣ ❤➻♥❤ ❤å❝ ❛r❜❡❧♦s ✳ ✲ ❚r➻♥❤ ❜➔② ♠é✐ ❜➔✐ t ợ ỳ ữủ t t tr tü✿ ①✉➜t sù ❝õ❛ ❜➔✐ t♦→♥✱ ❝→❝❤ ❣✐↔✐ q✉②➳t ♠ỵ✐ ❝õ❛ ❜➔✐ t♦→♥ ✈➔ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✳ ✲ ❈→❝ ❦➳t ❧✉➟♥ ❦❤♦❛ ❤å❝ rót r❛ tø ❝→❝ ❜➔✐ t♦→♥ ✈➔ →♣ ❞ö♥❣ ✤➸ ❣✐↔✐ t♦→♥ ❤å❝ s✐♥❤ ❣✐ä✐ tổ ỗ ữù ỹ ❝❤✉②➯♥ ✤➲ ❦❤â ð tr÷í♥❣ ❚❍❈❙ ✈➔ ❚❍P❚ ❣â♣ ♣❤➛♥ ✤➔♦ t↕♦ ❤å❝ s✐♥❤ ❤å❝ ❣✐ä✐ ♠æ♥ ❍➻♥❤ ❤å❝✳ download by : skknchat@gmail.com ✷ ✷✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐✱ ♥❤ú♥❣ ✈➜♥ ✤➲ ❝➛♥ ❣✐↔✐ q✉②➳t ❚r➻♥❤ ❜➔② ♠ët ❝→❝❤ ❤➺ t❤è♥❣ ❝→❝ ❜➔✐ t♦→♥ ♥â✐ tr➯♥✱ →♣ ❞ư♥❣ ✤÷đ❝ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝ ✈➔♦ ❝→❝ ❜➔✐ t♦→♥ ❦❤→❝✳ ◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ❝❤✐❛ ❧➔♠ ✸ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✳ ❚ø ❜➔✐ t♦→♥ ❚❤❡❜❛✉❧t ✤➳♥ ❜➔✐ t♦→♥ ❋❡✉❡r❜❛❝❤ ❳➨t ❤❛✐ ❜➔✐ t♦→♥ ✿ ❜➔✐ t♦→♥ ❚❤❡❜❛✉❧t✱ ❜➔✐ t♦→♥ ❋❡✉❡r❜❛❝❤ ✈➔ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝❤ó♥❣✳ ❇➔✐ t♦→♥ ❋❡✉❡r❜❛❝❤ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❜➔✐ t♦→♥ ✤➭♣ ✤➩ ♥❤➜t ❝õ❛ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ❊✉❝❧✐❞❡ tr↔✐ q✉❛ ♥❤✐➲✉ ♥➠♠ t❤→♥❣ ợ ự ữỡ ỗ ●✐ỵ✐ t❤✐➺✉ ✈➲ ❤❛✐ ❜➔✐ t♦→♥✿ ❜➔✐ t♦→♥ ❚❤❡❜❛✉❧t ✈➔ ❜➔✐ t♦→♥ ❋❡✉❡r❜❛❝❤ ✶✳✷✳ ❇➔✐ t♦→♥ ❝ì ❜↔♥ ✶✳✸✳ ⑩♣ ữỡ t tt ợ t t ▼❛❧❢❛tt✐ ✈➔ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐ ❣è❝✳ ❚r➻♥❤ ❜➔② ❝❤✐ t✐➳t ❧í✐ ❣✐↔✐ ❜➔✐ t♦→♥ t♦→♥ ▼❛❧❢❛tt✐ ❝❤♦ t❛♠ ❣✐→❝ ❜➜t ❦ý✱ ❣✐↔✐ t❤➼❝❤ ✤➛② ✤õ t↕✐ s❛♦ ❝→❝ ✤÷í♥❣ trá♥ ▼❛❧❢❛tt✐ ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐❣è❝ ✈➔ ✤➙✉ ❧➔ ♥❣❤✐➺♠ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✤â✳ ❈❤÷ì♥❣ ♥➔② ❜❛♦ ỗ s ợ t t tt ✷✳✷✳ ▲í✐ ❣✐↔✐ ❝õ❛ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐ ❣è❝ ✷✳✸✳ ▲í✐ ❣✐↔✐ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐ ✷✳✹✳ ▼ët sè ❜➔✐ t♦→♥ ❦✐➸✉ ▼❛❧❢❛tt✐ ❣è❝✳ ❈❤÷ì♥❣ ✸✳ ✣÷í♥❣ trá♥ t✐➳♣ ①ó❝ tr♦♥❣ ❤➻♥❤ ❤å❝ ❛r❜❡❧♦s ❍➻♥❤ ❤å❝ ❛r❜❡❧♦s ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♥û❛ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝✱ ❝❤ú ✧❛r❜❡❧♦s✧ ✤÷đ❝ ❣❤➨♣ tø ✼ ❝❤ú ❝→✐ α, , β, η, λ, θ, ς t❤➔♥❤ ✭α rs ỷ ữớ trỏ ợ ❝→❝ ✤÷í♥❣ ❦➼♥❤ tr➯♥ ♠ët ✤÷í♥❣ t❤➥♥❣✳ ❚❤❡♦ q✉❛♥ ✤✐➸♠ trü❝ q✉❛♥✱ ♥❣÷í✐ t❛ ❣å✐ ❛r❜❡❧♦s ❧➔ ✧❤➻♥❤ ❝♦♥ ❞❛♦ ❝õ❛ t❤ñ ✤â♥❣ ❣✐➛②✧✳ download by : skknchat@gmail.com ✹✵ ❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ ❤➻♥❤ ✸✳✶❜✮✳ ❚❛ ❝â ❝→❝ ❣â❝ ✈✉æ♥❣ AU C = ADB = CV B = 1v ♥➯♥ ❜❛ ❣â❝ U , D, V ❝õ❛ tù ❣✐→❝ CU DV ✤➲✉ ❜➡♥❣ 1v ✳ ❙✉② r❛ tù ❣✐→❝ CU DV ❧➔ ❤➻♥❤ ❝❤ú ♥❤➟t✳ ❇➔✐ t♦→♥ ✸✳✸✳ ●✐↔ t❤✐➳t ♥❤÷ ❜➔✐ t♦→♥ tr➯♥✱ ❦❤✐ ✤â ✤÷í♥❣ t❤➥♥❣ U V ❧➔ t✐➳♣ t✉②➳♥ ❝õ❛ ❤❛✐ ♥û❛ ✤÷í♥❣ trá♥ (AC) ✈➔ (CB)✳ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ O ❧➔ tr✉♥❣ ✤✐➸♠ CD✳ ❚❛ ❝â✧ AU C = 900 , U AC = 900 − U CA ▲↕✐ ❝â U CD = 1v − U CA ♥➯♥ AU C = U CD✱ ✈➟② ∆U O1 C ∼ ∆U CD ✈➔ s✉② r❛ U O1 C = U OD✳ ❱➻ C, O, D t❤➥♥❣ ❤➔♥❣ ♥➯♥ CU O + U OD = 1800 ❤❛② U O1 C + U OC = 1800 ♠➔ O1 CO = 900 ♥➯♥ O1 U C = 900 ❤❛② O1 U U O ữ t U V t ú ợ ỷ ✤÷í♥❣ trá♥ (AC) t↕✐ U ✳ ❍♦➔♥ t♦➔♥ t÷ì♥❣ tü✱ U V t ú ợ ỷ ữớ trỏ (CB) t V ✳ ✸✳✷ ✣÷í♥❣ trá♥ ♥ë✐ t✐➳♣ tr♦♥❣ ❛r❜❡❧♦s ❚❛ ①➨t ✤à♥❤ ♥❣❤➽❛ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣✱ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè t➼♥❤ ❝❤➜t ✤➸ tø ✤â ❝â ❝→❝ ❝→❝❤ ❞ü♥❣ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ tr♦♥❣ ♠ët ❆r❜❡❧♦s✳ ✸✳✷✳✶ ❚➼♥❤ ❝❤➜t ❝õ❛ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ tr♦♥❣ ❆r❜❡❧♦s ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳ ❈❤♦ ❆r❜❡❧♦s ❆❇❈✳ ✣÷í♥❣ trá♥ t✐➳♣ ①ó❝ ♥❣♦➔✐ ✈ỵ✐ ✭❇❈✮✱ ✭❈❆✮ t↕✐ ❳✱ ❨ ✈➔ t✐➳♣ ①ó❝ tr ợ t ữủ ữớ trỏ ♥ë✐ t✐➳♣ ❝õ❛ ❛r❜❡❧♦s ❆❇❈✳ ❇❛ ✤✐➸♠ ❳✱ ❨✱ ❩ ❧➔ ❝→❝ t✐➳♣ ✤✐➸♠ ▼➺♥❤ ✤➲ ✸✳✶✳ ✣÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❛r❜❡❧♦s ❆❇❈ ❝â ❜→♥ ❦➼♥❤ ρ= ab(a + b) a2 + ab + b2 ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ ω ❧➔ t➙♠ ✈➔ ρ ❧➔ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐ t t OO2 = ỵ ổs →♣ ❞ö♥❣ ✈➔♦ ∆O1 ωO, ∆O2 ωO✿ O1 ω = Oω + OO12 + 2Oω.OO1 cosθ O2 ω = Oω + OO22 + 2Oω.OO2 cosθ download by : skknchat@gmail.com ✹✶ ❍➻♥❤ ✸✳✷✿ ✣÷í♥❣ trá♥ ♥ë✐ t✐➳♣ rs tữỡ ữỡ ợ (a + )2 = (a + b − ρ)2 + b2 + 2b(a + b − ρ)cosθ (b + ρ)2 = (a + b − ρ)2 + a2 + 2a(a + b − ρ)cosθ ❑❤û cosθ t❛ ✤÷đ❝ a(a + ρ)2 + b(b + ρ)2 + b2 = (a + b)(a + b − ρ)2 + ab2 + ba2 tr ữợ t ữủ ữỡ tr t ố ợ a3 + b3 + 2(a2 + b2 )ρ = (a + b)2 + ab(a + b) − 2(a + b)2 ρ ❤❛② ρ = ab(a + b) ✳ a2 + ab + b2 ❚r♦♥❣ ❬✺❪✱ P✳❲♦♦ ✤➣ ✤÷❛ r❛ ✸ ❝→❝❤ ❞ü♥❣ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❝õ❛ ❤➻♥❤ ❛r❜❡❧♦s r➜t ✤ì♥ ❣✐↔♥✱ t➜t ❝↔ ✤➲✉ s✉② tø ✈✐➺❝ ♣❤→t ❤✐➺♥ r❛ ✸ ✤✐➸♠ t❤✉ë❝ ✤÷í♥❣ trá♥✳ ◆❣❛② s❛✉ ✤➙② t❛ s➩ tr➻♥❤ ❜➔② ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣✳ ❚ø ✤â s✉② r❛ ❝→❝❤ ❞ü♥❣ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ rs ỵ tự t ●✐↔ sû Q1, Q2 ❧➔ tr✉♥❣ ✤✐➸♠ ✷ ♥û❛ ✤÷í♥❣ trỏ ợ ỵ ữ tr ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❆r❜❡❧♦s ❆❇❈ t❤➻ ✐✱ A, C, X, Z ♥➡♠ tr➯♥ ✤÷í♥❣ trá♥✱ t➙♠ ❧➔ Q1 ✐✐✱ B, C, Y, Z ♥➡♠ tr➯♥ ✤÷í♥❣ trá♥✱ t➙♠ ❧➔ Q2 download by : skknchat@gmail.com ỵ ❇❛♥❦♦❢❢ t❤ù ♥❤➜t ❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ ❤➻♥❤ ✸✳✸✳ ●å✐ D ỷ ữớ trỏ ữớ AB ợ ữớ t CtAB ữ ỵ r t õ AB.AC = AD2 t ợ ữớ trỏ ♥❣❤à❝❤ ✤↔♦ ❧➔ (A, AD)✳ ❍❛✐ ✤✐➸♠ B, C ❧➔ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ♥❤❛✉✱ ❝á♥ AB ❧➔ ✤÷í♥❣ t❤➥♥❣ ❦➨♣✳ ❷♥❤ ❝õ❛ ❝→❝ ♥û❛ ✤÷í♥❣ trá♥ (AB), (AC) t÷ì♥❣ ù♥❣ ữớ t , ổ õ ợ AB ❧➛♥ ❧÷đt ✤✐ q✉❛ C ✈➔ B ✳ ◆û❛ ✤÷í♥❣ trá♥ (AB) trü❝ ❣✐❛♦ ✈ỵ✐ AB ✭❦➨♣✮ ♥➯♥ ❝ơ♥❣ ❧➔ ♥û❛ ✤÷í♥❣ trá♥ ❦➨♣✳ ✣÷í♥❣ trá♥ ♥ë✐ t✐➳♣ (XY Z) t ữớ trỏ t ú ợ ỷ ữớ trá♥ (BC) ✈➔ ❝→❝ ✤÷í♥❣ t❤➥♥❣ , t÷ì♥❣ ù♥❣ t↕✐ ✤✐➸♠ P, Y , Z ✳ ❱➻ ♥û❛ ✤÷í♥❣ trá♥ (BC) ❦➨♣ ♥➯♥ ❝→❝ ✤✐➸♠ A, X, P t❤➥♥❣ ❤➔♥❣❀ ❝→❝ ✤✐➸♠ Y , Z ✱ ❝➛♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✤➸ BP Z ✈➔ CP Y ❧➔ ❝→❝ ✤÷í♥❣ t❤➥♥❣ t↕♦ ✈ỵ✐ AB ❝→❝ ❣â❝ 450 ✳ ❚❛ ❧↕✐ ❝â ✤÷í♥❣ t❤➥♥❣ BP Z ✤✐ q✉❛ tr✉♥❣ ✤✐➸♠ L ❝õ❛ ✤÷í♥❣ trá♥ (AB)✳ ❷♥❤ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ♥â ❧➔ ✤÷í♥❣ ✤✐ trá♥ ✤✐ q✉❛ A, C, X, Z ✳ ❱➻ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❜↔♦ t♦➔♥ ❣â❝ ♥➯♥ ✤÷í♥❣ trá♥ ♥➔② ❝ơ♥❣ t↕♦ ✈ỵ✐ AB ❣â❝ 450 ✳ ❉♦ ✤â t➙♠ ❝õ❛ ♥â ❧➔ tr✉♥❣ ✤✐➸♠ Q1 ❝õ❛ ✤÷í♥❣ trá♥ (AC)✳ P❤➛♥ t❤ù ❤❛✐ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü download by : skknchat@gmail.com ✹✸ ❍➻♥❤ ✸✳✹✿ ❇❛ ❝→❝❤ ❞ü♥❣ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❛r❜❡❧♦s ❆❇❈ ▼➺♥❤ ✤➲ ✸✳✸✳ ❈→❝ ✤÷í♥❣ t❤➥♥❣ ❆❳✱ ❇❨✱ ❈❩ ❝➢t ♥❤❛✉ t↕✐ ✤✐➸♠ ❙ tr➯♥ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ✭❳❨❩✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ ❤➻♥❤ ✸✳✹ ❜✮ ✳ ❚❛ ❧✉æ♥ ❝â A, X, Q2 t❤➥♥❣ ❤➔♥❣✱ B, Y, Q1 t❤➥♥❣ ❤➔♥❣✳ ●å✐ S = AQ2 ∩ (XY Z) t ợ ữớ trỏ ✤↔♦ ❧➔ A(AD)✳ ❷♥❤ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ S ❧➔ S = AQ2 (Q2 Y Z ) ữ ỵ r AS Z = Q2 S Z = Q2 Y Z = 450 = ABZ ♥➯♥ A, B, S , Z t❤✉ë❝ ♠ët ✤÷í♥❣ trá♥✳ ❇➡♥❣ ❝→❝❤ ①➨t ↔♥❤ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ✤÷í♥❣ trá♥ ♥➔② t❛ rót r❛ CZ ❝❤ù❛ S ✳ ◆â✐ ❝→❝❤ ❦❤→❝ AQ2 ✈➔ CZ ❝➢t ♥❤❛✉ t↕✐ ✤✐➸♠ S tr➯♥ ✤÷í♥❣ trá♥ (XY Z)✳ ❈ơ♥❣ ❣✐è♥❣ ♥❤÷ ✈➟② ✤è✐ ✈ỵ✐ BQ1 ✈➔ CZ ✳ ▼➺♥❤ ✤➲ ✸✳✹✳ ●å✐ ▼ ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ ♥û❛ ✤÷í♥❣ trá♥ ✭❆❇✮ ✤è✐ ự ợ ỷ ữớ trỏ rs ✤â✱ ❝→❝ ✤✐➸♠ ❆✱ ❇✱ ❳✱ ❨ ♥➡♠ tr➯♥ ✤÷í♥❣ trá♥ t➙♠ ▼ ✈➔ ❈❩ ✤✐ q✉❛ ▼✳ ❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ ❤➻♥❤ ✸✳✹ ❝✮✳ ❱➻ C, Q2, Y ♥➡♠ tr➯♥ ữớ t t ợ AB õ 450 ✤↔♦ ❝õ❛ ♥â ❧➔ ♠ët ✤÷í♥❣ trá♥ ✤✐ q✉❛ A, B, X, Y ❝ơ♥❣ t↕♦ ✈ỵ✐ AB ❣â❝ 450 ✳ ❚➙♠ ❝õ❛ ✤÷í♥❣ trá♥ ♥➔② ♣❤↔✐ ❧➔ tr✉♥❣ ✤✐➸♠ M ỷ ữớ trỏ (AB) ố ự ợ ỷ (AB) ❝õ❛ ❛r❜❡❧♦s q✉❛ AB ✳ ◆è✐ AM ✱ ♥â ❝➢t ð M ✳ ❱➻ BAM = 450 = BZ M A, Z , B, M ỗ ❙û ❞ö♥❣ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ t❛ s✉② r❛ CZ ✤✐ q✉❛ M ✳ download by : skknchat@gmail.com ✹✹ ✸✳✷✳✷ ❈→❝❤ ❞ü♥❣ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❛r❜❡❧♦s ❆❇❈ ❚❛ ❝â ♥❤✐➲✉ ❝→❝❤ ❞ü♥❣ ✤÷í♥❣ trá♥ ♥ë✐ t➳♣ ♠ët ❛r❜❡❧♦s✿ ❈→❝❤ ❞ü♥❣ ✶✳ ✭❙✉② tø ♠➺♥❤ ✤➲ ✸✳✷✮✱ ❤➻♥❤ ✸✳✹ ❛✮✮ ✲ ❉ü♥❣ Q1 , Q2 ❧➔ tr✉♥❣ ✤✐➸♠ ❝→❝ ♥û❛ ✤÷í♥❣ trá♥ (AB), (CB) ✲ ❉ü♥❣ ✤÷í♥❣ trá♥ Q1 (Q1 A) ❝➢t ❝→❝ ♥û❛ ✤÷í♥❣ trá♥ (CB), (AB) ❧➛♥ ❧÷đt ð X, Y ✲ ❉ü♥❣ ✤÷í♥❣ trá♥ Q2 (Q2 B) ❝➢t ❝→❝ ♥û❛ ✤÷í♥❣ trá♥ (AC), (AB) ❧➛♥ ❧÷đt ð Y, Z ✲ ✣÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ ∆XY Z ❧➔ ✤÷í♥❣ trá♥ ❝➛♥ ❞ü♥❣✳ ❈→❝❤ ❞ü♥❣ ✷✳ ✭❙✉② tø ♠➺♥❤ ✤➲ ✸✳✸✮✱ ❤➻♥❤ ✸✳✹ ❜✮✮ ✲ ❉ü♥❣ X = AQ2 ∩ (CB), Y = BQ1 = ∩(AC)✱ ❣å✐ S ❧➔ ❣✐❛♦ ❝õ❛ ❝→❝ ✤÷í♥❣ t❤➥♥❣ AQ2 , BQ1 ✲ ✣÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ ∆XY Z ❧➔ ✤÷í♥❣ trá♥ ❝➛♥ ❞ü♥❣✳ ❈→❝❤ ❞ü♥❣ ✸✳ ✭❙✉② tø ♠➺♥❤ ✤➲ ✸✳✹✮✱ ❤➻♥❤ ✸✳✹ ❝✮✮ ✲ ❉ü♥❣ M ❧➔ tr✉♥❣ ✤✐➸♠ ♥û❛ ✤÷í♥❣ trá♥ ✤è✐ ự ợ ỷ ữớ trỏ (AB) tr rs ỹ ✤÷í♥❣ trá♥ M (M A)✱ ♥â ❝➢t ❝→❝ ♥û❛ ✤÷í♥❣ trá♥ (CB), (AC) ❧➛♥ ❧÷đt ð X, Y ✲ ❉ü♥❣ ✤÷í♥❣ t❤➥♥❣ M C ✱ ♥â ❝➢t ♥û❛ ✤÷í♥❣ trá♥ (AB) ð Z ✲ ✣÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ ∆XY Z ❧➔ ✤÷í♥❣ trá♥ ❝➛♥ ❞ü♥❣✳ ❚❛ ❧↕✐ t❤➜② r➡♥❣ t➙♠ ❝õ❛ (XY Z) ❝❤➼♥❤ ❧➔ ❣✐❛♦ ❝õ❛ ❝→❝ ✤÷í♥❣ t❤➥♥❣ ♥è✐ X, Y, Z ❧➛♥ ❧÷đt ❧➔ t➙♠ ❝→❝ ♥û❛ ✤÷í♥❣ trá♥ (BC), (AC), (AB)✳ ❚✉② ♥❤✐➯♥ t❛ ❝â t❤➸ ❞ü♥❣ t➙♠ ♠ët ❝→❝❤ ✤ì♥ ❣✐↔♥ ❤ì♥✱ ❜➡♥❣ ❝→❝❤ ❝❤➾ ỹ ổ ỵ t➙♠ ❝õ❛ (XY Z) ❧➔ ω ✤➸ sû ❞ö♥❣ ✈➲ s❛✉✳ download by : skknchat@gmail.com ✹✺ ❍➻♥❤ ✸✳✺✿ ❈→❝❤ ❞ü♥❣ t❤ù t÷ ❝õ❛ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ✸✳✸ ❈→❝ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s tr♦♥❣ ❛r❜❡❧♦s ❈→❝ ❝➦♣ ✤÷í♥❣ trá♥ ❝â t➼♥❤ ❝❤➜t ❣✐è♥❣ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s ✤÷đ❝ ♣❤→t ❤✐➺♥ ✈➔ ❝æ♥❣ ❜è tr♦♥❣ ♥❤✐➲✉ ❝→❝ ❜➔✐ ❜→♦ ❦❤♦❛ ❤å❝ ❣➛♥ ✤➙②✳ ❈➦♣ ✤÷í♥❣ trá♥ t❤ù ♥❤➜t ❞♦ ❝❤➼♥❤ ❆r❝❤✐♠❡❞❡s t➻♠ r❛✱ ỉ♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝❤ó♥❣ ❝â ❜→♥ ❦➼♥❤ ❜➡♥❣ ♥❤❛✉ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ✈à tr➼ ❝õ❛ ✤✐➸♠ C tr➯♥ AB ✳ ✸✳✸✳✶ ❈➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù t tự ỵ rs ữớ trỏ t ú ợ ợ ỷ ữớ trá♥ ❖✭❛✰❜✮ ✈➔ ♠ët tr♦♥❣ ❤❛✐ ♥û❛ ✤÷í♥❣ trá♥ O1(a), O2(b) ❝â ❜→♥ ❦➼♥❤ t = a ab ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ✈à tr➼ ❝õ❛ ❈ tr➯♥ ❆❇ +b ❈❤ù♥❣ t ữớ trỏ t ú ợ ỷ ữớ trỏ O(a + b), O1 (a) CD ỵ t ❧➔ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥✳ ❇➡♥❣ ❝→❝❤ t➼♥❤ ❦❤♦↔♥❣ tứ t ữớ trỏ tợ AB t ❝→❝❤ t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ (a + b − t)2 − (a − b − t)2 = (a + t)2 − (a − t)2 ab ✳ a+b ❉♦ t➼♥❤ ✤è✐ ①ù♥❣ ❝õ❛ ❜✐➸✉ t❤ù❝ ✤è✐ ✈ỵ✐ a, b t❛ s✉② r❛ ✤÷í♥❣ trá♥ t❤ù ❤❛✐ ❝ơ♥❣ ❝â ❜→♥ ❦➼♥❤ t✳ ❚ø ✤â✱ t = download by : skknchat@gmail.com ✹✻ ❍➻♥❤ ✸✳✻✿ ❈➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ♥❤➜t ✈➔ t❤ù ❤❛✐ ❍❛✐ ✤÷í♥❣ trá♥ ♥â✐ tr➯♥ ❧➔ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ♥❤➜t✳ P❤➨♣ ❞ü♥❣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s ✤÷đ❝ t❤ü❝ t ữợ s ỹ Q1 , Q2 ❧➔ tr✉♥❣ ✤✐➸♠ ♥û❛ ✤÷í♥❣ trá♥ (AB) ✈➔ CB ✲ ❉ü♥❣ K = O1 Q2 ∩ O2 Q1 ✱ t❤➻ K ∈ CD ✈➔ KC = KC = ab ✳ ú ỵ r a+b ab = t ❝õ❛ ❝→❝ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s a+b ✲ ❉ü♥❣ M1 , M2 ∈ AB s❛♦ ❝❤♦ CM1 = CM2 = KC ✲ ❉ü♥❣ W1 = O1 (O1 M2 ) ∩ M1 u ✈ỵ✐ M1 u⊥AB ✳ ✣â ❧➔ t➙♠ ❆r❝❤✐♠❡❞❡s t❤ù ♥❤➜t ✲ ❉ü♥❣ W1 = O2 (O2 1M1 ) ∩ M2 v ✈ỵ✐ M1 v⊥AB ✳ ✣â ❧➔ t➙♠ ❆r❝❤✐♠❡❞❡s tự r t ỵ (W1 ), (W1 )✳ ❙❛✉ ❆r❝❤✐♠❡❞❡s ♥❣÷í✐ t❛ t➻♠ ✤÷đ❝ ab ❦❤→ ♥❤✐➲✉ ❝→❝ ❝➦♣ ✤÷í♥❣ trá♥ ❝â ❜→♥ ❦➼♥❤ ✈➔ ❝â t➼♥❤ ❝❤➜t t✐➳♣ ①ó❝ a+b ❣✐è♥❣ ♥❤÷ t❤➳✳ ❈❤ó♥❣ tỉ✐ s➩ ❧➛♥ ❧÷đt tr➻♥❤ ❜➔② ♠ët sè ❝➦♣✱ ❝â ❝↔ ♥❤ú♥❣ ❝➦♣ ✤÷đ❝ ♣❤→t ❤✐➺♥ tr♦♥❣ ♥❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙②✳ ❈➦♣ ✤÷í♥❣ trá♥ (W2 ), (W2 ) ❧➛♥ ❧÷đt ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ (W1 ), (W1 ) ❧➯♥ AB ✳ ✣â ❧➔ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ❤❛✐✱ ❝→❝ ❝➦♣ ✤÷í♥❣ trá♥ (W1 ), (W2 ) download by : skknchat@gmail.com ✹✼ ❝â t✐➳♣ t✉②➳♥ ❝❤✉♥❣ ✤✐ q✉❛ B ✱ ❝á♥ ❝→❝ ❝➦♣ (W1 ), (W2 ) ❝â t✐➳♣ t✉②➳♥ ❝❤✉♥❣ ✤✐ q✉❛ A✱ ❤➻♥❤ ✸✳✻✳ ❈➦♣ ♥➔② ✤÷đ❝ ♣❤→t ❤✐➺♥ ❜ð✐ ❈✳❲✳ ❉♦❞❣❡✱ ❝æ♥❣ ❜è tr♦♥❣ t↕♣ ❝❤➼ ▼❛t❤✳ ▼❛❣✳✱✼✷✭✶✾✾✾✮✳ ỵ tự ỵ tự sỷ ữớ trỏ t✐➳♣ ❝õ❛ ❛r❜❡❧♦s [ABC] t✐➳♣ ①ó❝ ❤❛✐ ♥û❛ ✤÷í♥❣ trá♥ ✭❆❈✮ ✈➔ ✭❈❇✮ t÷ì♥❣ ù♥❣ t↕✐ ❳✱❨✳ ❑❤✐ ✤â ✤÷í♥❣ trá♥ ✤✐ q✉❛ ❈✱ ❳✱ ❨ ❝ô♥❣ ❝â ❜→♥ ❦➼♥❤ ❜➡♥❣ t = a ab +b ❈❤ù♥❣ ♠✐♥❤✳ ❘ã r➔♥❣ ✤÷í♥❣ trá♥ (CXY ) ❧➔ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❝õ❛ t❛♠ ❣✐→❝ ωO1 O2 ✈➔ ωX = ωY = t, O1 X = O1 C = a, O2 Y = O2 C = b ◆û❛ ❝❤✉ ✈✐ ❝õ❛ t❛♠ ❣✐→❝ CO1 O2 ❜➡♥❣ (a + b)2 ab(a + b) = a + b + t = (a + b) + a + ab + b2 a + ab + b2 S ❇→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ t❛♠ ❣✐→❝ ✤÷đ❝ t➼♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ r = p r= abt = a+b+t ab ab.ab(a + b) = (a + b)3 a+b ✣â ❝❤➼♥❤ ❧➔ ❜→♥ ❦➼♥❤ t ❝õ❛ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s✳ ✣÷í♥❣ trá♥ CXY ✤â ❝â t➯♥ ❣å✐ ❧➔ ✤÷í♥❣ trá♥ ❇❛♥❦♦❢❢✱ ❤➻♥❤ ✸✳✼✳ ❑➳t q✉↔ ♥➔② ❝❤➾ r❛ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❛r❜❡❧♦s ữớ trỏ ụ ữớ trỏ rs ỗ t❤í✐ ✤÷í♥❣ trá♥ ❇❛♥❦♦❢❢ ❧↕✐ ❧➔ ✤÷í♥❣ trá♥ ♥ë✐ t✐❡♣s ❝õ❛ t❛♠ ❣✐→❝ ωO1 O2 ✳ download by : skknchat@gmail.com ✹✽ ✸✳✸✳✷ ❈➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ❜❛ ✈➔ t❤ù t÷ ❍➻♥❤ ✸✳✽✿ ❈➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ❜❛ ✈➔ tự tữ ỵ t I tr AB ữớ ổ õ ợ AB q O, C ❧➛♥ ❧÷đt ❝➢t Q1 Q2 ð I, J ✳ ❑❤✐ ✤â t❛ ❝â CJ = 2t ✈➔ ✈➻ O ✈➔ C ✤è✐ ①ù♥❣ ♥❤❛✉ q✉❛ tr✉♥❣ ✤✐➸♠ ❝õ❛ O1 O2 ♥➯♥ t❤❡♦ t➼♥❤ ❝❤➜t ✤÷í♥❣ tr✉♥❣ ❜➻♥❤ tr♦♥❣ ❤➻♥❤ t❤❛♥❣ t❛ ❝â✿ OI = (a + b) − 2t t II = 2t ữ ỵ r OQ1 = OQ2 ✈➔ ✈➻ I ✈➔ J ❧↕✐ ✤è✐ ①ù♥❣ ♥❤❛✉ q✉❛ tr✉♥❣ ✤✐➸♠ ❝õ❛ Q1 Q2 ♥➯♥ ❝â JJ = II = 2t✳ ❚ø ✤â s✉② r❛✿ ❤❛✐ ✤÷í♥❣ trá♥ t➙♠ (W3 ), (W3 ) ♠é✐ ✤÷í♥❣ trá♥ ✤✐ q I, J t ú ợ ỷ ữớ trỏ ❧ỵ♥ ♥❤➜t ❝õ❛ ❛r❜❡❧♦s ✤➲✉ ❝â ❜→♥ ❦➼♥❤ ❜➡♥❣ t✳ ✣â ❧➔ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ❜❛ ❝õ❛ ❛r❜❡❧♦s [ABC]✱ ①❡♠ ❤➻♥❤ ✸✳✽❛✳ ❈➦♣ ♥➔② ✤÷đ❝ ♣❤→t ❤✐➺♥ ❜ð✐ s r ữ ỵ r ❝â r➜t ♥❤✐➲✉ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s tr♦♥❣ ❤➻♥❤ ❆r❜❡❧♦s✳ ab ●✐↔ sû t = ♥❤÷ tr➯♥✳ ◆➳✉ U V ❧➔ t✐➳♣ t✉②➳♥ ❝❤✉♥❣ ♥❣♦➔✐ ❝õ❛ ❤❛✐ a+b ♥û❛ ✤÷í♥❣ trá♥ ♥❤ä tr♦♥❣ ❤➻♥❤ ❛r❜❡❧♦s ✈➔ t✐➳♣ ①ó❝ ✈ỵ✐ ❞➙② ❝✉♥❣ HK ỷ ữớ trỏ ợ W4 = O1 W ∩ O2 U ✳ ❱➻ O1 U = a, O2 V = b ✈➔ O1 C a ab = ♥➯♥ W4 = = t✳ ✣✐➲✉ ✤â ♥❣❤➽❛ ❧➔ ✤÷í♥❣ trá♥ W4 (t) ✤✐ CO2 b a+b q✉❛ C ✈➔ t✐➳♣ ①ó❝ ✈ỵ✐ HK ð ✤✐➸♠ N ✳ ●å✐ M ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ HK ✳ ❱➻ O ✈➔ C ✤è✐ ①ù♥❣ ♥❤❛✉ q✉❛ tr✉♥❣ ✤✐➸♠ ❝õ❛ O1 O2 ♥➯♥ OM + CN = O1 U + O2 V = a + b✳ ❚ø ✤â s✉② r❛ (a + b) − OM = CN = 2t✳ ◆❣❤➽❛ ❧➔ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝ ✈ỵ✐ ❞➙② HK ✈➔ ❝✉♥❣ HK ❝â ❜→♥ t ữớ trỏ t W4 t ú ợ ♥û❛ ✤÷í♥❣ trá♥ (AB) ð ✤✐➸♠ Q✳ ❈➦♣ ✤÷í♥❣ trá♥ t➙♠ W4 (t), (W4 (t) ❣å✐ ❧➔ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù t÷✱ ❤➻♥❤ ✸✳✽❜✮ ✳ download by : skknchat@gmail.com ✹✾ ✸✳✸✳✸ ❈→❝ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ♥➠♠ ✈➔ t❤ù s→✉ ❍➻♥❤ ✸✳✾✿ ❈➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ♥➠♠ ✈➔ t❤ù s→✉ ◆➠♠ ✷✵✵✺✱ ❋r❛♥❦ P♦✇❡r ✤➣ ♣❤→t ❤✐➺♥ r❛ ✷ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s (W5 ), (W5 ) ✈➔ (W6 ), (W6 )✱ ①❡♠ ❬✸❪✳ ▼➺♥❤ ✤➲ ✸✳✼✳ ữớ trỏ t ú tr ợ ỷ ữớ trỏ (AB) ✈➔ t✐➳♣ ①ó❝ ✈ỵ✐ OQ1 ð Q1 ✭❍♦➦❝ t✐➳♣ ①ó❝ ✈ỵ✐ OQ2 ð Q2 ✮ ❝â ❜→♥ ❦➼♥❤ t = a ab ✳ +b ❈❤ù♥❣ ♠✐♥❤✳ ❈â ❤❛✐ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝ ✈ỵ✐ OQ1 ð Q1✱ tr➯♥ ❤➻♥❤ ✸✳✾ t❛ ỵ t W5 W5 t ữớ trá♥ t➙♠ W5 ✱ ❜→♥ ❦➼♥❤ r✳ ❚❛ ❝â ❝→❝ t❛♠ ❣✐→❝ ✈✉æ♥❣ OQQ1 ✈➔ OW5 Q1 ♥➯♥✿ OQ21 = O1 Q21 + OO12 = a2 + b2 , t❤❛② ✈➔♦ ✤➥♥❣ t❤ù❝ s❛✉ OW52 = Q1 W52 + OQ21 ⇐⇒ (a − b − r)2 = (a2 + b2 ) + r2 ab ✳ ❚➼♥❤ t♦→♥ ♥❤÷ t❤➳ t❤✉ ✤÷đ❝ (W5 ) ❝ơ♥❣ ❝â ❜→♥ ❦➼♥❤ t✳ ❚ø ✤â✱ r = a+b ❍♦➔♥ t♦➔♥ t÷ì♥❣ tü✱ t❛ ❝â t❤➯♠ ❝➦♣ (W6 ), (W6 )✳ ❈➦♣ ✤÷í♥❣ trá♥ (W5), (W5) ✈➔ (W6), (W6) ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ❝➦♣ ✤÷í♥❣ trá♥ Pr s ợ t t ữ ✈➟②✳ download by : skknchat@gmail.com ✺✵ ✸✳✸✳✹ ❈→❝ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ❜↔② ✈➔ t❤ù t→♠ ●å✐ M ❧➔ tr CD ỵ t ố ữớ ữớ trỏ CD ổ õ ợ OM U1 , U2 ú ỵ r OC = (a − b)2 ✈➔ ✈➻ CD = ab ♥➯♥ OD2 = a2 − ab + b2 ✈➔ OU12 = a2 + b2 ❍➻♥❤ ✸✳✶✵✿ ❈➦♣ ✤÷í♥❣ trá♥ t❤ù ❜↔②✱ t❤ù t→♠ ❇➙② ❣✐í ①➨t ❝➦♣ ✤÷í♥❣ trá♥ ❜➡♥❣ ♥❤❛✉✱ ♠é✐ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝ O(a+b) ✈➔ t✐➳♣ ①ó❝ ✈ỵ✐ ♥❤❛✉ t↕✐ U1 ✈➔ U2 ✳ ❇→♥ ❦➼♥❤ r ❝õ❛ ❝→❝ ✤÷í♥❣ trá♥ ♥➔② t❤ä❛ ♠➣♥ (a + b − r)2 = OU12 + r2 ✳ ❚❤❛② ❝→❝ ✤➥♥❣ t❤ù tr➯♥ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ab t❤✉ ✤÷đ❝✿ r = ✳ ❱➟② t❛ ❝â ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù ❜↔② ❦✐➸✉ a+b P♦✇❡r (W7 ), (W7 )✳ ❇➡♥❣ ❝→❝❤ ❧➜② ✤è✐ ①ó♥❣ q✉❛ OM t❛ ❝â ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s t❤ù t→♠ ❦✐➸✉ P♦✇❡r (W8 ), (W8 )✱ ❤➻♥❤ ✸✳✶✵✳ ❍❛✐ ❝➦♣ ♥➔② ✤÷đ❝ ♣❤→t ❤✐➺♥ ❜ð✐ ❋❧♦♦r ✈❛♥ ▲❛♠♦❡♥ ✭❙t✳ ❲✐❧✐❜r♦r❞❝♦❧❧❡❣❡✱ ❋r✉✐t❧❛❛♥ ✸✱ ✹✹✻✷ ❊P ●♦❡s✱ ❚❤❡ ◆❡t❤❡r❧❛♥❞s✮✳ ◆➠♠ ✷✵✶✹✱ ❉❛♦ ❚❤❛♥❤ ❖❛✐ ✈➔ ❚r❛♥ ◗✉❛♥❣ ụ ợ t ữớ trỏ rs tr ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✿ ❝➦♣ (W9 ), (W9 ) ✈➔ ❝➦♣ (W10 ), (W10 ) tr➯♥ ❤➻♥❤ ✸✳✶✶ ❝õ❛ ❉❛♦ ❚❤❛♥❤ ❖❛✐❀ ❝➦♣ (W11 ), (W11 ) ✈➔ ❝➦♣ (W12 ), (W12 ) tr➯♥ ❤➻♥❤ ✸✳✶✷ ❝õ❛ ❚r❛♥ ◗✉❛♥❣ ❍✉♥❣✳ + b)2 ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❞✐➺♥ t➼❝❤ ∆IO1O2 ❜➡♥❣ aab(a ✈➔ + ab + b2 ■ ❝→❝❤ ❆❇ ♠ët ❦❤♦↔♥❣ ❜➡♥❣ 2ρ✳ ❇➔✐ t♦→♥ ✸✳✹✳ download by : skknchat@gmail.com ✺✶ ❇➔✐ t♦→♥ ✸✳✺✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❝→❝ t✐➳♣ ✤✐➸♠ ❝õ❛ I(ρ) ợ ỷ ữớ trỏ õ t ❝→❝❤✿ ❳✱❩ ❧➔ ❣✐❛♦ ❝õ❛ Q1(Q1A) ✈ỵ✐ ❤❛✐ ♥û❛ O1 (a), O(a+b)✱ ❝á♥ ❳✱ ❩ ❧➔ ❣✐❛♦ ❝õ❛ Q2 (Q2 B) ✈ỵ✐ ❤❛✐ ♥û❛ O2 (b), O(a+b)✳ ❍➻♥❤ ✸✳✶✶✿ ❈➦♣ ✤÷í♥❣ trá♥ t❤ù ❝❤➼♥ ✈➔ ❝➦♣ t❤ù ♠÷í✐ ❇➔✐ t♦→♥ ✸✳✻✳ ✭❚✳ ❖✳ ❉❛♦✮ ❚r➯♥ ❤➻♥❤ ✸✳✶✶✱ ❣✐↔ sû ❆✬✳ ❇✬ ❧➔ ❝→❝ ❤➻♥❤ ❝❤✐➳✉ ✈✉æ♥❣ ❣â❝ ❝õ❛ ❉ tr➯♥ t✐➳♣ t✉②➳♥ t↕✐ ❑ ✈➔ ❍ ❝õ❛ ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ ❆❇✱ t÷ì♥❣ ù♥❣✳ ❈→❝ ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ ❉❆✬ ✈➔ ❉❇✬ ❧➔ ❝→❝ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s✳ ❇➔✐ t♦→♥ ✸✳✼✳ ✭❚✳ ❖✳ ❉❛♦✮ ❚r➯♥ ❤➻♥❤ ✸✳✶✶✱ ❣✐↔ sû A1A2 ✈➔ B1B2 ❧➔ t✐➳♣ t✉②➳♥ ❝õ❛ ❤❛✐ ✤÷í♥❣ trá♥ ✭❆❈✮✱ ✭❈❇✮ ✈ỵ✐ A1, B1 ∈ AB ✈➔ A1A2 = a, B1 B2 = b✳ ●å✐ W10 = CQ1 ∩ A1 B2 , W10 = CQ2 ∩ B1 A2 ✳ ❑❤✐ ✤â ❝→❝ ✤÷í♥❣ trá♥ t➙♠ W10, W10 ✤✐ q✉❛ ❈ ❧➔ ❝→❝ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s✳ ❇➔✐ t♦→♥ ✸✳✽✳ ✭◗✳ ❍✳ ❚r❛♥✮ r ữớ t ổ õ ợ t↕✐ O1, O2 ❝➢t ✭❆❇✮ t÷ì♥❣ ù♥❣ t↕✐ ❊✱ ❋✳ ỵ W11 = AF (AC), W11 = BE ∩ (CB) t❤➻ ❝→❝ ✤÷í♥❣ trá♥ t➙♠ W11 , W11 t✐➳♣ ①ó❝ ✈ì✐ ❈❉ ❧➔ ❝→❝ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s✳ download by : skknchat@gmail.com ✺✷ ❇➔✐ t♦→♥ ✸✳✾✳ ✭◗✳ ❍✳ ❚r❛♥✮ ❚r➯♥ ❤➻♥❤ ✸✳✶✷✱ ❣✐↔ sû W12 ❧➔ ❣✐❛♦ ❝õ❛ ❆❉ ợ ỷ ữớ trỏ (AO2) W12 ợ ỷ ữớ trỏ (BO1 ) ữớ trỏ t➙♠ W12 ✈➔ W12 t✐➳♣ ①ó❝ ✈ỵ✐ ❈❉ ❧➔ ❝→❝ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s✳ ❍➻♥❤ ✸✳✶✷✿ ❈➦♣ t❤ù ♠÷í✐ ♠ët ✈➔ ❝➦♣ t❤ù ♠÷í✐ ❤❛✐ ❇➔✐ t♦→♥ ✸✳✶✵✳ ✭✣÷í♥❣ trá♥ ❝õ❛ ❙❝❤♦❝❤✮ ✣÷í♥❣ trá♥ ❈ ♥ë✐ t✐➳♣ tr♦♥❣ t❛♠ ❣✐→❝ ❝♦♥❣ ợ ỷ ữớ trỏ ✤÷í♥❣ trá♥ ❆✭✷❛✮✱ ❇✭✷❜✮✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❈ ❧➔ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s✳ download by : skknchat@gmail.com ✺✸ ❑➳t ❧✉➟♥ ❝õ❛ ❧✉➟♥ ✈➠♥ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉ ✶✳ ❚r➻♥❤ ❜➔② ❤❛✐ ❜➔✐ t♦→♥ ♥ê✐ t✐➳♥❣ ợ ữợ ự ợ t t t♦→♥ ❋❡✉❡r❜❛❝❤ ①✉➜t ♣❤→t tø ❜➔✐ t♦→♥ ❝ì ❜↔♥ ✈➔ tê♥❣ q✉→t ❤â❛✳ ✷✳ ❚r➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐ ✈➲ ❞ü♥❣ ✸ ✤÷í♥❣ trá♥ ✈➔ tr➻♥❤ ❜➔② t÷í♥❣ ♠✐♥❤ ✈➲ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ▼❛❧❢❛tt✐ ❣è❝ ❝ò♥❣ ❜❛ ❜➔✐ t♦→♥ ❦✐➸✉ ▼❛❧❢❛tt✐ ✭❝❤♦ t❛♠ ❣✐→❝ ✤➲✉✱ ❤➻♥❤ ✈✉æ♥❣ ✈➔ ✤÷í♥❣ trá♥✮✳ ✸✳ P❤→t ❜✐➸✉ ✈➔ tr➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ✈➲ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝ tr♦♥❣ ❤➻♥❤ ❤å❝ ❆r❜❡❧♦s✳ ✣÷❛ r❛ ❝→❝ ❦➳t q✉↔ ✈➲ ỹ ữớ trỏ t rs ABC ợ t❤✐➺✉ ✶✷ ❝➦♣ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s ❝õ❛ ❆r❜❡❧♦s✳ ❈→❝ ❦➳t q ợ ữủ ổ ố tr ❣➛♥ ✤➙②✿ ❬✷❪✱ ❬✹❪✱ ❬✺❪✱ ❬✼❪✳ ❈❤ó♥❣ tỉ✐ ♥❤➟♥ t❤➜② ữợ ự t t t ❝→❝ ❜➔✐ t♦→♥ ù♥❣ ❞ö♥❣ ❦➳t q✉↔ ❝õ❛ ❝→❝ ✤à♥❤ ỵ tr s t ❤å❝ ❆r❜❡❧♦s✳ ✲ ❙û ❞ö♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ❤➻♥❤ t❤➼❝❤ ❤đ♣ ❤♦➦❝ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë ✤➸ ♥❣❤✐➯♥ ❝ù✉ s➙✉ ✈➲ ❝→❝ ❜➔✐ t♦→♥ ✤❛♥❣ ①➨t✳ ▼➦❝ ❞ò ✤➣ r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ t rt sỹ õ ỵ ❜ê s✉♥❣ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝õ❛ ❝→❝ ỗ t q ự ❝❤➾♥❤ ✈➔ ❝â ➼❝❤ ❤ì♥✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ download by : skknchat@gmail.com ✺✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ t ỳ ỵ tr ❜➔✐ t♦→♥ →♣ ❞ö♥❣✱ ◆❳❇ ●✐→♦ ❞ö❝ ❱✐➺t ♥❛♠✳ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ✈➔ ❝→❝ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ❑♦st❛♥❞✐♥♦✈ ❊✳✱ ✭✷✵✶✸✮ ▼❛❧❢❛tt✐✬s Pr♦❜❧❡♠s✱ ▼❡❡t✐♥❣ ✐♥ ▼❛t❤❡♠❛t✐❝s ✷♥❞ ❡❞✐t✐♦♥✱ ❇✉❧❣❛r✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s✳ ❬✸❪ P♦✇❡r ❋✳ ✭✷✵✵✺✮✱ ❙♦♠❡ ▼♦r❡ ❆r❡❝❤✐♠❡❞❡❛♥ ❈✐r❝❧❡s ✐♥ t❤❡ ❛r❜❡❧♦s✱ ❱♦❧✲ ✉♠❡ ✺✱ ✶✸✸✲✶✸✹✱ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✳ ❬✹❪ ❙❝❤❡❧❧❜❛❝❤✱ ✭✶✾✾✽✮✱ ▼❛❧❢❛tt✐✬s Pr♦❜❧❡♠✱✶✾✾✽✳ ❱♦❧✉♠❡ ✹✺✱ ❈r❡❧❧❡✬s ❏♦✉r✲ ♥❛❧✳ ❬✺❪ ❲♦♦ P✳ ❨✳✱ ✭✷✵✵✶✮✱ ❙✐♠♣❧❡ ❈♦♥str✉❝t✐♦♥s ♦❢ ❱♦❧✉♠❡ ✶✱✶✸✸✲✶✸✻✱ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✳ t❤❡ ■♥❝✐r❝❧❡ ♦❢ ❛♥ ❛r❜❡❧♦s✱ ❬✻❪ ❩❛❧❣❛❧❧❡r ❱✳❆✱ ▲♦s ●✳❆✱ ✭✶✾✾✹✮✱ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ ▼❛❧❢❛tt✐✬s Pr♦❜❧❡♠✱ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s ❱♦❧✳ ✼✷✱ N0 ✹ ✭♣✸✶✻✸✲✸✶✼✼✮✳ ❚✐➳♥❣ ◆❣❛ ❬✼❪ Ïðỵịàđỵâ ➶✳ ủỵựốồủ ợờúổớợủũố ũ ềồỏợ ọợ ễồộ ồỏừ õớũ ợỡồ ✶✶✲✶✾✼✺✳ download by : skknchat@gmail.com ... HỌC KHOA HỌC  - VŨ THỊ NĂM MỘT SỐ BÀI TOÁN VỀ ĐƯỜNG TRỊN TIẾP XÚC Chun ngành: Phương pháp Tốn sơ cấp Mã số: 8460113 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC PGS.TS Nguyễn... ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s ❝õ❛ ♠ët ❛r❜❡❧♦s✱ ❝➟♣ ♥❤➟t ✤÷đ❝ ♥❤ú♥❣ ♣❤→t ❤✐➺♥ tr♦♥❣ ♥❤ú♥❣ ♥➠♠ ỗ ởt số t♦→♥ ✤ì♥ ❣✐↔♥ ✸✳✷✳ ✣÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ✈➔ ✤÷í♥❣ trá♥ ❆r❝❤✐♠❡❞❡s✳ ✸✳✸✳ ❉ü♥❣ ❝→❝ ❝➦♣ ✤÷í♥❣ trá♥... ✧❍➺ t❤è♥❣ ❤â❛ ❤➻♥❤ ❤å❝✧✱ ♠ð ✤➛✉ ❝❤♦ ❤➻♥❤ ❤å❝ ❤✐➺♥ ✤↕✐✳ ◆❣➔② ♥❛② tr♦♥❣ ❝→❝ ♣❤➨♣ ❝❤ù♥❣ ♠✐♥❤ ỵ r số sỷ ổ ữ ỵ Pt tờ qtữ ❝ơ♥❣ ❝â ♥❤ú♥❣ ♣❤➨♣ ❝❤ù♥❣ ♠✐♥❤ ❤♦➔♥ t♦➔♥ ❝➜♣✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ♣❤➨♣ ❝❤ù♥❣