Tứ giác ngoại tiếp và các vấn đề liên quan

ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - BÙI ĐỨC HUY TỨ GIÁC NGOẠI TIẾP VÀ CÁC VẤN ĐỀ LIÊN QUAN LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2019 ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - BÙI ĐỨC HUY TỨ GIÁC NGOẠI TIẾP 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 NGUYÊN - 2019 ✐ ❉❛♥❤ ♠ö❝ ❤➻♥❤ ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✶✳✺ ✶✳✻ ✶✳✼ ✶✳✽ ✶✳✾ ✶✳✶✵ ✶✳✶✶ ✶✳✶✷ ✶✳✶✸ ✶✳✶✹ ✶✳✶✺ ✶✳✶✻ ✶✳✶✼ ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ✷✳✺ ✷✳✻ ✷✳✼ ✷✳✽ ✷✳✾ ✷✳✶✵ ✷✳✶✶ ✷✳✶✷ ✸✳✶ ✸✳✷ ✣à♥❤ ỵ Ptt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ët ❜➜t ✤➥♥❣ t❤ù❝ ❤➻♥❤ ❤å❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ự ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈→❝ ❣â❝ tr♦♥❣ ✤➦❝ tr÷♥❣ ■♦s✐❢❡s❝✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣✐➲✉ ❦✐➺♥ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❝õ❛ ❲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍❛✐ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝ ✷ ❝↕♥❤✱ ✶ ✤÷í♥❣ ❝❤➨♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈→❝ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝ ð ❝→❝ ♣❤➼❛ ❤❛✐ ✤÷í♥❣ ❝❤➨♦ ✳ ✳ ✳ ✳ ✳ ✳ ❈→❝ t✐➳♣ ✤✐➸♠ ❝õ❛ ✹ ✤÷í♥❣ trá♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ●✐↔ t❤✉②➳t ❝õ❛ ❈❤r✐st♦♣❤❡r ❇r❛❞❧❡② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣➦❝ tr÷♥❣ ❱❛✐♥s❤t❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 1 1 + = + ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ R1 R R2 R4 ❈→❝ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ ❝õ❛ ❈❤r✐st♦♣❤❡r ❇r❛❞❧❡② ✳ ✳ ✳ ✳ ✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ t❤ù ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ t❤ù ✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ t❤ù ✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈→❝ ✤÷í♥❣ ❝❛♦ h1, h2, h3, h4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚ù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ♥➔② ❧➔ ♠ët tù ❣✐→❝ ❝→♥❤ ❞✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣÷í♥❣ trá♥ ♥ë✐ t✐➳♣ tr♦♥❣ t❛♠ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇è♥ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ tr♦♥❣ ❝→❝ t❛♠ ❣✐→❝ ♥❤ä ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ t❛♠ ❣✐→❝ ✤è✐ ❞✐➺♥ ✤➾♥❤ C ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇è♥ ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ ❜è♥ t❛♠ ❣✐→❝ ♥❤ä ✤è✐ ❞✐➺♥ ✤➾♥❤ P ❚ù ❣✐→❝ s♦♥❣ t➙♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤✐♥❛ ❲❡st❡r♥ ▼❛t❤❡♠❛t✐❝❛❧ ❖❧②♠♣✐❛❞ ✷✵✵✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ABCD ♥ë✐ t✐➳♣ ✤÷đ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ∆IJK ❧➔ t❛♠ ❣✐→❝ ✈✉ỉ♥❣ ✣÷í♥❣ t❤➥♥❣ ◆❡✇t♦♥ ❝õ❛ ABCD ✈➔ W XY Z ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍➻♥❤ t❤❛♥❣ ❝➙♥ ♥❣♦↕✐ t✐➳♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ●â❝ α ❣✐ú❛ ❝➦♣ ❝↕♥❤ ✤è✐ ❞✐➺♥ ❝õ❛ tù ❣✐→❝ KLM N ✳ ✳ ✳ ✳ ✳ ✳ ✣ë ❞➔✐ ❝→❝ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ W X ✈➔ Y Z ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❉➙② ❝✉♥❣ t✐➳♣ ①ó❝ W X, Y Z ✤✐ q✉❛ ❣✐❛♦ ✤✐➸♠ ✷ ✤÷í♥❣ ❝❤➨♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✺ ✻ ✼ ✾ ✶✶ ✶✷ ✶✺ ✶✻ ✶✼ ✶✽ ✶✾ ✷✷ ✷✸ ✷✻ ✷✼ ✷✾ ✸✺ ✸✻ ✸✼ ✸✽ ✸✾ ✹✵ ✹✶ ✹✷ ✹✸ ✹✹ ✹✺ ✹✻ ✹✽ ✺✵ ✐✐ ✸✳✸ ✸✳✹ ✸✳✺ ✸✳✻ ✸✳✼ ✸✳✽ ✸✳✾ ●â❝ ϕ ❣✐ú❛ ✷ ❞➙② ❝✉♥❣ W X ✈➔ Y Z ❚ù ❣✐→❝ t✐➳♣ ①ó❝ W XY Z ✳ ✳ ✳ ✳ ✳ ✳ ự ỵ ss ❚➼♥❤ s✐♥ ❝õ❛ ♠ët ♥û❛ ❣â❝ A ✳ ✳ ✳ ✳ ❱➼ ❞ö ✸✳✸✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❱➼ ❞ö ✸✳✸✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❱➼ ❞ö ✸✳✸✳✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✺✷ ✺✸ ✺✺ ✺✻ ✺✼ ✺✽ ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✐✈ ▼ð ✤➛✉ ỵ Ptt tữỡ ữỡ ỵ ỡ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❈→❝ ✤✐➲✉ ❦✐➺♥ ✈➲ ❝↕♥❤ ✈➔ ✤÷í♥❣ ❝❤➨♦ ✳ ✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✹ t❛♠ ❣✐→❝ ✳ ✣➦❝ tr÷♥❣ ✈➲ ❣â❝ ✈➔ ✤÷í♥❣ trá♥ ✳ ✳ ✳ ✳ ✳ ✷ ❚ù ❣✐→❝ ❝→♥❤ ❞✐➲✉ ✈➔ tù ❣✐→❝ s♦♥❣ t➙♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶ ❚ù ❣✐→❝ ❝→♥❤ ❞✐➲✉ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✶ ▼ët sè ❤➺ t❤ù❝ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✸ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❜è♥ t❛♠ ❣✐→❝ ✳ ✷✳✷ ❚ù ❣✐→❝ s♦♥❣ t➙♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ▼ët sè ✤➦❝ tr÷♥❣ ❝õ❛ tù s t trữ ợ ❝õ❛ tù ❣✐→❝ s♦♥❣ t➙♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✳ ✸ ✳ ✶✷ ✳ ✶✸ ✳ ✷✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✶ ✸✶ ✸✷ ✸✻ ✹✶ ✹✶ ✹✷ ✸ ❈→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✹✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✻✶ ✸✳✶ ✣♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥ ✈➔ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✷ ❚ù ❣✐→❝ t✐➳♣ ①ó❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✸ ❚ù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✈➔ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✐✈ ▲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 ▲✉➟♥ ✈➠♥ ❇ị✐ ✣ù❝ ❍✉② ✶ ▼ð ✤➛✉ ✶✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ♥➔② ❧➔✿ − ◆❣❤✐➯♥ ❝ù✉ s➙✉ t❤➯♠ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ t❤÷í♥❣ ➼t ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❝→❝ s→❝❤ ❤➻♥❤ ❤å❝ ð ❱✐➺t ♥❛♠✱ ♥➳✉ ❝â ❝ô♥❣ ❝❤➾ ♥â✐ ✤➳♥ ỵ Ptt tr t t tự t ữủ ợ t tữớ r ỏ ❝â ❧ỵ♣ ❝→❝ tù ❣✐→❝ ✤➦❝ ❜✐➺t ❝õ❛ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ ❣✐↔✐ t♦→♥✳ ●✐ỵ✐ t❤✐➺✉ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❝ị♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ t õ ỵ t ❝õ❛ tæ✐✳ − ❙❛✉ ❦❤✐ tr➻♥❤ ❜➔② ❣➛♥ ✷✵ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝ị♥❣ ❝→❝ t➼♥❤ ❝❤➜t ✭❝ơ♥❣ ❧➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ✮ ❝õ❛ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✱ ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ tù ❣✐→❝ ❝→♥❤ ❞✐➲✉ ✈➔ ❝õ❛ tù ❣✐→❝ s♦♥❣ t➙♠ ❝❤ó♥❣ tỉ✐ ♠✉è♥ ❦❤➥♥❣ ✤à♥❤ sü ♣❤♦♥❣ ♣❤ó ✈➔ s➙✉ s➢❝ ❝õ❛ ❤➻♥❤ ❤å❝ ❝➜♣ ❦❤✐ ❝❤ó♥❣ t❛ ❜✐➳t tê♥❣ ❤đ♣✱ ❦❤❛✐ t❤→❝ ❝→❝ ❦❤➼❛ ❝↕♥❤ ❝õ❛ ❦❤→✐ ♥✐➺♠ ❜➡♥❣ ❝→❝ ❝ỉ♥❣ ❝ư s➤♥ õ ỗ ữù ỹ ❦❤â ð tr÷í♥❣ ❚❍❈❙ ✈➔ ❚❍P❚ ❣â♣ ♣❤➛♥ ✤➔♦ t↕♦ ❤å❝ s✐♥❤ ❤å❝ ❣✐ä✐ ♠æ♥ ❍➻♥❤ ❤å❝✳ ✷✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐✱ ♥❤ú♥❣ ✈➜♥ ✤➲ ❝➛♥ ❣✐↔✐ q✉②➳t ❚r➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët tù ỗ tự t õ t ✷ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✿ ❚ù ❣✐→❝ ❝→♥❤ ❞✐➲✉✱ tù ❣✐→❝ s♦♥❣ t➙♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❤ó♥❣✳ P❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❤➺ t❤ù❝ ❧✐➯♥ q✉❛♥✳ ◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ❝❤✐❛ ữỡ ữỡ ỵ Ptt ✤✐➲✉ ❦✐➺♥ t÷ì♥❣ ✤÷ì♥❣ ❙❛✉ ❦❤✐ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ t ỵ ỡ tự ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✭t❤❛♠ ❦❤↔♦ ✈➔ ❜ê s✉♥❣ ❝❤✐ t✐➳t tr♦♥❣ ❬✶❪✱ ❬✻❪✮ ❧✉➟♥ ✈➠♥ ✷ tr➻♥❤ ❜➔② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ♥ú❛ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❝❤✐❛ ❧➔♠ ❝→❝ ❞➜✉ ❤✐➺✉ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝↕♥❤✱ ✤÷í♥❣ ❝❤➨♦✱ ❧✐➯♥ q✉❛♥ ✤➳♥ ❞✐➺♥ t➼❝❤✱ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ✈➔ ❜➔♥❣ t✐➳♣✱✳✳✳ ❈❤÷ì♥❣ ♥➔② ❜❛♦ ỗ ỵ ỡ tự ♥❣♦↕✐ t✐➳♣ ✶✳✷✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ✈➲ ❝↕♥❤ ✈➔ ✤÷í♥❣ ❝❤➨♦ ✶✳✸✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❜è♥ t❛♠ ❣✐→❝ ✶✳✹✳ ✣➦❝ tr÷♥❣ ✈➲ ❣â❝ ✈➔ ✤÷í♥❣ trá♥✳ ❈❤÷ì♥❣ ✷✳ ❚ù ❣✐→❝ ❝→♥❤ ❞✐➲✉ ✈➔ tù ❣✐→❝ s♦♥❣ t➙♠ ✣➙② ❧➔ ❤❛✐ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ❱ỵ✐ ♥❤ú♥❣ ❣✐↔ t❤✐➳t ✤➦❝ ❜✐➺t t❛ t❤✉ ✤÷đ❝ ❝→❝ ❞➜✉ ❤✐➺✉ ✤➦❝ tr÷♥❣ ❝õ❛ tù ❣✐→❝ ❝→♥❤ ❞✐➲✉ ✈➔ tù ❣✐→❝ s♦♥❣ t➙♠ ❝ò♥❣ ❝→❝ t➼♥❤ ❝❤➜t ữỡ ỗ s ự ❣✐→❝ ❝→♥❤ ❞✐➲✉ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✷✳✷✳ ❚ù ❣✐→❝ s♦♥❣ t➙♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t✳ ❈❤÷ì♥❣ ✸✳ ❈→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ❇➯♥ ❝↕♥❤ ❦❤→✐ ♥✐➺♠ tù ❣✐→❝ ♥❣♦↕✐ t ợ trữớ ủ t õ õ r➜t ♥❤✐➲✉ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t❛ ✤➲ ❝➟♣ ✤➳♥ ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❤❛② ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❣✐↔✐ t♦→♥✱ ✤â ❧➔✿ ✸✳✶✳ ✣♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥ ✈➔ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ ✸✳✷✳ ❚ù ❣✐→❝ t✐➳♣ ①ó❝ ✸✳✸✳ ❚ù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✈➔ ♣❤➨♣ ữỡ ỵ Ptt tữỡ ữỡ ỵ ỡ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❚❛ ♥❤➢❝ ❧↕✐ tù ❣✐→❝ t ữớ trỏ tự ỗ tt t ú ợ ởt ữớ trỏ tự t tỗ t tỗ t ởt ữớ trỏ t tr tự ữ ỵ r➡♥❣ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ✤â ❧➔ ❞✉② ♥❤➜t✳ ❚r♦♥❣ t♦➔♥ ❜ë ❧✉➟♥ ✈➠♥ ❝❤ó♥❣ tỉ✐ s➩ sû ❞ư♥❣ ✏tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✑ ✬ t❤❛② ❝❤♦ ❝→❝❤ ♥â✐ ✏tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ♠ët ✤÷í♥❣ trá♥✑✳ ❉➵ t❤➜② ❦❤ỉ♥❣ ♣❤↔✐ ♠å✐ tự ỗ tự t ✤â✱ ♠✉è♥ ♠ët tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❝➛♥ ♣❤↔✐ ❝â t❤➯♠ ♠ët ✭❤♦➦❝ ♠ët sè✮ ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤â✱ ♠➔ t❛ ❣å✐ ❧➔ ✏✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✑✳ ❉➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ♠ët tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ①✉➜t ❤✐➺♥ sỵ♠ ✈➔ ✤â♥❣ ✈❛✐ trá q trồ ỵ Ptt r Ptt ♠ët ❦ÿ s÷ ♥❣÷í✐ P❤→♣ ✤➣ ❝ỉ♥❣ ❜è ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ❝ô♥❣ ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ♠ët tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ♥❣❛② tø ♥➠♠ ✶✼✷✺✱ ♣❤➨♣ ❝❤ù♥❣ ♠✐♥❤ ✤➛✉ t✐➯♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ ❜ð✐ ♥❤➔ t♦→♥ ❤å❝ ❚❤ư② s tr ỵ ✳ ❚ù ❣✐→❝ ABCD ✈ỵ✐ ❝→❝ ❝↕♥❤ a, b, c, d ♥❣♦↕✐ t✐➳♣ ✭P✐t❤♦t✮ ✤÷í♥❣ trá♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ AB + CD = BC + DA✱ a + c = b + d ✭✶✳✶✮ ABCD ♥❣♦↕✐ t✐➳♣ ✤÷í♥❣ trá♥ (I)✱ ❝→❝ t✐➳♣ ❝↕♥❤ AB, BC, CD, DA ❧➔ M, N, P, Q✳ ❙✉② r❛✿ ❈❤ù♥❣ ♠✐♥❤✳ ✭❬✶❪✮✱ ●✐↔ sû ✤✐➸♠ t❤ù tü tr➯♥ ❝→❝ tù❝ ❧➔ ✹ AM = AQ✱ BM = BN ✱ CN = CP, DP = DQ✳ AB + CD = BC + DA✳ ❈ë♥❣ ✈➳ ợ t õ ỵ Ptt ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû tù ❣✐→❝ ABCD t❤ä❛ ♠➣♥ AB + CD = BC + DA✳ ❑❤æ♥❣ ♠➜t t➼♥❤ ❝❤➜t tê♥❣ q✉→t t❛ ❝♦✐ AB ≤ AD✳ ❉♦ AB + CD = BC + DA ♥➯♥ BC ≤ DC ✳ õ tỗ t Q AD, P DC s❛♦ ❝❤♦ AB = AQ ✈➔ CB = CP ✱ s✉② r❛ DP = DQ✳ ❚ø ✤â✱ ❝→❝ t❛♠ ❣✐→❝ ABQ, CBP, DP Q ❧➔ ♥❤ú♥❣ t❛♠ ❣✐→❝ ❝➙♥ ✈➔ ❝→❝ ✤÷í♥❣ ❝❛♦ tø ❜❛ ✤➾♥❤ A, C, D ❧➔ tr trỹ t P ỗ q t ♠ët ✤✐➸♠ I ✳ ❚❛ ❝â I ❝→❝❤ ✤➲✉ ❝→❝ AD, DC, CB, AB tự tỗ t ữớ trỏ t I t ú ợ tự ú ỵ ỏ õ t q ỡ ỵ Ptt ụ ự ỵ Ptt sỷ ABCD ởt tự tũ ỵ õ ữớ trỏ t ú ợ AB, AD, BC ỗ tớ t DC t↕✐ ❤❛✐ ✤✐➸♠✳ ❑❤✐ ✤â AB + DC ≥ AD + BC ✳ ❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ABCD tự t t ỵ ữ ❍➻♥❤ ✶✳✷ t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ trð t❤➔♥❤ x + y + z ≥ c + d✳ ỵ ữỡ t ữớ trá♥ (O; R) ✈➔ ✤✐➸♠ M ❝è ✤à♥❤✳ ▼ët ✤÷í♥❣ t❤➥♥❣ t❤❛② ✤ê✐ q✉❛ M ❝➢t ✤÷í♥❣ trá♥ t↕✐ ❤❛✐ ✤✐➸♠ A ✈➔ B ✳ ❑❤✐ ✤â M A.M B = M O2 − R2 = d2 − R2 ✳ ✹✼ ❈❤÷ì♥❣ ✸ ❈→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ●✐↔ sû ABCD ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✤÷í♥❣ trá♥ t➙♠ I ✱ ❜→♥ ❦➼♥❤ r✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t❛ s➩ ❧➛♥ ❧÷đt ①➨t ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥✱ ①✉➜t ❤✐➺♥ tr♦♥❣ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ✣â ❧➔✿ ◗✉❛♥ ❤➺ ❣✐ú❛ ✤ë ❞➔✐ ✤♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥ ✈➔ ❞➙② ❝✉♥❣ ♥è✐ ✷ t✐➳♣ ✤✐➸♠ ✤è✐ ❞✐➺♥✱ ❝á♥ ❣å✐ ❧➔ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝❀ ●â❝ ❣✐ú❛ ❤❛✐ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝❀ ❚ù ❣✐→❝ ♠➔ ✹ ✤➾♥❤ ❧➔ ✹ t✐➳♣ ✤✐➸♠ ❤❛② tù ❣✐→❝ t✐➳♣ ①ó❝ ✈➔ ❝→❝ ❜➔✐ t♦→♥ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✤÷đ❝ ❣✐↔✐ ❜➡♥❣ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦✳ ✸✳✶ ✣♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥ ✈➔ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ ●✐↔ sû ❆❇❈❉ ❧➔ ♠ët tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✤÷í♥❣ trá♥ (I, r)✳ ❚❛ ❣å✐ ❝→❝ ❦❤♦↔♥❣ ❝→❝❤ tø ✹ ✤➾♥❤ ✤➳♥ ❝→❝ t t t t ỵ ❜ð✐ e, f, g ✈➔ h✱ ❍➻♥❤ ✸✳✶ ❚❛ ❝â t❤➸ t➼♥❤ ❜→♥ ❦➼♥❤ r ✈➔ ❞✐➺♥ t➼❝❤ S ❝õ❛ tù ❣✐→❝ ABCD t❤❡♦ ❝→❝ ❝æ♥❣ t❤ù❝ s❛✉ r= ef g + f gh + ghe + hef e+f +g+h ✭✸✳✶✮ ❙û ❞ö♥❣ S = 12 r · (a + b + c + d) = r(e + f + g + h) ✈➔ ✭✸✳✶✮ t❛ ❝â✿ S= (e + f + g + h)(ef g + f gh + ghe + hef ) ✭✸✳✷✮ ❚r♦♥❣ ❬✶❪✱ ❍❛❥❥❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ✷ ✤÷í♥❣ ❝❤➨♦ ✹✽ p = BD, q = AC t❤❡♦ ❝→❝ ✤♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥✿ p= e+g [(e + g)(f + h) + 4f h] f +h ✭✸✳✸✮ q= f +h [(e + g)(f + h) + 4eg] e+g ✭✸✳✹✮ ✿ ✣ë ❞➔✐ ❝→❝ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ W X ✈➔ Y Z ❍➻♥❤ ✸✳✶ ❙❛✉ ✤➙② t❛ ①➨t t❤➯♠ ♠ët sè ❝æ♥❣ t❤ù❝ ❜✐➸✉ ❞✐➵♥✳ ●✐↔ sû ✤÷í♥❣ trá♥ (I, r) ♥ë✐ t✐➳♣ tù ❣✐→❝ ABCD ✈ỵ✐ ❝→❝ t✐➳♣ ✤✐➸♠ W ∈ AB ✱ X ∈ BC ✱ Y ∈ CD✱ Z ∈ DA✳ ❚❛ s➩ ❣å✐ ✷ ❞➙② ❝✉♥❣ W Y ✈➔ XZ ❧➔ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝✳ ▼ët t➼♥❤ ❝❤➜t ❦❤→ ❤❛② ✈➲ ❝→❝ ❞➙② ❝✉♥❣ ♥➔② ❧➔ ❝❤ó♥❣ ❝➢t ♥❤❛✉ t↕✐ ❣✐❛♦ ✷ ✤÷í♥❣ ❝❤➨♦ ✭❬✽❪✮✳ ❚r♦♥❣ ❜➔✐ ❜→♦ ❙♦♠❡ ♣r♦♦❢s ♦❢ ❛ t❤❡♦r❡♠ ♦♥ q✉❛❞r✐❧❛t❡r❛❧✱ ▼❛t❤✳ ▼❛❣✳✱ ✸✺✭✶✾✻✷✮✱ ✷✽✾✲✷✾✹✱ ❑✳❚❛♥ ✤➣ ✤÷❛ r❛ ✾ ♣❤➨♣ ❝❤ù♥❣ ♠✐♥❤ ❦❤→❝ ♥❤❛✉✳ ▼➺♥❤ ✤➲ ✸✳✶✳ ❈→❝ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ k = W Y ✈➔ l = XZ ✤÷đ❝ t➼♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ k= l= 2(ef g + f gh + ghe + hef ) (e + f )(g + h)(e + g)(f + h) 2(ef g + f gh + ghe + hef ) (e + h)(f + g)(e + g)(f + h) ✹✾ ◆➳✉ I ❧➔ t➙♠ ♥ë✐ t✐➳♣✱ β; γ ❧➔ ❝→❝ ❣â❝ tr➯♥ ❍➻♥❤ ✸✳✶✳ ⑩♣ ❞ö♥❣ ✣à♥❤ ỵ s t W Y I ự ♠✐♥❤✳ k = 2r2 − 2r2 cos(2β + 2γ) = 2r2 (1 − cos(2β + 2γ)) ❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❝ë♥❣ ♥❤➟♥ ✤÷đ❝ k2 = − cos 2β cos 2γ + sin 2β sin 2γ 2r2 ❚ø ❝æ♥❣ t❤ù❝ ❣â❝ ♥❤➙♥ ✤æ✐ t❛ ❝â − tan2 β r2 − r2 tan2 β r2 − f = = + tan β r + r2 tan2 β r2 + f 2rf tan β = sin 2β = + tan2 β r2 + f cos = tữỡ tỹ ợ γ ✱ t❤❛② f ❜ð✐ g✳ ◆❤÷ ✈➟② t❛ ❝â k2 r2 − f r2 − g 2rf 2rg (f + g)2 = 1− · + · = 2r · 2r2 r + f r2 + g r2 + f r2 + g (r + f ) (r2 + g ) ❉♦ ✤â✱ k2 = 2r2 (r2 +(ff 2+) (rg)2 + g2) ✳ ❇➙② ❣✐í t❛ ♣❤➙♥ t➼❝❤ r2 + f ✈ỵ✐ r t➼♥❤ t❤❡♦ ✭✸✳✶✮✿ ef g + f gh + ghe + hef + f (e + f + g + h) e+f +g+h e f g + f h + gh + f + f gh + f + f g + f h = e+f +g+h (e + f )[g(f + h) + f (h + f )] = e+f +g+h (e + f )(f + g)(f + h) = e+f +g+h + g)(g + h) t÷ì♥❣ tü✱ r2 + g2 = (e +eg)(f ✳ ◆❤÷ ✈➟②✱ +f +g+h r2 + f = ❍♦➔♥ t♦➔♥ (f + g)2 (e + f + g + h)2 (e + f )(f + g)(f + h)(e + g)(f + g)(g + h) e+f +g+h k = 2r2 · ✳ ❉ü❛ ✈➔♦ ✭✸✳✶✮✱ (e + f )(f + h)(h + g)(g + e) k = 2r2 ❑➨♦ t❤❡♦ r t❤✉ ✤÷đ❝ k= 2(ef g + f gh + ghe + hef ) (e + h)(f + g)(e + g)(f + h) ❦❤û ✺✵ ❈ỉ♥❣ t❤ù❝ ✤è✐ ✈ỵ✐ ❧ ữủ s ỵ f h tr♦♥❣ ❝æ♥❣ t❤ù❝ t➼♥❤ k ✳ ✿ ❉➙② ❝✉♥❣ t✐➳♣ ①ó❝ W X, Y Z ✤✐ q✉❛ ❣✐❛♦ ✤✐➸♠ ✷ ✤÷í♥❣ ❝❤➨♦ ❍➻♥❤ ✸✳✷ ❚ø ♠➺♥❤ ✤➲ tr➯♥ t❛ ❝â ♥❣❛② ✷ ❤➺ q✉↔ q✉❛♥ trå♥❣ ❍➺ q✉↔ ✸✳✶✳ ❚r♦♥❣ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✈ỵ✐ ❝↕♥❤ a, b, c, d t✛ sè ❤❛✐ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ t❤ä❛ ♠➣♥ k l = bd ✳ ac ❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ❦➳t q✉↔ tr♦♥❣ ▼➺♥❤ ✤➲ ✸✳✶✱ s❛✉ ✤â rót ❣å♥✿ (e + h)(f + g) k = = l (e + f )(h + g) db ac ❙✉② r❛ ❤➺ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✸✳✷✳ ◆➳✉ e, f, g, h ❧➔ ❝→❝ ✤♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥ tr♦♥❣ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ t❤➻ ❣â❝ ϕ ❣✐ú❛ ❤❛✐ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ ✤÷đ❝ t➼♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ sin ϕ = (e + f + g + h)(ef g + f gh + ghe + hef ) (e + f )(f + g)(g + h)(h + e) ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ s➩ t❤❛② ❣â❝ ϕ ❜ð✐ ❝→❝ ❣â❝ x, y, z, w ♥❤÷ tr➯♥ ❍➻♥❤ ✸✳✸✳ ❚ø tê♥❣ ❝→❝ ❣â❝ tr♦♥❣ ❝→❝ tù ❣✐→❝ BW P X ✈➔ DY P Z t❛ ❝â w + x + ϕ + B = 2π ❀ y + z + ϕ + D = 2π ✳ ❈ë♥❣ ❧↕✐ w + x + y + z + 2ϕ + B + D = 4π ✭✸✳✼✮ ✺✶ ✿ ●â❝ ϕ ❣✐ú❛ ✷ ❞➙② ❝✉♥❣ W X ✈➔ Y Z ❍➻♥❤ ✸✳✸ ❉♦ w + y = π ✈➔ x + z = π ✳ ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t❤➻ ✤÷đ❝ 2π + 2ϕ + B + D = 4π ⇐⇒ B + D = 2π − 2ϕ √ B+D ♥➯♥ ❱➻ ❞✐➺♥ t➼❝❤ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❜➡♥❣ S = abcd sin √ √ S = abcd sin(π − ϕ) = abcd sin ϕ✳ ❉♦ ✤â✱ sin ϕ = √ S = abcd ✭✸✳✽✮ (e + f + g + h)(ef g + f gh + ghe + hef ) (e + f )(f + g)(g + h)(h + e) ❈æ♥❣ t❤ù❝ ❝✉è✐ t❛ ✤➣ sû ❞ö♥❣ ✭✸✳✷✮✳ ❍➺ q✉↔ ✸✳✷✳ ❍❛✐ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ tr♦♥❣ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✈✉ỉ♥❣ ❣â❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tù ❣✐→❝ ❧➔ s♦♥❣ t➙♠✳ ❚r♦♥❣ ♠å✐ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ t❛ ❝â B + D = 2π − 2ϕ✳ ▼➦t π ❦❤→❝✱ W Y ⊥ XZ ⇐⇒ ϕ = ⇐⇒ B + D = π ✳ ❚ù ❣✐→❝ ♥➔② ❧➔ tù ❣✐→❝ ❈❤ù♥❣ ♠✐♥❤✳ t t ủ ợ t ữủ ❧➔ tù ❣✐→❝ s♦♥❣ t➙♠✳ ✸✳✷ ❚ù ❣✐→❝ t✐➳♣ ①ó❝ ❚ù ❣✐→❝ ❝â ❝→❝ ✤➾♥❤ ❧➔ ❝→❝ t✐➳♣ ✤✐➸♠ W, X, Y, Z ✤÷đ❝ ❣å✐ ❧➔ tù ❣✐→❝ t✐➳♣ ①ó❝✱ ❍➻♥❤ ✸✳✹✳ ❚❛ s➩ t➻♠ ❝æ♥❣ t❤ù❝ t➼♥❤ ❞✐➺♥ t➼❝❤ tù ❣✐→❝ t✐➳♣ ①ó❝ t❤❡♦ ❝→❝ ✤♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥✳ ✺✷ ✿ ❚ù ❣✐→❝ t✐➳♣ ①ó❝ W XY Z ❍➻♥❤ ✸✳✹ ▼➺♥❤ ✤➲ ✸✳✸✳ ◆➳✉ e, f, g ✈➔ h ❧➔ ❝→❝ ✤♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥ tr♦♥❣ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ t❤➻ tù ❣✐→❝ t✐➳♣ ①ó❝ ❝â ❞✐➺♥ t➼❝❤ (e + f + g + h)(ef g + f gh + ghe + hef )5 Stx = (e + f )(e + g)(e + h)(f + g)(f + h)(g + h) ❚❛ ❝â SABCD = pq sin θ✳ ❚÷ì♥❣ tü✱ SW XY Z = 12 k.l sin ϕ✱ tr♦♥❣ ✤â k, l ❧➔ ✷ ✤÷í♥❣ ❝❤➨♦✱ ϕ ❧➔ ❣â❝ ❣✐ú❛ ✷ ✤÷í♥❣ ❝❤➨♦ W Y, XZ ✳ ❚ø ❝→❝ ❝æ♥❣ t❤ù❝ ✤➣ ❝â t❛ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❈â ♠ët ❦➳t q✉↔ r➜t ✤è✐ ①ù♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐✱ ♥❣♦↕✐ t✐➳♣ ❝õ❛ tù ❣✐→❝ s♦♥❣ t➙♠✳ ✣â ❧➔ ❤➺ t❤ù❝ ❋✉ss✿ ▼➺♥❤ ỵ ss ự s t ợ r, R ❧➛♥ ❧÷đt ❧➔ ❜→♥ ❈❤ù♥❣ ♠✐♥❤✳ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣✱ ♥❣♦↕✐ t✐➳♣✱ d ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ✷ t➙♠✳ ❑❤✐ ✤â 1 + = (R + d)2 (R − d)2 r2 ✭✸✳✾✮ ❈â ♥❤✐➲✉ ự ỵ t ữỡ q t ỵ Pt ữỡ ✤↔♦✱✳✳✳ Ð ✤➙② ❝❤ó♥❣ tỉ✐ ❝❤å♥ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ✤ì♥ ❣✐↔♥ ♥❤➜t ❝❤➾ ❞ị♥❣ ❦✐➳♥ t❤ù❝ ❤➻♥❤ ❤å❝ ❊✉❝❧✐❞ ✤ì♥ t❤✉➛♥✳ P❤➨♣ ❝❤ù♥❣ ♠✐♥❤ ♥➔② ✤÷đ❝ ❏✳❈✳❙❛❧❛③❛r tr➻♥❤ ❜➔② tr♦♥❣ ❋✉ss t❤❡♦r❡♠✱ ▼❛t❤✳ ●❛③❡tt❡✱ ✾✵✭✷✵✵✻✮✱ ✸✵✻✲✸✵✼✳ ❍➻♥❤ ✸✳✺✳ ❱➻ A + C = 180◦ ♥➯♥ α + β = 90◦✳ sỷ ữớ trỏ (I, r) t ú ợ AB, BC ð J, K ✳ ❚❛ ❝â ✷ t❛♠ ❣✐→❝ ổ ỗ AJI IKC ự ự ỵ ss õ AJI = IKC = 90 JAI = KIC ữ ỵ r➡♥❣ IJ = IK = r✮✳ ◆➳✉ ❞ü♥❣ t❛♠ ❣✐→❝ ✈✉ỉ♥❣ JIC ′ , IC ′ = IC ♥❤÷ ❍➻♥❤ ✸✳✺ t❛ ✤÷đ❝ t❛♠ ❣✐→❝ AIC ′ ✈✉ỉ♥❣ ð I ✳ ❚ø ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝ ♥➔②✿ 2SAIC ′ = r.AC ′ = AI.IC ′ ⇔ r.AC = AI.IC ✳ ❱➟② r2 AC = AI · IC ❤❛② r2 AI + IC = AI IC ✳ ❙✉② r❛ AI + IC 1 = = + ✭✸✳✶✵✮ 2 2 r AI · IC AI IC ❈→❝ ✤÷í♥❣ t❤➥♥❣ AI ✈➔ IC ❝➢t (O, R) t÷ì♥❣ ù♥❣ ð F ✈➔ E ✳ ❱➻ F OD + EOD = 2(F AD + ECD) = 2(α + β) = 180◦ ♥➯♥ EF ❧➔ ✤÷í♥❣ ❦➼♥❤ ❝õ❛ (O, R)✳ ❚❤❡♦ ❝ỉ♥❣ t❤ù❝ ❜➻♥❤ ♣❤÷ì♥❣ ❝→❝ ✤÷í♥❣ tr✉♥❣ t✉②➳♥ tr♦♥❣ t❛♠ ❣✐→❝ t❛ ❝â ✿ IE + IF = 2IO2 + EF = R2 + d2 ◆❤í ❝ỉ♥❣ t❤ù❝ ♣❤÷ì♥❣ t➼❝❤✿ ✭✸✳✶✶✮ I = CI.EI = R2 − d2 ⇒ AI F I = CI EI = R2 − d2 ✱ ❦➳t ❤đ♣ ✈ỵ✐ ✭✸✳✶✶✮ t❛ ❝â R + d2 EI EI + F I F I2 + = = + = AI CI (R2 − d2 ) (R2 − d2 )2 (R2 − d2 )2 (R2 − d2 )2 ✺✹ ❚ø ✤â✱ t❤❛② ✈➔♦ ✸✳✶✵✿ R + d2 (R + d)2 + (R − d)2 1 = + = = r2 (R + d)2 (R − d)2 (R2 − d2 )2 (R2 − d2 )2 ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚❛ ♠✉è♥ t➻♠ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ❝→❝ ❣â❝ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ t❤❡♦ ❝→❝ t t t ổ tự ỗ ❦➲♥❤ ♥❤÷♥❣ ✤➲✉ ❝â q✉② ❧✉➟t r✐➯♥❣✳ ✣✐➲✉ q✉❛♥ trå♥❣ ❧➔ tø ✤➙② t❛ ❝â t❤➸ rót r❛ ✤÷đ❝ ❝→❝ t❤ỉ♥❣ t✐♥ ❤➻♥❤ ❤å❝ tr♦♥❣ ♠ët sè tr÷í♥❣ ❤đ♣ ❝ư t❤➸✳ ▼➺♥❤ ✤➲ ✸✳✺✳ ABCD ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✈ỵ✐ ✤ë ❞➔✐ ❝→❝ ✤♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥ e, f, g, h ①✉➜t ♣❤→t tø ❝→❝ ✤➾♥❤✳ ❑❤✐ ✤â ❈❤ù♥❣ ♠✐♥❤✳ sin A = ef g + f gh + ghe + hef (e + f )(e + g)(e + h) sin B = ef g + f gh + ghe + hef (f + e)(f + g)(f + h) sin C = ef g + f gh + ghe + hef (g + e)(g + f )(g + h) sin D = ef g + f gh + ghe + hef (h + e)(h + f )(h + g) ỵ ữ ỵ ổ s W ZI 4e2 r2 W Z = 2r (1 − cos 2α) = r + e2 tr♦♥❣ ✤â✱ t❛ ❞ị♥❣ ❝ỉ♥❣ t❤ù❝ ✸✳✺ ✈ỵ✐ t❤❛② ✤ê✐ f ←→ e✳ ❚✐➳♣ t❤❡♦✱ sû g)(e + h) ✱ ❞♦ t➼♥❤ ✤è✐ ①ù♥❣ t❛ t❤❛② ❞ö♥❣ ✭✸✳✶✮ ✈➔ r2 + e2 = (e +ef+)(ef + +g+h ✤ê✐ g ←→ e t❤➻ ✤÷đ❝✿ ef g + f gh + ghe + hef e+f +g+h W Z = 4e2 · · e+f +g+h (e + f )(e + g)(e + h) ❉♦ ✤â✱ W Z = 2e · ❝õ❛ s✐♥✿ ef g + f gh + ghe + hef ✳ ❈✉è✐ ❝ò♥❣✱ tø ✤à♥❤ ♥❣❤➽❛ (e + f )(e + g)(e + h) WZ A sin = = e ef g + f gh + ghe + hef (e + f )(e + g)(e + h) ✺✺ P❤➛♥ ❝á♥ ❧↕✐ t÷ì♥❣ tü ❤♦➦❝ sû ❞ư♥❣ t➼♥❤ ❝❤➜t ✤è✐ ①ù♥❣✳ ✿ ❚➼♥❤ s✐♥ ❝õ❛ ♠ët ♥û❛ ❣â❝ A ❍➻♥❤ ✸✳✻ ✸✳✸ ❚ù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✈➔ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ◆❤✐➲✉ ❜➔✐ t♦→♥ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✤÷đ❝ ❣✐↔✐ ❦❤→ ♣❤ù❝ t↕♣ ❦❤✐ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝➜♣ ✭t❛♠ ❣✐→❝ ỗ trử ữỡ ỹ ố ỹ tr ♥➯♥ ✤ì♥ ❣✐↔♥ ✈➔ ❤✐➸♥ ♥❤✐➯♥ ♥➳✉ t❛ ❜✐➳t sû ❞ö♥❣ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦✳ ✣â ❧➔ ✈➻ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝â ♥❤✐➲✉ t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t ❦❤✐ t❛ ❧➔♠ ✈✐➺❝ ợ ữớ trỏ ợ ỹ ♣❤÷ì♥❣ t➼❝❤ t❤➼❝❤ ❤đ♣ t❛ ❝â t❤➸ ❝❤✉②➸♥ ❜➔✐ t♦→♥ ✤÷í♥❣ trá♥ t❤➔♥❤ ❜➔✐ t♦→♥ ✤÷í♥❣ t❤➥♥❣✱ ❝→❝ ✤÷í♥❣ trá♥ t✐➳♣ ①ó❝ t❤➔♥❤ ❝→❝ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣✱✈✈✳✳✳Ð ✤➙② ❝❤ó♥❣ tæ✐ ❦❤æ♥❣ ❝â ✤✐➲✉ ❦✐➺♥ tr➻♥❤ ❜➔② t♦➔♥ ❜ë ♥ë✐ ❞✉♥❣ ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ♠➔ ❝❤➾ ♥➯✉ ♠ët sè ✈➼ ❞ö ✤➸ ♣❤➙♥ t➼❝❤ ❝→❝❤ t✐➳♣ ❝➟♥ ❝→❝ ❜➔✐ t♦→♥ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤à❝❤ ✤↔♦✳ ❈ơ♥❣ ❝❤➼♥❤ tø ù♥❣ ❞ư♥❣ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ♠➔ t❛ ❝â t❤➸ s→♥❣ t↕♦ r❛ ❝→❝ ❜➔✐ t♦→♥ ♠ỵ✐✳ tự t ỵ E = AB ∩ CD✱ F = AD ∩ BC ✳ ❑❤✐ õ tỗ t ởt ữớ trỏ t ú ợ ố ✤÷í♥❣ trá♥ (EAD), (EBC), (F AB), (F CD)✳ ❱➼ ❞ư ✸✳✸✳✶✳ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ X, Y, Z, T ❧➔ ❝→❝ t✐➳♣ ✤✐➸♠ t÷ì♥❣ ù♥❣ ❝õ❛ AB ✱ BC ✱ CA✱ AD ợ ữớ trỏ (I) t ỹ I ữỡ t ỵ f2I ✳ ❚❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ f2I ✱ ❝→❝ ✤÷í♥❣ trá♥ (EAD)✱ (EBC)✱ (F AB)✱ (F CD) t÷ì♥❣ ù♥❣ t❤➔♥❤ ❝→❝ ✤÷í♥❣ trá♥ ❊✉❧❡r ❝õ❛ ❝→❝ t❛♠ ❣✐→❝ s❛✉✿ ∆T XZ, ∆Y XZ, ∆T XY, ∆T ZY ✳ ❍➻♥❤ ✸✳✼✿ ❱➼ ❞ư ✸✳✸✳✶ ❉♦ ❝→❝ ✤÷í♥❣ trá♥ (EAD)✱ (EBC)✱ (F AB)✱ (F CD) ỗ q t q tự t♦➔♥ ♣❤➛♥ ABCDEF ♥➯♥ ❝→❝ ✤÷í♥❣ trá♥ ❊✉❧❡r ❝õ❛ ❝→❝ t❛♠ ❣✐→❝ ∆T XZ, ∆Y XZ, ∆T XY, ∆T ZY ỗ q t J õ t ✤÷í♥❣ trá♥ ♥➔② ✤➲✉ ❝â ❜→♥ ❦➼♥❤ r ✷ ♥➯♥ ữớ trỏ (J, r) t ú ợ ố ữớ trá♥ ❊✉❧❡r ♥â✐ tr➯♥✳ ⑩♣ ❞ö♥❣ ♣❤➨♣ −1 t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ♥❣❤à❝❤ ✤↔♦ f2I ❈❤♦ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ABCD✱ E = AB ∩ CD✱ F = AD∩BC ✳ ●å✐ M ❧➔ ✤✐➸♠ ▼✐q✉❡❧ ❝õ❛ tù ❣✐→❝ t♦➔♥ ♣❤➛♥ ABCDEF ✳ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝ü❝ M ✱ ữỡ t tũ ỵ A A B → B ′ ✱ C → C ′ , D → D′ ✳ ❈❤ù♥❣ ♠✐♥❤ A′ B ′ C ′ D′ ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ❱➼ ❞ö ✸✳✸✳✷✳ ❈❤ù♥❣ t tỗ t ởt ữớ trỏ (C) t ú ợ ữớ trỏ (M AB)✱ (M BC)✱ (M CD)✱ (M DA) ♥➯♥ ↔♥❤ ❝õ❛ (C) q✉❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ✤â s➩ ❧➔ ✤÷í♥❣ trá♥ t ú ợ ữớ t A B B ′ C ′ ✱ C ′ D′ ✱ D′ A′ ✳ ◆❣❤➽❛ ❧➔ A′ B ′ C ′ D′ ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ❱➼ ❞ö ✸✳✸✳✸✳ ❈❤♦ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ABCD ✈➔ P = AC ∩ BD✳ P❤➨♣ ✺✼ ♥❣❤à❝❤ ✤↔♦ ❝ü❝ P ♣❤÷ì♥❣ t➼❝❤ ❜➜t ❦ý ❜✐➳♥ A, B, C, D ❧➛♥ ❧÷đt t❤➔♥❤ A′ , B ′ , C ′ , D′ ✳ ❑❤✐ ✤â A′ B ′ C ′ D′ ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ AB ✱ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦✿ A′ B ′ = k P A.P B BC CD DA B ′C ′ = k , C ′D′ = k , D ′ A′ = k ✳ ❙✉② r❛ P B.P C PC · PD PD · PA CD AB + P A.P B P C · P D 2AB k 2CD = sin AP B + P A.P B sin AP B P C.P D · sin DP C AB CD k + = sin AP B · SAP B SDP C A′ B ′ + C ′ D′ = k ❍➻♥❤ ✸✳✽✿ ❱➼ ❞ư ✸✳✸✳✸ ●å✐ t✐➳♣ ✤✐➸♠ ❝õ❛ (I) ✈ỵ✐ AB ✱ BC ✱ CD✱ DA ❧➛♥ ❧÷đt ❧➔ X, Y, Z, T ỵ AX = AT = x BX = BY = y ✱ CY = CZ = z ✱ DZ = DT = t✳ ❚ø A ❦➫ Ag DC ✱ Ag ∩ XZ = J ✳ ❉➵ t❤➜② AXJ = DZX = AJX SAP B AP AJ x = = = ✳ ❚÷ì♥❣ tü✱ t❛ ❝â SBP C PC CZ z t SDP A x = , = ❚ø ✤â s✉② r❛ x SAP B y ♥➯♥ AX = AJ = x✳ ❙✉② r❛ SBP C y SCP D = , SCP D t SDP A SBP C SCP D SDP A SAP B = = = = q xy yz zt tx ✺✽ CD x+y z+t 1 1 = + = · + + + ✳ SDP C qxy qzt q x y z t AP B ◆❤÷ ✈➟②✱ A′B ′ + C ′D′ 2qk sin AP B x1 + y1 + z1 + 1t ✳ ❍♦➔♥ t♦➔♥ t÷ì♥❣ tü s✉② r❛✿ A′B ′ + C ′D′ = A′D′ + B ′C ′✳ ❚ù ❣✐→❝ A′ B ′ C ′ D′ ♥❣♦↕✐ t ữủ t ỵ Ptt ữủ SAB + tỗ t ởt ữớ trỏ t ú ợ ữớ t AB B C C ′ D′ ✱ D′ A′ ♥➯♥ ↔♥❤ ❝õ❛ γ ′ q✉❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝ü❝ P s➩ ❧➔ ✤÷í♥❣ trỏ t ú ợ ữớ trỏ t t ❣✐→❝ P AB ✱ P BC ✱ P CA✱ P AD ♥➯♥ t❛ ❝â ❜➔✐ t♦→♥✿ ❈❤♦ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ABCD✳ ●å✐ P = AC ∩ BD ✳ ❍➣② ự tọ tỗ t ởt ữớ trỏ t ú ợ ❝→❝ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ ∆P AB, ∆P BC, ∆P CA, ∆P DA✳ ❱➼ ❞ö ✸✳✸✳✹✳ ❚✐➳♣ t❤❡♦ ♥➳✉ ❣å✐ X, Y, Z, T ❧➛♥ ❧÷đt ❧➔ t✐➳♣ ✤✐➸♠ ❝õ❛ (I) ✈➔ AB ✱ BC ✱ CD✱ DA✳ ❳➨t ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝ü❝ I ✱ ♣❤÷ì♥❣ t➼❝❤ R2 , fRI : A → A′ , B → B ′ , C → C ′ , D → D′ , P → P ′ ✳ ❉➵ t❤➜② A′ , B ′ , C ′ , D′ t❤ù tü ❧➔ tr✉♥❣ ✤✐➸♠ T X, XY, Y Z, ZT ỵ K = T X ∩ Y Z ✱ J = XY ZT ỵ rr P I tr JK ỗ tớ P ′ ❧➔ ✤✐➸♠ ▼✐q✉❡❧ ❝õ❛ tù ❣✐→❝ t♦➔♥ ♣❤➛♥ XY ZT KJ ✳ ❚❛ ❝â ❤❛✐ ❜➔✐ t♦→♥✿ ❱➼ ❞ö ✸✳✸✳✺ ✈➔ ❱➼ ❞ö ✸✳✸✳✻ s❛✉✿ ❱➼ ❞ö ✸✳✸✳✺✳ ❈❤♦ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ABCD✱ E = AB ∩ CD✱ ❍➻♥❤ ✸✳✾✿ ❱➼ ❞ö ✸✳✸✳✺ ✺✾ F = AD∩BC ✳ ●å✐ M ❧➔ ✤✐➸♠ ▼✐q✉❡❧ ❝õ❛ tù ❣✐→❝ t♦➔♥ ♣❤➛♥ ABCDEF ✱ ❝→❝ ✤✐➸♠ X, Y, Z, T ❧➛♥ ❧÷đt ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ AB, BC, CD, DA✳ ❑❤✐ õ tỗ t ởt ữớ trỏ t ú ợ ữớ trá♥ ♥❣♦↕✐ t✐➳♣ ❝→❝ t❛♠ ❣✐→❝ ∆M XY, ∆M Y Z, M ZT, M T X ợ ỵ ♥❤÷ tr♦♥❣ ✈➼ ❞ư ✸✳✸✳✺✳ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝ü❝ M ✱ ♣❤÷ì♥❣ t➼❝❤ ❜➜t ❦ý✱ ❜✐➳♥ X, Y, Z, T t÷ì♥❣ ù♥❣ t❤➔♥❤ X ′ , Y ′ , Z ′ , T ′ ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ X ′ Y ′ Z ′ T ′ ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ❱➼ ❞ư ✸✳✸✳✻✳ ❈❤÷ì♥❣ ❜❛ ❝❤➾ tr➻♥❤ ❜➔② ✸ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ✣â ❧➔✿ ✤♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥✱ tù ❣✐→❝ t✐➳♣ ①ó❝ ✈➔ ù♥❣ ❞ö♥❣ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ✈➔♦ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ t q ữủ tự ợ õ ❦❤✐ t➼♥❤ t♦→♥ ❝→❝ ②➳✉ tè ❝õ❛ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣❀ ♠✐♥❤ ❤å❛ ✤÷đ❝ ÷✉ t❤➳ ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❦❤✐ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ✻✵ ❑➳t ❧✉➟♥ ❝õ❛ ❧✉➟♥ ✈➠♥ ▲✉➟♥ ✈➠♥ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉ ✶✳ ❍➺ t❤è♥❣ ❧↕✐ ❝→❝ ❞➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ✣➙② ❧➔ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ♠ð rë♥❣ ✈➔ ❜ê s✉♥❣ s➙✉ s➢❝ ✈➔♦ ❝→❝ ❦✐➳♥ t❤ù❝ ❝❤✉♥❣ ✈➲ tù ❣✐→❝ ❝õ❛ t→❝ ❣✐↔ ❧✉➟♥ ✈➠♥✳ ✷✳ ●✐ỵ✐ t❤✐➺✉ ❤❛✐ ❧♦↕✐ tù ❣✐→❝ ✤➦❝ ❜✐➺t✿ ❚ù ❣✐→❝ ❝→♥❤ ❞✐➲✉ ✈➔ tù ❣✐→❝ s♦♥❣ t➙♠✳ ❑❤✐ t➻♠ ❤✐➸✉ ✈➲ ❤❛✐ ❧♦↕✐ tù ❣✐→❝ ♥➔②✱ ❧✉➟♥ ✈➠♥ ✤➣ ❝➟♣ ♥❤➟t ❝→❝ ❞➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ✤÷đ❝ ♣❤→t ❤✐➺♥ tr♦♥❣ ♥❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙② ✭tø ✷✵✵✾ ✤➳♥ ✷✵✶✹✮ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ▼✐♥❝✉❧❡t❡ tr♦♥❣ ❬✻❪✱ ❏♦s❡❢ss♦♥ tr♦♥❣ ❬✷❪✱ ❬✹❪✱ ❬✺❪✳ ✸✳ ❚r➻♥❤ ❜➔② ♠ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ tù ❣✐→❝ t✐➳♣ ①ó❝✱ ✤➳♥ ù♥❣ ú ỵ ❝æ♥❣ t❤ù❝ t➼♥❤ t♦→♥ tr➯♥ ❝→❝ ✤ë ❞➔✐ ❝õ❛ tù ❣✐→❝ ✈➔ ❝→❝ ù♥❣ ❞ö♥❣ ❝â ❤✐➺✉ q✉↔ ❝õ❛ ♣❤➨♣ ú tổ t õ ữợ ❝ù✉ t✐➳♣ t❤❡♦✿ − ❚➻♠ t❤➯♠ ✈➲ ❝→❝ ❜➔✐ t♦→♥ t÷ì♥❣ tü tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥✿ ❚ù ❞✐➺♥ trü❝ t➙♠✱ tù ❞✐➺♥ ♥ë✐ ♥❣♦↕✐ t✐➳♣✱✈✈✳✳✳✳ − ❙û ❞ö♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ❤➻♥❤ t❤➼❝❤ ❤ñ♣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ s➙✉ ✈➲ ❝→❝ ❜➔✐ t♦→♥ ✤❛♥❣ ①➨t✳ ▼➦❝ ❞ị ✤➣ r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳✱ ❦❤✐➳♠ ❦❤✉②➳t✳ rt sỹ õ ỵ s t ổ ỗ ❧➔♠ ❝❤♦ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❤♦➔♥ ❝❤➾♥❤ ✈➔ ❝â ➼❝❤ ❤ì♥✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ✻✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▼♦✇❛❢❢❛q ❍❛❥❥❛ ✭✷✵✵✽✮✱ ✧❆ ❝♦♥❞✐t✐♦♥ ❢♦r ❛ ❝✐r❝✉♠s❝r✐♣t✐❜❧❡ ◗✉❛♥✲ ❞r✐❧❛t❡r❛❧ t♦ ❜❡ ❝②❝❧❝✐✧✱ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠ ✱ ✽✱ ♣♣✳ ✶✵✸✲✶✵✻ ❬✷❪ ▼❛rt✐♥ ❏♦s❡❢ss♦♥ ✭✷✵✶✵✮✱✧❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❇✐❝❡♥tr✐❝ ◗✉❛♥❞r✐✲ ❧❛t❡r❛❧✧✱ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✱ ✶✵✱ ♣♣✳ ✶✻✺✲✶✼✸ ❬✸❪ ▼❛rt✐♥ ❏♦s❡❢ss♦♥ ✭✷✵✶✵✮✱ ✧❈❛❧❝✉❧❛t✐♦♥s ❝♦♥❝❡r♥✐♥❣ t❤❡ t❛♥❣❡♥t ❧❡♥❣t❤s ❛♥❞ t❛♥❣❡♥❝② ❝❤♦r❞s ♦❢ ❛ t❛♥❣❡♥t✐❛❧ q✉❛❞r✐❧❛t❡r❛❧✧✱ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✱ ✶✵✱ ♣♣✳ ✶✶✾✲✶✸✵✳ ❬✹❪ ▼❛rt✐♥ ❏♦s❡❢ss♦♥ ✭✷✵✶✶✮✱ ✧❲❤❡♥ ✐s ❛ ❚❛♥❞❣❡♥t✐❛❧ ◗✉❛♥❞r✐❧❛t❡r❛❧ ❛ ❑✐t❡✧✱ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✱ ✶✶✱ ♣♣✳ ✶✻✺✲✶✼✹ ❬✺❪ ▼❛rt✐♥ ❏♦s❡❢ss♦♥ ✭✷✵✶✹✮✱ ✧❆♥❣❧❡ ❛♥❞ ❈✐r❝❧❡ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❚❛♥❣❡♥t✐❛❧ ◗✉❛♥❞r✐❧❛t❡r❛❧s✧✱ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✱ ✶✹✱ ♣♣✳ ✶✲✶✸ ❬✻❪ ◆✐❝✉s♦r ▼✐♥❝✉❧❡t❡ ✭✷✵✵✾✮✱ ✧❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❛ ❚❛♥❞❣❡♥t✐❛❧ ◗✉❛♥❞r✐❧❛t❡r❛❧✧✱ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✱ ✾✱ ♣♣✳ ✶✶✸✲✶✶✽ ❬✼❪ ■✳ ❱❛✐♥s❤t❡✐♥ ✭✶✾✾✻✮✱ ✧Pr♦❜❧❡♠ ▼✶✺✷✹✧✱ ❑✈❛♥t ✭✐♥ ❘✉ss✐❛♥✮ ♥♦✳✸✱ ♣♣✳ ✷✺✲✷✻✳ ❬✽❪ P✳ ❨✐✉ ✭✶✾✾✽✮✱ ✧❊✉❝❧✐❞❡❛♥ ●❡♦♠❡tr② ◆♦t❡s✧✱ ❋❧♦r✐❞❛ ❆t❧❛♥t✐❝ ❯♥✐✲ ✈❡rs✐t② ▲❡❝t✉r❡ ◆♦t❡s✳ ...ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - BÙI ĐỨC HUY TỨ GIÁC NGOẠI TIẾP VÀ CÁC VẤN ĐỀ LIÊN QUAN Chuyên ngành: Phương pháp Toán sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN