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 R3 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 ✳ ✳ ✳ ✳ ✳ ✳ ❈❤ù♥❣ 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 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 ú ✤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 ▲✉➟♥ ✈➠♥ ❇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✐ ✤➳♥ ✣à♥❤ ỵ Ptt tr t t tự t ữủ ợ t tữớ r ỏ õ ❧ỵ♣ ❝→❝ tù ❣✐→❝ ✤➦❝ ❜✐➺t ❝õ❛ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ ❣✐↔✐ t♦→♥✳ ●✐ỵ✐ t❤✐➺✉ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❝ị♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ✤➦❝ t õ ỵ t 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➤♥ ❝â✳ ỗ ữù ỹ õ 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♥❣ trỏ t t ữỡ 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 ợ ỳ tt t t t ữủ ❝→❝ ❞➜✉ ❤✐➺✉ ✤➦❝ 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 rt ♥❤✐➲✉ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t❛ ✤➲ ❝➟♣ ✤➳♥ ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❤❛② ✤÷đ❝ sû ❞ö♥❣ tr♦♥❣ ❣✐↔✐ t♦→♥✱ ✤â ❧➔✿ ✸✳✶✳ ✣♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥ ✈➔ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ ✸✳✷✳ ❚ù ❣✐→❝ t✐➳♣ ①ó❝ ✸✳✸✳ ❚ù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✈➔ ♣❤➨♣ ♥❣❤à❝❤ 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✐ tù ỗ tự t õ ♠✉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 ởt ❦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♥❣ ✈➳ ✈ỵ✐ t õ ỵ Ptt 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 ❝õ❛ tù ❣✐→❝✳ ❱➟② tỗ t ữớ trỏ t I t ú ợ ❝↕♥❤ tù ❣✐→❝✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû tù ❣✐→❝ ABCD tọ ú ỵ ỏ õ t q ỡ ỵ Ptt ụ ABCD ởt t ú ợ AB, AD, BC ỗ tớ t AB + DC ≥ AD + BC ✳ ❉➜✉ ❜➡♥❣ ự ỵ Ptt sỷ tự tũ ỵ õ ữớ trỏ DC t↕✐ ❤❛✐ ✤✐➸♠✳ ❑❤✐ ✤â ❦❤✐ ABCD ❧➔ tù t r t ỵ ữ ❍➻♥❤ ✶✳✷ t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ trð t❤➔♥❤ x + y + z ≥ c + d✳ ỵ ữỡ t ữớ trá♥ M ❝➢t ✤÷í♥❣ M A.M B = M O − R = d2 − R2 ✳ ❝è ✤à♥❤✳ ▼ët ✤÷í♥❣ t❤➥♥❣ t❤❛② ✤ê✐ q✉❛ A ✈➔ B✳ ❑❤✐ ✤â (O; R) ✈➔ ✤✐➸♠ M 2 trá♥ t↕✐ ❤❛✐ ✤✐➸♠ ✹✼ ❈❤÷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❝ (I, r)✳ ❚❛ ❣å✐ ❝→❝ ●✐↔ sû ❆❇❈❉ ❧➔ ♠ët tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✤÷í♥❣ trá♥ ❦❤♦↔♥❣ ❝→❝❤ 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= ❙û ❞ö♥❣ ef g + f gh + ghe + hef e+f +g+h S = r · (a + b + c + d) = r(e + f + g + h) S= ✭✸✳✶✮ ✈➔ ✭✸✳✶✮ t❛ ❝â✿ (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✐ Y ∈ CD✱ Z ∈ DA✳ ❚❛ s➩ ❣å✐ ✷ W ∈ AB ✱ X ∈ BC ✱ W Y ✈➔ XZ ❧➔ ❞➙② ❝✉♥❣ ❝→❝ t✐➳♣ ✤✐➸♠ ❞➙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 t❤❡♦ ❝æ♥❣ 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) ✤÷đ❝ t➼♥❤ ✹✾ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ I ❧➔ t➙♠ t ỵ s t ; γ ❧➔ ❝→❝ ❣â❝ tr➯♥ ❍➻♥❤ ✸✳✶✳ ⑩♣ W Y I✿ 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 r2 − r2 tan2 β r2 − f − tan2 β = = + tan β r + r2 tan2 β r2 + f 2 tan β 2rf sin 2β = = + tan2 β r2 + f cos = tữỡ tỹ ợ t f ❜ð✐ g✳ ✭✸✳✺✮ ✭✸✳✻✮ ◆❤÷ ✈➟② t❛ ❝â r2 − f r2 − g 2rf 2rg (f + g)2 k2 = 1− · + · = 2r · 2r2 r + f r2 + g r2 + f r2 + g (r + f ) (r2 + g ) ❉♦ ✤â✱ (f + g)2 ✳ (r2 + f ) (r2 + g ) 2 ♣❤➙♥ t➼❝❤ r + f ✈ỵ✐ r t➼♥❤ k = 2r2 ❇➙② ❣✐í 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 (e + g)(f + g)(g + h) 2 t÷ì♥❣ tü✱ r + g = ✳ ◆❤÷ ✈➟②✱ e+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 tự t f ←→ h tr♦♥❣ ❝ỉ♥❣ 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♥✿ k (e + h)(f + g) = = 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❝ sin ϕ = ϕ ❣✐ú❛ ❤❛✐ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ ✤÷đ❝ t➼♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ (e + f + g + h)(ef g + f gh + ghe + hef ) (e + f )(f + g)(g + h)(h + e) 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♥❣ ❧↕✐ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ s➩ t❤❛② ❣â❝ ϕ ❜ð✐ ❝→❝ ❣â❝ 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 abcd sin ♥➯♥ ❱➻ ❞✐➺♥ t➼❝❤ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❜➡♥❣ S = √ √ 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) ❈ỉ♥❣ t❤ù❝ ❝✉è✐ t❛ ✤➣ sû ❞ư♥❣ ✭✸✳✷✮✳ ❍➺ q✉↔ ✸✳✷✳ ❍❛✐ ❞➙② ❝✉♥❣ t✐➳♣ ①ó❝ tr♦♥❣ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✈✉æ♥❣ ❣â❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tù ❣✐→❝ ❧➔ s♦♥❣ t➙♠✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚r♦♥❣ ♠å✐ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ t❛ ❝â ❦❤→❝✱ π W Y ⊥ XZ ⇐⇒ ϕ = ⇐⇒ B + D = π ✳ B + D = 2π − 2ϕ✳ ▼➦t ❚ù ❣✐→❝ ♥➔② ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦➳t ❤đ♣ ✈ỵ✐ ♥❣♦↕✐ t✐➳♣ ✤÷đ❝ ♥➯♥ ❧➔ tù ❣✐→❝ s♦♥❣ t➙♠✳ ✸✳✷ ❚ù ❣✐→❝ t✐➳♣ ①ó❝ ❚ù ❣✐→❝ ❝â ❝→❝ ✤➾♥❤ ❧➔ ❝→❝ t✐➳♣ ✤✐➸♠ W, X, Y, Z ✤÷đ❝ ❣å✐ ❧➔ tù ❣✐→❝ t✐➳♣ ①ó❝✱ ❍➻♥❤ ✸✳✹✳ ❚❛ s➩ t➻♠ ❝ỉ♥❣ 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) ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â tr♦♥❣ ✤â k, l SABCD = ❧➔ ✷ ✤÷í♥❣ ❝❤➨♦✱ p sin θ✳ ❚÷ì♥❣ tü✱ SW XY Z = k.l sin ϕ✱ q ϕ ❧➔ ❣â❝ ❣✐ú❛ ✷ ✤÷í♥❣ ❝❤➨♦ W Y, XZ ✳ ❚ø ❝→❝ ❝ỉ♥❣ 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 ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣✱ ♥❣♦↕✐ t✐➳♣✱ d r, R ❧➛♥ ❧÷đt ❧➔ ❜→♥ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ✷ t➙♠✳ ❑❤✐ ✤â 1 + = (R + d)2 (R − d)2 r2 ✭✸✳✾✮ ❈❤ù♥❣ ♠✐♥❤✳ ❈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❛♠ ❣✐→❝ ✈✉æ♥❣ ỗ AJI IKC ự ỵ ss õ AJI = IKC = 90◦ ◆➳✉ ❞ü♥❣ t❛♠ ❣✐→❝ ✈✉æ♥❣ JAI = KIC ữ ỵ r IJ = IK = r JIC , IC = IC ♥❤÷ ❍➻♥❤ ✸✳✺ t❛ ✤÷đ❝ t❛♠ ✈➔ AIC ✈✉æ♥❣ ð I ✳ ❚ø ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝ ♥➔②✿ 2SAIC = r.AC = AI.IC ⇔ r.AC = AI.IC ✳ ❱➟② r2 AC = AI · IC 2 ❤❛② r 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 1 F I2 EI EI + F I + = + = = AI CI (R2 − d2 ) (R2 − d2 )2 (R2 − d2 )2 (R2 − d2 )2 ✺✹ ❚ø ✤â✱ t❤❛② ✈➔♦ ✸✳✶✵✿ R2 + 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➻♠ ❝æ♥❣ t❤ù❝ t➼♥❤ ❝→❝ ❣â❝ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ t❤❡♦ ❝→❝ ✤♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥✳ ❈→❝ ❝æ♥❣ t❤ù❝ ♥➔② ỗ ữ õ q 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 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) ự 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 (e + f )(e + g)(e + h) 2 ❞ö♥❣ ✭✸✳✶✮ ✈➔ r + e = ✱ ❞♦ t➼♥❤ ✤è✐ ①ù♥❣ t❛ t❤❛② e+f +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 ❞ö♥❣ t➼♥❤ ❝❤➜t ✤è✐ ①ù♥❣✳ ✿ ❚➼♥❤ s✐♥ ❝õ❛ ♠ët ♥û❛ ❣â❝ A ❍➻♥❤ ✸✳✻ ✸✳✸ ❚ù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✈➔ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ◆❤✐➲✉ ❜➔✐ t♦→♥ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✤÷đ❝ ❣✐↔✐ ❦❤→ ♣❤ù❝ t↕♣ ❦❤✐ sû ữỡ sỡ t ỗ trử ✤➥♥❣ ♣❤÷ì♥❣✱ ❝ü❝✲ ✤è✐ ❝ü❝✱✈✈✳✳✳✮ ❧↕✐ trð ♥➯♥ ✤ì♥ ❣✐↔♥ ✈➔ ❤✐➸♥ ♥❤✐➯♥ ♥➳✉ t❛ ❜✐➳t sû ❞ö♥❣ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦✳ ✣â ❧➔ ✈➻ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝â ♥❤✐➲✉ t➼♥❤ t t t ợ ữớ trỏ ợ ỹ ữỡ 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♥❣ tỉ✐ ❦❤ỉ♥❣ ❝â ✤✐➲✉ ❦✐➺♥ tr➻♥❤ ❜➔② t♦➔♥ ❜ë ♥ë✐ ❞✉♥❣ ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ♠➔ ❝❤➾ ♥➯✉ ♠ët sè ✈➼ ❞ö ✤➸ ♣❤➙♥ t➼❝❤ ❝→❝❤ t✐➳♣ ❝➟♥ ❝→❝ ❜➔✐ t♦→♥ ✈➲ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤à❝❤ ✤↔♦✳ ❈ơ♥❣ ❝❤➼♥❤ tø ù♥❣ ❞ö♥❣ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ♠➔ t❛ ❝â t❤➸ s→♥❣ t↕♦ r❛ ❝→❝ ❜➔✐ t♦→♥ ♠ỵ✐✳ ❱➼ ❞ư ✸✳✸✳✶✳ tự t ỵ E = AB ∩ CD✱ F = AD ∩ BC ✳ ❑❤✐ õ tỗ t ởt ữớ trỏ (EAD), (EBC), (F AB), (F CD) trỏ t ú ợ ố ữớ X, Y, Z, T ❧➔ ❝→❝ t✐➳♣ ✤✐➸♠ t÷ì♥❣ ù♥❣ ❝õ❛ AB BC CA AD ợ ữớ trỏ (I) t ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝ü❝ I ✱ ♣❤÷ì♥❣ t➼❝❤ ✷✱ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ ✺✻ 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 ỵ (EAD) (EBC) (F AB) (F CD) ỗ q t ▼✐q✉❡❧ ❝õ❛ tù ❣✐→❝ t♦➔♥ ♣❤➛♥ ABCDEF ♥➯♥ ❝→❝ ✤÷í♥❣ trá♥ ❊✉❧❡r ❝õ❛ ❝→❝ t❛♠ ❣✐→❝ ∆T XZ, ∆Y XZ, T XY, T ZY ỗ q t J ❝→❝ ✤÷í♥❣ trá♥ ♥➔♦ ✤â✳ ▼➦t ❦❤→❝✱ ❝→❝ ✤÷í♥❣ trá♥ ♥➔② ✤➲✉ ❝â ❜→♥ ❦➼♥❤ r ✷ ♥➯♥ ✤÷í♥❣ (J, r) t ú ợ ố ữớ trỏ r õ tr➯♥✳ ⑩♣ ❞ö♥❣ ♣❤➨♣ I −1 t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ♥❣❤à❝❤ ✤↔♦ f2 trá♥ ❱➼ ❞ö ✸✳✸✳✷✳ ❈❤♦ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ABCD✱ E = AB ∩ CD✱ F = AD∩BC ✳ ●å✐ M ❧➔ ✤✐➸♠ ▼✐q✉❡❧ ❝õ❛ tù ❣✐→❝ t♦➔♥ ♣❤➛♥ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝ü❝ M✱ C C ,D D ữỡ t tũ ỵ ❜✐➳♥ ❈❤ù♥❣ ♠✐♥❤ ABC D ABCDEF ✳ A → A✱ B → B✱ ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ❈❤ù♥❣ ♠✐♥❤✳ t tỗ t ởt ữớ trỏ t ú ợ ữớ trỏ (C) (C) (M AB)✱ (M BC)✱ (M CD)✱ (M DA) ♥➯♥ ↔♥❤ q✉❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ✤â s➩ ❧➔ ✤÷í♥❣ trá♥ t✐➳♣ ú ợ ữớ t AB BC C D DA✳ ◆❣❤➽❛ ❧➔ ABC D ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ❱➼ ❞ö ✸✳✸✳✸✳ ❈❤♦ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ABCD ✈➔ P = AC ∩ BD✳ P❤➨♣ ✺✼ ♥❣❤à❝❤ ✤↔♦ ❝ü❝ A ,B ,C ,D ✳ P ♣❤÷ì♥❣ t➼❝❤ ❜➜t ❦ý ❜✐➳♥ ❑❤✐ ✤â ABC D A, B, C, D ❧➛♥ ❧÷đt t❤➔♥❤ ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ AB ✱ AB = k P A.P B BC CD DA BC =k ,C D = k ,D A = k ✳ ❙✉② r❛ P B.P C PC · PD PD · PA ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦✿ CD AB + P A.P B P C · P D k 2AB 2CD = sin AP B + P A.P B sin AP B P C.P D · sin DP C CD k AB + = sin AP B · SAP B SDP C AB +C D =k ✿ ❱➼ ❞ư ✸✳✸✳✸ ❍➻♥❤ ✸✳✽ (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 ♥➯♥ AX = AJ = x✳ ❙✉② r❛ SBP C PC CZ z SBP C y SCP D t SDP A x = , = , = ❚ø ✤â s✉② r❛ SCP D t SDP A x SAP B y ●å✐ t✐➳♣ ✤✐➸♠ ❝õ❛ SAP B SBP C SCP D SDP A = = = = q xy yz zt tx ✺✽ AB 1 1 CD x+y z+t + = · + + + + = ✳ SAP B SDP C qxy qzt q x y z t 1 1 k sin AP B + + + ✳ ◆❤÷ ✈➟②✱ A B + C D 2q x y z t ❍♦➔♥ 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 tỗ t ởt ữớ trỏ CDDA t ú ợ ữớ t P s ✤÷í♥❣ trá♥ P AB ✱ P BC ✱ P CA✱ q ỹ t ú ợ ữớ trỏ ♥❣♦↕✐ t✐➳♣ ❝→❝ t❛♠ ❣✐→❝ P AD AB✱BC ✱ ♥➯♥ t❛ ❝â ❜➔✐ t♦→♥✿ ❱➼ ❞ö ✸✳✸✳✹✳ ❈❤♦ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ABCD✳ ●å✐ P = AC ∩ BD ✳ ự tọ tỗ t ởt ữớ trỏ t ú ợ ữớ trỏ t P AB, P BC, ∆P CA, ∆P DA✳ (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❤❡♦ ♥➳✉ ❣å✐ X, Y, Z, T ❧➛♥ ❧÷đt ❧➔ t✐➳♣ ✤✐➸♠ ❝õ❛ ❤❛✐ ❜➔✐ 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 trỏ t t M XY, ∆M Y Z, ∆M ZT, ∆M T X ✳ ❱➼ ợ ỵ ữ tr P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝ü❝ M✱ ♣❤÷ì♥❣ t➼❝❤ ❜➜t ❦ý✱ ❜✐➳♥ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ XY ZT X, Y, Z, T t÷ì♥❣ ù♥❣ t❤➔♥❤ X ,Y ,Z ,T ✳ ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ❈❤÷ì♥❣ ❜❛ ❝❤➾ tr➻♥❤ ❜➔② ✸ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣✳ ✣â ❧➔✿ ✤♦↕♥ t❤➥♥❣ t✐➳♣ t✉②➳♥✱ tù ❣✐→❝ t✐➳♣ ①ó❝ ✈➔ ù♥❣ ❞ư♥❣ ♣❤➨♣ ♥❣❤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♥❣ ❞ư♥❣ ❝õ❛ ú ỵ ổ tự t➼♥❤ t♦→♥ tr➯♥ ❝→❝ ✤ë ❞➔✐ ❝õ❛ tù ❣✐→❝ ✈➔ ❝→❝ ù♥❣ ❞ö♥❣ ❝â ❤✐➺✉ q✉↔ ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦✳ ú tổ t õ ữợ ự t t❤❡♦✿ − ❚➻♠ t❤➯♠ ✈➲ ❝→❝ ❜➔✐ t♦→♥ t÷ì♥❣ tü tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥✿ ❚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... B, C t❛ ❝â ❤➺ t❤ù❝ s❛✉ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ❤➺ t❤ù❝ ✭✶✳✶✹✮✿ 1 1 =− + + Ra hb hc t ỵ ữớ tứ hA , hC A, C ✭✶✳✶✾✮ ①✉è♥❣ ✤÷í♥❣ ❝❤➨♦ BD tữỡ ự tữỡ tỹ ố ợ ữớ ❝❛♦ ❤↕ ①✉è♥❣ ❆❈ ✭❍➻♥❤ ✶✳✶✸✮ t❤➻... ①❡♠ ①➨t ✤➳♥ ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ ❝õ❛ ❝→❝ t❛♠ ❣✐→❝ ❦❤→❝ t❛ ❝á♥ t❤✉ ✤÷đ❝ t q s q ỵ ởt tự ỗ ữ tứ õ t ✤÷đ❝ ♠ët ❦➳t q✉↔ tèt✳ ✷✺ ▼➺♥❤ ✤➲ ✶✳✶✶✳ ❚ù ỗ ABCD ợ P = AC BD t✐➳♣ ♠ët ✤÷í♥❣ trá♥ ❦❤✐