Luận văn thạc sĩ bài toán chứng minh tính vuông góc, song song trong hình học

ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - NGỌC THỊ HÀ BÀI TOÁN CHỨNG MINH TÍNH VNG GĨC, SONG SONG TRONG HÌNH HỌC LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2019 ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - NGỌC THỊ HÀ BÀI TỐN CHỨNG MINH TÍNH VNG GĨC, SONG SONG TRONG HÌNH HỌC Chun ngành: Phương pháp Toán sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC PGS.TS Trịnh Thanh Hải THÁI NGUYÊN - 2019 ✐ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝➠♥ ❜↔♥ ✶✳✶ ❈→❝ ỵ t ổ õ s s tr♦♥❣ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✷ ❈→❝ t➼♥❤ ❝❤➜t ✈➲ t➼♥❤ ✈✉æ♥❣ ❣â❝✱ s♦♥❣ s♦♥❣ tr♦♥❣ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✸ ✾ ❈→❝ ✤à♥❤ ỵ t s s ổ õ tr♦♥❣ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ✶ ✸ ✾ ▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ ✈✉æ♥❣ ❣â❝✱ s♦♥❣ s♦♥❣ tr♦♥❣ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷ ❈→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ ✈✉æ♥❣ ❣â❝ tr♦♥❣ ❝→❝ ✤➲ t❤✐ ❍å❝ s✐♥❤ ❣✐ä✐ ✸✺ ✸ ❈→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ s♦♥❣ s♦♥❣ tr♦♥❣ ❝→❝ ✤➲ t❤✐ ❍å❝ s✐♥❤ ❣✐ä✐ ✻✷ ❑➳t ❧✉➟♥ ✼✾ t ỡ rữợ t✐➯♥ ❡♠ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s s t tợ P r ữớ t ✈ỵ✐ ❧á♥❣ ♥❤✐➺t ❤✉②➳t ✤➣ ❧✉ỉ♥ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ tứ ỳ t ỗ tớ ữ r❛ ♥❤ú♥❣ ❧í✐ ❦❤✉②➯♥ ❜ê ➼❝❤ ❣✐ó♣ ❡♠ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❊♠ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❝→❝ t❤➛② ❝ỉ✱ t➟♣ t❤➸ ❝→♥ ❜ë ❦❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ỗ r t ữợ tữớ t ◆✐♥❤✱ ❝ị♥❣ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❧ỵ♣ ❝❛♦ ❤å❝ ❚♦→♥ ❑✶✶❉✱ ✤➣ ❦❤æ♥❣ ❝❤➾ tr❛♥❣ ❜à ❝❤♦ ❡♠ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❜ê ➼❝❤ ♠➔ ❝á♥ ❧✉ỉ♥ ❣✐ó♣ ✤ï✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ tr♦♥❣ q✉→ tr➻♥❤ ❡♠ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❈✉è✐ ❝ị♥❣ ❡♠ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ♥❣÷í✐ t❤➙♥ ❧➔ ♥❤ú♥❣ ♥❣÷í✐ ❧✉ỉ♥ õ♥❣ ❤ë✱ ✤ë♥❣ ✈✐➯♥ ❡♠ ✈÷đt q✉❛ ♥❤ú♥❣ ❦❤â ❦❤➠♥ ✤➸ ❡♠ ❤♦➔♥ t❤➔♥❤ tèt ❧✉➟♥ ✈➠♥✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✻ t❤→♥❣ ✸ ♥➠♠ ✷✵✶✾ ✶ ▼ð ✤➛✉ ❚r♦♥❣ ❤➻♥❤ ❤å❝ ♣❤➥♥❣✱ ❝→❝ ❞↕♥❣ ❜➔✐ t➟♣ ✈➲ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ s♦♥❣ s♦♥❣ ❤❛② ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ✈✉æ♥❣ ❣â❝ ❧✉æ♥ ❧➔ ❝→❝ ❜➔✐ t➟♣ t❤ó ✈à ♥❤÷♥❣ t❤÷í♥❣ r➜t ❦❤â✳ ✣➦❝ ❜✐➺t ❧➔ ♥❤ú♥❣ ❜➔✐ t♦→♥✱ ✤➲ t❤✐ ❞➔♥❤ ❝❤♦ ❤å❝ s✐♥❤ ❣✐ä✐ t❤➻ ❤å❝ s✐♥❤ ♣❤↔✐ ♥➢♠ ✤÷đ❝ ❝→❝ ❦✐➳♥ t❤ù❝ ♥➙♥❣ ❝❛♦✱ ỵ t t ữỡ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ ❦❤ỉ♥❣ ❝â tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ✤↕✐ tr➔ ❝ơ♥❣ ♥❤÷ ❝❤÷ì♥❣ tr➻♥❤ ♥➙♥❣ ❝❛♦ ð ❜➟❝ ❝ì sð✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ ✈ø❛ q✉❛✱ ✤➣ ❝â ♥❤✐➲✉ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝ ❧ü❛ ❝❤å♥ ❝→❝ ❝❤õ ✤➲ ✈➲ ❤➻♥❤ ❤å❝ ✤➸ tr✐➸♥ ❦❤❛✐ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ♥❤ú♥❣ ❝❤÷❛ ❝â ❤å❝ ✈✐➯♥ ♥➔♦ ♥❣❤✐➯♥ ❝ù✉ ♠ët ❝→❝❤ ❤➺ t❤è♥❣ ✈➲ ❝→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ s♦♥❣ s♦♥❣✱ ✈✉æ♥❣ ❣â❝ ✤➸ ♣❤→t tr✐➸♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝❤✉②➯♥ ♥❣➔♥❤ Pữỡ t sỡ ợ ố t ỵ t t ụ ữ ữỡ ự ♠✐♥❤ t➼♥❤ s♦♥❣ s♦♥❣✱ t➼♥❤ ✈✉æ♥❣ ❣â❝ q✉❛ ♠ët sè ❜➔✐ t♦→♥✱ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ✤➸ ❧➔♠ t➔✐ ❧✐➺✉ ❝❤♦ ✈✐➺❝ ❣✐↔♥❣ ❞↕② ❝õ❛ ❜↔♥ t❤➙♥ ✈➔ ❧➔♠ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤♦ ❤å❝ s✐♥❤ tü ❤å❝✱ tæ✐ ❝❤å♥ ❝❤õ ✤➲✿ P❤÷ì♥❣ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ s♦♥❣ s♦♥❣✱ t➼♥❤ ✈✉æ♥❣ ❣â❝ q✉❛ ✈✐➺❝ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥✱ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ❝❤♦ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ ♠➻♥❤✳ ▲✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ s ã ỵ t ❝❤➜t ❧✐➯♥ q✉❛♥ ✤➳♥ ✤✐➲✉ ❦✐➺♥ ✤➸ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ s s ổ õ ợ ụ ữ ❤➺ q✉↔ ❝â ✤÷đ❝ tø ✈✐➺❝ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s ổ õ ã ữ t t ❧✉②➺♥ t❤✐ ✤ë✐ t✉②➸♥ ❤å❝ s✐♥❤ ❣✐ä✐✱ ❝→❝ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ t♦→♥ ✈➲ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ s♦♥❣ s♦♥❣✱ t➼♥❤ ✈✉ỉ♥❣ ❣â❝✳ • ❚r➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ❧✉②➺♥ ❤å❝ s✐♥❤ ❣✐ä✐✱ ❝→❝ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ t♦→♥ ✈➲ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ s♦♥❣ s♦♥❣✱ t➼♥❤ ✈✉æ♥❣ ❣â❝✳ ❚r♦♥❣ ✤â ❝è ❣➢♥❣ ✤÷❛ r❛ ❧í✐ ❣✐↔✐ t÷í♥❣ ♠✐♥❤ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ❜➔✐ t♦→♥✱ ✤➲ t❤✐ ♠➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➾ ❝â ❧í✐ ❣✐↔✐ ✈➢♥ t➢t ❤♦➦❝ ✤à♥❤ ữợ ã ố ợ ởt t♦→♥✱ ❝è ❣➢♥❣ ✤÷❛ r❛ ♥❤✐➲✉ ❧í✐ ❣✐↔✐ ✤➸ ♠✐♥❤ ❤å❛ t➼♥❤ ❧✐♥❤ ❤♦↕t tr♦♥❣ ✈✐➺❝ ✈➟♥ ❞ö♥❣ ❝→❝ t➼♥❤ t ỵ ự t t s♦♥❣ s♦♥❣✱ t➼♥❤ ✈✉ỉ♥❣ ❣â❝✳ ❱ỵ✐ ♠ư❝ t✐➯✉ ♥❣❤✐➯♥ ❝ù✉ ữ ố ỗ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ tố t t ỵ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❜➔✐ t♦→♥ ✈➲ t➼♥❤ ✈✉ỉ♥❣ ❣â❝ ✭✤÷í♥❣ t❤➢♥❣✱ ❣â❝✮ ✈➔ t➼♥❤ s♦♥❣ s♦♥❣ tr♦♥❣ ỵ t t ỡ ữ ỵ s ỵ Ptrs ỵ ự ữớ t ổ ởt s s ỗ q ỵ s tr t tự ỵ rt t ữủ tứ ❝→❝ ✤÷í♥❣ t❤➥♥❣ ✈✉ỉ♥❣ ❣â❝ ♥➡♠ tr➯♥ ❝→❝ ❝↕♥❤ ❝õ❛ t ỗ tớ ụ ữ r ởt số t ỵ tr ự ♠✐♥❤ t➼♥❤ ✈✉ỉ♥❣ ❣â❝ ✈➔ s♦♥❣ s♦♥❣✳ ❈❤÷ì♥❣ ✷✳ ❈→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ✈✉æ♥❣ ❣â❝ tr♦♥❣ ❝→❝ ✤➲ t❤✐ ❍å❝ s✐♥❤ ❣✐ä✐ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ởt tữớ ỵ t➼♥❤ ❝❤➜t ✤➸ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ ✈✉ỉ♥❣ ❣â❝✳ ❙÷✉ t➛♠ ❝→❝ ❜➔✐ t♦→♥ ❧✉②➺♥ t❤✐ ✤ë✐ t✉②➸♥ ❤å❝ s✐♥❤ ❣✐ä✐✱ ❝→❝ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ t♦→♥ ✈➲ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ ✈✉ỉ♥❣ ❣â❝✳ ❈❤÷ì♥❣ ✸✳ ❈→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ s♦♥❣ s♦♥❣ tr♦♥❣ ❝→❝ ✤➲ t❤✐ ❍å❝ s✐♥❤ ❣✐ä✐ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✸ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ t÷í♥❣ ♠✐♥❤ ✈✐➺❝ ✈➟♥ ❞ư♥❣ ❝→❝ ✤à♥❤ ỵ t t ự ởt sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ s♦♥❣ s♦♥❣✳ ❙÷✉ t➛♠ ❝→❝ ❜➔✐ t♦→♥ ❧✉②➺♥ t❤✐ ✤ë✐ t✉②➸♥ ❤å❝ s✐♥❤ ❣✐ä✐✱ ❝→❝ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ t♦→♥ ✈➲ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ s♦♥❣ s♦♥❣✳ ❱➻ ✤✐➲✉ ❦✐➺♥ t❤í✐ ❣✐❛♥ ❣✐ỵ✐ ❤↕♥ ♥➯♥ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❧✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ ❝❤õ ②➳✉ ❧➔ ❝→❝ ❜➔✐ t♦→♥ t❤✉ë❝ ❍➻♥❤ ❤å❝ ♣❤➥♥❣✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✻ t❤→♥❣ ✸ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ◆❣å❝ ❚❤à ❍➔ ✸ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝➠♥ ❜↔♥ ✶✳✶ ❈→❝ ✤à♥❤ ỵ t ổ õ s s tr tự rữợ t ❝❤ó♥❣ t❛ s➩ ♥❤➢❝ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✤➣ ✤÷đ❝ ✤➲ ❝➟♣ tr♦♥❣ ❝→❝ ❝❤÷ì♥❣ tr➻♥❤ ❣✐→♦ ❞ư❝ ♣❤ê t❤ỉ♥❣ ✈➲ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣✱ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ ✈✉ỉ♥❣ ❣â❝ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝❤ó♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❍❛✐ ✤÷í♥❣ t❤➥♥❣ xx , yy ❝➢t ♥❤❛✉ ✈➔ tr♦♥❣ ❝→❝ ❣â❝ t↕♦ t❤➔♥❤ ❝â ♠ët ❣â❝ ✈✉ỉ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ ✈✉ỉ♥❣ ❣â❝ ữủ ỵ xx yy ữớ t❤➥♥❣ ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ ♠ët ✤♦↕♥ t❤➥♥❣ t↕✐ tr✉♥❣ ✤✐➸♠ ❝õ❛ ♥â ✤÷đ❝ ❣å✐ ❧➔ ✤÷í♥❣ tr✉♥❣ trü❝ ❝õ❛ ✤♦↕♥ t❤➥♥❣ ➜②✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❍❛✐ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ ❧➔ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ ❦❤ỉ♥❣ ❝â ✤✐➸♠ ❝❤✉♥❣✳ ❍❛✐ ✤÷í♥❣ t❤➥♥❣ ♣❤➙♥ ❜✐➺t t❤➻ ❤♦➦❝ ❝➢t ♥❤❛✉ ❤♦➦❝ s♦♥❣ s♦♥❣ ✈ỵ✐ ♥❤❛✉✳ ✹ ◆❤➟♥ ①➨t ✶✳✶✳ ✭✐✮ ❍❛✐ ❣â❝ A1 ứ ữợ ú t ❝➦♣ ❣â❝ s❛✉ ✤➙② ✈➔ B3 ❝ơ♥❣ ♥❤÷ ❤❛✐ ❣â❝ A4 ✈➔ B2 ✤÷đ❝ ❣å✐ ❧➔ ❤❛✐ ❣â❝ s♦ ❧❡ tr♦♥❣✳ ✭✐✐✮ ❈➦♣ ❣â❝ A1 ✈➔ B1 ✤÷đ❝ ❣å✐ ❧➔ õ ỗ ữỡ tỹ t õ õ ỗ A2 B2 A3 ✈➔ B3 ❀ A4 ✈➔ B4 ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ◆➳✉ ✤÷í♥❣ t❤➥♥❣ c ❝➢t ❤❛✐ ✤÷í♥❣ t❤➥♥❣ a, b ✈➔ tr♦♥❣ ❝→❝ ❣â❝ t↕♦ t❤➔♥❤ ❝â ♠ët ❝➦♣ ❣â❝ s♦ ❧❡ tr♦♥❣ ❜➡♥❣ ♥❤❛✉ ✭❤♦➦❝ ♠ët ❝➦♣ ❣â❝ ỗ t a b s s ✈ỵ✐ ♥❤❛✉✳ ❚✐➯♥ ✤➲ ✶✳✶ ✭❚✐➯♥ ✤➲ ❊✉❝❧✐❞❡✮✳ ◗✉❛ ♠ët ✤✐➸♠ ð ♥❣♦➔✐ ♠ët ✤÷í♥❣ t❤➥♥❣ ❝❤➾ ❝â ♠ët ✤÷í♥❣ t s s ợ ữớ t õ t AB ✈➔ CD ❣å✐ ❧➔ t➾ ❧➺ ✈ỵ✐ ❤❛✐ ✤♦↕♥ t❤➥♥❣ AB ✈➔ CD ♥➳✉ ❝â t➾ ❧➺ t❤ù❝ AB AB = CD CD ❤♦➦❝ AB CD = AB CD ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❈❤♦ ✤÷í♥❣ t❤➥♥❣ d✳ P❤➨♣ ❜✐➳♥ ❤➻♥❤ ❜✐➳♥ ♠é✐ ✤✐➸♠ M t❤✉ë❝ d t❤➔♥❤ ❝❤➼♥❤ ♥â✱ ❜✐➳♥ ♠é✐ ✤✐➸♠ M ❦❤æ♥❣ t❤✉ë❝ d t❤➔♥❤ M s❛♦ ❝❤♦ d ❧➔ ✤÷í♥❣ tr✉♥❣ trü❝ ❝õ❛ ✤♦↕♥ t❤➥♥❣ M M ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ✤è✐ ①ù♥❣ q✉❛ ✤÷í♥❣ t❤➥♥❣ d ❤❛② ♣❤➨♣ ✤è✐ ①ù♥❣ trö❝ d✳ P❤➨♣ ✤è✐ ①ù♥❣ trư❝ t❤÷í♥❣ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ✣d✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❈❤♦ ✤✐➸♠ I ✳ P❤➨♣ ❜✐➳♥ ❤➻♥❤ ❜✐➳♥ ✤✐➸♠ I t❤➔♥❤ ❝❤➼♥❤ ♥â✱ ❜✐➳♥ ♠é✐ ✤✐➸♠ M ❦❤→❝ I t❤➔♥❤ M s❛♦ ❝❤♦ I ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ ✤♦↕♥ t❤➥♥❣ ✺ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ✤è✐ ①ù♥❣ t➙♠ I ✳ P❤➨♣ ✤è✐ ①ù♥❣ t➙♠ t❤÷í♥❣ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ✣I MM ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❈❤♦ ✤✐➸♠ O ✈➔ ❣â❝ ❧÷đ♥❣ ❣✐→❝ α✳ P❤➨♣ ❜✐➳♥ ❤➻♥❤ ❜✐➳♥ O t❤➔♥❤ ❝❤➼♥❤ ♥â✱ ❜✐➳♥ ♠é✐ ✤✐➸♠ M ❦❤→❝ O t❤➔♥❤ ✤✐➸♠ M s❛♦ ❝❤♦ OM = OM ✈➔ ❣â❝ ❧÷đ♥❣ ❣✐→❝ (OM, OM ) = α ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ q✉❛② t➙♠ O ❣â❝ α✳ P❤➨♣ q✉❛② t➙♠ O ❣â❝ α t❤÷í♥❣ ✤÷đ❝ Q(O,) trữợ ♠ët ✤✐➸♠ O ✈➔−−sè→t❤ü❝ k−−=→0✳ P❤➨♣ ❜✐➳♥ ❤➻♥❤ ❜✐➳♥ ♠å✐ ✤✐➸♠ M t❤➔♥❤ ✤✐➸♠ M s❛♦ ❝❤♦ OM = kOM ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ✈à tü t➙♠ O t➾ sè k ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ V(O,k)✳ ✣✐➸♠ M ✤÷đ❝ ❣å✐ ❧➔ ↔♥❤ ❝õ❛ ✤✐➸♠ M, M ✤÷đ❝ ❣å✐ ❧➔ t↕♦ ↔♥❤ ❝õ❛ M , O ❧➔ t➙♠ ❝õ❛ ♣❤➨♣ ✈à tü✱ k ❧➔ t➾ sè ✈à tü✳ ◆❤➟♥ ①➨t ✶✳✷✳ P❤➨♣ ✈à tü t➾ sè ❦ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿ ✭✐✮ ❇✐➳♥ ❜❛ ✤✐➸♠ t❤➥♥❣ ❤➔♥❣ t❤➔♥❤ ❜❛ ✤✐➸♠ t❤➥♥❣ ❤➔♥❣ ✈➔ ❜↔♦ t♦➔♥ t❤ù tü ❣✐ú❛ ❝→❝ ✤✐➸♠ ✤â✳ ✭✐✐✮ ❇✐➳♥ ✤÷í♥❣ t❤➥♥❣ t❤➔♥❤ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ ❤♦➦❝ trị♥❣ ✈ỵ✐ ♥â✱ ❜✐➳♥ t✐❛ t❤➔♥❤ t✐❛✱ ❜✐➳♥ ✤♦↕♥ t❤➥♥❣ t❤➔♥❤ ✤♦↕♥ t❤➥♥❣✳ ✭✐✐✐✮ ❇✐➳♥ t❛♠ ❣✐→❝ t❤➔♥❤ t ỗ ợ õ õ t õ ❜➡♥❣ ♥â✳ ✻ ✣à♥❤ ♥❣❤➽❛ ✶✳✽✳ ❈❤♦ ✤÷í♥❣ trá♥ (O; R) ✈➔ ✤✐➸♠ M ❝è ✤à♥❤✱ OM = d✳ ▼ët ✤÷í♥❣ t❤➥♥❣ t❤❛② ✤ê✐ q✉❛ M ❝➢t ✤÷í♥❣ trá♥ t↕✐ ❤❛✐ ✤✐➸♠ A ✈➔ B ✳ ❑❤✐ ✤â✱ M A.M B = M O2 − R2 = d2 − R2 ✣↕✐ ❧÷đ♥❣ ❦❤ỉ♥❣ ✤ê✐ M A.M B = M O2 − R2 = d2 − R2 ❣å✐ ❧➔ ♣❤÷ì♥❣ t M ố ợ ữớ trỏ (O; R) PM/(O) t q ỵ s ✤➙② t❤÷í♥❣ ✤÷đ❝ ❞ị♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❜➔✐ t♦→♥ tr♦♥❣ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ✈➲ t➼♥❤ s♦♥❣ s♦♥❣ ✈➔ ✈✉æ♥❣ ❣â❝✱ ❝❤ó♥❣ t❛ s➩ ❜ä q✉❛ ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ỵ t ABC tự ữủ tr♦♥❣ t❛♠ ❣✐→❝ ✈✉ỉ♥❣✮ ✈✉ỉ♥❣ t↕✐ A✱ ✤÷í♥❣ ❝❛♦ AH ✱ t❛ ❝â AB = BC.BH, AC = BC.HC, AH = BH.CH, BC.AH = AC.AH, 1 = + AH AB AC ỵ M ữớ trỏ (O) t ữủ t t M T tợ ữớ trỏ ❑❤✐ ✤â PM/(O) = M A.M B = M T 2✳ ✻✼ O N C = O CN ✭✈➻ OBC ❝➙♥ t↕✐ O ✮ ✈➔ OBC = OCB ✭✈➻ OBC ❝➙♥ t↕✐ O✮✳ ❙✉② r❛ OBC = O N C õ ỗ õ AB M N ✳ ❑❤✐ ✤â ❇➔✐ t♦→♥ ✸✳✼ ✳ ❚ø ✤✐➸♠ M ♥➡♠ ♥❣♦➔✐ ✤÷í♥❣ trá♥ t➙♠ O✱ ✈➩ ✤➳♥ ❤❛✐ t✐➳♣ t✉②➳♥ ✭❚✉②➸♥ s✐♥❤ ❧ỵ♣ ✶✵ ❝❤✉②➯♥ ❚♦→♥ ✕ ỗ M A, M B ✈➔ ❝→t t✉②➳♥ M CD ✭C ♥➡♠ ❣✐ú❛ M ✈➔ D✮✳ ●å✐ H ❧➔ ❣✐❛♦ ✤✐➸♠ M O ✈➔ AB ✳ ✣÷í♥❣ t❤➥♥❣ CH ❝➢t (ω) t↕✐ ✤✐➸♠ t❤ù ❤❛✐ E ❦❤→❝ C ✳ ❈❤ù♥❣ ♠✐♥❤ AB DE ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❣✐↔ t❤✐➳t M A, M B ❧➔ t✐➳♣ t✉②➳♥ ❝õ❛ (ω) ♥➯♥ AH ⊥ M O ✈➔ M A2 = M C.M D✳ ❉♦ ✤â✱ AH ❧➔ ✤÷í♥❣ ❝❛♦ ❝õ❛ t❛♠ ❣✐→❝ AM O✱ →♣ ❞ư♥❣ ❤➺ t❤ù❝ ❧÷đ♥❣✱ t❛ ❝â AM = M H.M O M C.M D = M H.M O✳ ❱➟② tù ❣✐→❝ CDOH ♥ë✐ t✐➳♣✳ ❉➵ t❤➜② M O ⊥ AB ✳ ự AB DE tữỡ ữỡ ợ ❝❤ù♥❣ ♠✐♥❤ M O ⊥ DE ✳ ❚❤❡♦ ❤➺ t❤ù❝ ❧÷đ♥❣ tr♦♥❣ t❛♠ ❣✐→❝ M AO ✱ t❛ ❝â ❉♦ ✤â AH = AH.HB = M H.HO ▼➔ tù ❣✐→❝ ACBE ❝â AB ❣✐❛♦ CE t↕✐ H ♥➯♥ AH.HB = CH.HE M H.HO = CH.HE ✳ M OE = M CE ❤❛② ❚ø ✭✸✳✸✮ ✈➔ ✭✸✳✹✮ s✉② r❛ t✐➳♣✱ s✉② r❛ ✭✸✳✹✮ ❉♦ ✤â tù ❣✐→❝ M OE = M CH ❱➻ tù ❣✐→❝ CDOH ✭✸✳✸✮ M COE ♥ë✐ ✭✸✳✺✮ ♥ë✐ t✐➳♣ ♥➯♥ t❛ ❝â HOD = M CH ❤❛② M OD = M CH ✭✸✳✻✮ ✻✽ M OD = M OE ✳ ❑➳t ❤đ♣ ✈ỵ✐ OD = OE ❝❤ó♥❣ M D = M E ✳ ❱➟②✱ M O ❧➔ tr✉♥❣ trü❝ ✤♦↕♥ CE ♥➯♥ M O ⊥ DE ✳ ❚ø ✭✸✳✺✮ ✈➔ ✭✸✳✻✮ s✉② r❛ s✉② r❛ t❛ ❇➔✐ t♦→♥ ✸✳✽ ✳ ❈❤♦ ✤÷í♥❣ trá♥ (O) ✤÷í♥❣ ❦➼♥❤ AB✱ ❞➙② CD ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ AB t↕✐ ✭❚✉②➸♥ s✐♥❤ ✈➔♦ ❧ỵ♣ ✶✵ ❈❤✉②➯♥ ◗✉↔♥❣ ◆❛♠ ♥➠♠ ❤å❝ ✷✵✶✻ ✕ ✷✵✶✼✮ ♥➡♠ ❣✐ú❛ O ✈➔ A✳ ▲➜② ✤✐➸♠ E ❜➜t ❦➻ tr➯♥ ❝✉♥❣ ♥❤ä BD✱ ❣å✐ M ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ B ❧➯♥ CE ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ HM AE ✳ H ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❣✐↔ t❤✐➳t✱ t❛ ❝â CM B = CHB = 90◦ ♥➯♥ CHM B ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ❙✉② r❛ BHM = BCM = BCE ACBE ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ♥➯♥ BCE = BAE ✳ ❑➳t ❤ñ♣ H ∈ BA s✉② r❛ BHM = BAE tø ✤â ❞➝♥ ✤➳♥ HM AE ✳ ▼➦t ❦❤→❝✱ ✈➔ ❞♦ ❇➔✐ t♦→♥ ✸✳✾ ✭✸✳✼✮ ✈ỵ✐ ✭✷✳✺✮ ✳ ❈❤♦ ✭❈❤å♥ ✤ë✐ t✉②➸♥ ❑✐➯♥ ●✐❛♥❣ ♥➠♠ ❤å❝ ✷✵✶✷ ✕ ✷✵✶✸✮ ❤➻♥❤ t❤❛♥❣ ABCD ❝â ✤→② ❧ỵ♥ ❧➔ CD✳ ◗✉❛ A ✈➩ AK BC (K ∈ CD) ✈➔ q✉❛ B ❦➫ BI AD (I ∈ CD❀ BI ❝➢t AC t↕✐ F, AK ❝➢t BD t↕✐ E ✳ ❈❤ù♥❣ ♠✐♥❤ KD = CI ✈➔ EF AB.✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❣✐↔ t❤✐➳t AK BC, BI AD ✈➔ ABCD ❧➔ ❤➻♥❤ t❤❛♥❣ ♥➯♥ AB CD✱ s✉② r❛ ❝→❝ tù ❣✐→❝ ABCK, ABID ❧➔ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤✳ ❉♦ ✤â DI = CK ✭❝ò♥❣ ❜➡♥❣ AB ✮✱ s✉② r❛ DI + IK = CK + IK ✳ ❱➟② DK = CI ✻✾ ❱➻ AEB ∼ KED (g.g) ♥➯♥ AE AB = EK KD ❱➻ AF B ∼ CF I (g.g) ♥➯♥ AF AB = FC CI ▼➦t ❦❤→❝✱ KD = CI ♥➯♥ AF AE = EF EK FC KC ỵ s ✤↔♦✮✳ ❇➔✐ t♦→♥ ✸✳✶✵ ✳ ❈❤♦ ✭❈❤å♥ ❤å❝ s✐♥❤ ❣✐ä✐ ❆♥ ●✐❛♥❣ ♥➠♠ ❤å❝ ✷✵✶✸ ✕ ✷✵✶✹✮ ✤÷í♥❣ trá♥ t➙♠ O ✤÷í♥❣ ❦➼♥❤ AB ✳ ▲➜② ♠ët ✤✐➸♠ M tr➯♥ ữớ trỏ s BAM = 30 t ợ ✤÷í♥❣ trá♥ t↕✐ ✤✐➸♠ A ✈➔ ✤✐➸♠ M ❝➢t ♥❤❛✉ t↕✐ C ✳ CM ❝➢t AB t↕✐ D✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ BM OC ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ✤➲ ❜➔✐ t❛ ❝â BAM = 30◦✱ t❛♠ ❣✐→❝ AM B ✈✉æ♥❣ t↕✐ M ✭❣â❝ ♥ë✐ t✐➳♣ ❝❤➢♥ ♥û❛ ✤÷í♥❣ trá♥✮ s✉② r❛ M BO = 60◦ ✭✸✳✽✮ M OB ❝➙♥ ❝â M BO = 60◦ ♥➯♥ t❛♠ ❣✐→❝ M OB ❧➔ t❛♠ ❣✐→❝ ✤➲✉✱ ◦ s✉② r❛ AOM = 120 ✳ ❚❛ ❝â CA, Cm ❧➔ ❤❛✐ t✐➳♣ t✉②➳♥ ❝ò♥❣ ①✉➜t ♣❤→t tø ✤✐➸♠ C ♥➯♥ CO ❧➔ ✤÷í♥❣ ♣❤➙♥ ❣✐→❝ ❝õ❛ ❣â❝ ACM ✱ ❤❛② CO ❧➔ ♣❤➙♥ ❣✐→❝ ❝õ❛ ❣â❝ AOM ♥➯♥ ❚❛♠ ❣✐→❝ COA = 60◦ ❚ø ✭✸✳✽✮ ✈➔ ✭✸✳✾✮ s✉② r❛ ❇➔✐ t♦→♥ ✸✳✶✶ BM OC ✭✸✳✾✮ ✭❤❛✐ ❣â❝ ỗ õ ❈❤♦ t❛♠ ❣✐→❝ ABC ✱ ✤÷í♥❣ t❤➥♥❣ d ❝➢t AB ✱ AC ✈➔ tr✉♥❣ t✉②➳♥ AM t❤❡♦ t❤ù tü t↕✐ E, F, N ✳ ●✐↔ sû ✤÷í♥❣ t❤➥♥❣ d BC ✳ ❚r➯♥ t✐❛ ✤è✐ ❝õ❛ t✐❛ F B ❧➜② ✤✐➸♠ K ✱ ✤÷í♥❣ t❤➥♥❣ KN ❝➢t AB t↕✐ P ✱ ✤÷í♥❣ t❤➥♥❣ KM ❝➢t AC t↕✐ Q ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ PQ BC ✼✵ ❈❤ù♥❣ ♠✐♥❤✳ ❑❤✐ d BC ⇒ EF BC ❞♦ ✤â N ✈ỵ✐ AB ❝➢t KP F ❦➫ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ t↕✐ L✳ ❚❛ ❝â N F L (c.g.c) ⇒ EP = LF ✳ ❉♦ ✤â EP LF KF = = PB PB KB ❚ø B ữớ t s s ợ AC t KM t↕✐ H ✳ ❚❛ ❝â CM Q (c.g.c) ⇒ BH = QC ❉♦ ✤â FQ KF FQ = = QC BH KB EP FQ ❚ø ✭✸✳✶✵✮ ✈➔ ✭✸✳✶✶✮ s✉② r❛ = ⇒ P Q BC ✳ PB QC ❚ø ❇➔✐ t♦→♥ ✸✳✶✷ EF ✳ NF P = ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ ✭✸✳✶✵✮ BM H = ✭✸✳✶✶✮ ✳ ❈❤♦ ❤❛✐ ✭❈❤å♥ ✤ë✐ t✉②➸♥ ❍á❛ ❇➻♥❤ ♥➠♠ ❤å❝ ✷✵✶✻ ✲ ✷✵✶✼✮ ✤÷í♥❣ trá♥ (O1) ✈➔ (O2) ❝➢t ♥❤❛✉ t↕✐ A, B ✳ CD ❧➔ t✐➳♣ t✉②➳♥ ❝❤✉♥❣ ❝õ❛ ❤❛✐ ✤÷í♥❣ trá♥ (O1) ✈➔ (O2)❀ ✈ỵ✐ C t❤✉ë❝ (O1)✱ D t❤✉ë❝ (O2) ✈➔ B ❣➛♥ CD ❤ì♥ A✳ ●å✐ E ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ BC ✈➔ AD, F ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ DB ✈➔ AC ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ EF CD✳ ❈❤ù♥❣ ỵ r BCD = CAB BCD = BAD✳ ▼➦t ❦❤→❝✱ BCD + BDC + CBD = 180◦ ⇒ EAF + EBF = 180◦ ❙✉② r❛ tù ❣✐→❝ EF CD✳ AEBF ♥ë✐ t✐➳♣✱ ❞♦ ✤â EF B = BAE = BCD ❉♦ ✈➟②✱ ✼✶ ❇➔✐ t♦→♥ ✸✳✶✸ ✭❚r✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣ ❈❤✉②➯♥ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✳ ❈❤♦ t❛♠ ❣✐→❝ ABC ✳ ✣÷í♥❣ trá♥ (I) ♥ë✐ t✐➳♣ t❛♠ ♥➠♠ ❤å❝ ✷✵✶✻ ✲ ✷✵✶✻✮ ❣✐→❝ ABC t✐➳♣ ①ó❝ ✈ỵ✐ BC, CA, AB t↕✐ D, E, F ✳ ✣÷í♥❣ t❤➥♥❣ DI ❝➢t ✤÷í♥❣ trá♥ t➙♠ A ❜→♥ ❦➼♥❤ AE t↕✐ M, N ✭N ♥➡♠ ❣✐ú❛ M ✈➔ D✮✳ ❈→❝ ✤÷í♥❣ t❤➥♥❣ AD, EF ❝➢t ♥❤❛✉ t↕✐ P ✳ ❈→❝ ✤÷í♥❣ t❤➥♥❣ M A, N P ❝➢t ♥❤❛✉ t↕✐ Q✳ ●å✐ H ❧➔ ❣✐❛♦ ✤✐➸♠ t❤ù ❤❛✐ ❝õ❛ AD ✈➔ (I)✳ ✣÷í♥❣ t❤➥♥❣ q✉❛ tr✉♥❣ ✤✐➸♠ ❝õ❛ DH, DE ❝➢t AC t↕✐ L✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ DL EF ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ J, G ❧➛♥ ❧÷đt ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ HD, ED t❤➻ HE JG JL✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ HEF = JLD✳ ❚❤➟t ✈➟② AEH = ALJ = ADE s✉② r❛ tù ❣✐→❝ JDLE ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ♥➯♥ JLD = JED ✭✸✳✶✷✮ ◆❣♦➔✐ r❛✱ 180◦ − EJD = AJE = AF E = F DE = 180◦ − EF H ⇒ EJD = EHF ▼➦t ❦❤→❝ HF E = HDE ⇒ HEF = JDE ❚ø ✭✸✳✶✷✮ ✈➔ ✭✸✳✶✸✮ t❛ s✉② r❛ ❇➔✐ t♦→♥ ✸✳✶✹ DL ✭✸✳✶✸✮ EF ✭❈❤✉②➯♥ ▲❛♠ ❙ì♥ ✕ ❚❤❛♥❤ ❍â❛ ♥➠♠ ❤å❝ ✷✵✶✺ ✕ ✷✵✶✻✮ ✳ ❈❤♦ t❛♠ ❣✐→❝ ABC ✈✉æ♥❣ t↕✐ A ✈➔ (C) ❧➔ ✤÷í♥❣ trá♥ t➙♠ C ❜→♥ ❦➼♥❤ CA✳ ▲➜② ✤✐➸♠ D t❤✉ë❝ ✤÷í♥❣ trá♥ (C) ✈➔ ♥➡♠ tr♦♥❣ t❛♠ ❣✐→❝ ABC ✳ ●å✐ M ❧➔ ✤✐➸♠ tr➯♥ ❝↕♥❤ AB s❛♦ ❝❤♦ BDM = 21 ACD❀ N ❧➔ ❣✐❛♦ ữớ t M D ợ ữớ AH ❝õ❛ t❛♠ ❣✐→❝ ABC ❀ E ❧➔ ❣✐❛♦ ✤✐➸♠ t❤ù ữớ t BD ợ ữớ trỏ (C) ự ♠✐♥❤ r➡♥❣ M N AE ✳ ✼✷ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ✤÷í♥❣ trá♥ (C) t❛ ❝â AED = ACD ✭❣â❝ ♥ë✐ t✐➳♣ ❜➡♥❣ ♥û❛ ❣â❝ ð t➙♠ ❝ò♥❣ ❝❤➢♥ ❝✉♥❣ AD) ✭✸✳✶✹✮ ❚ø ✭✸✳✶✹✮ ✈➔ ✭✸✳✶✺✮ s✉② r❛ BDM = ACD AED = BDM ❞♦ ✭✸✳✶✺✮ ✤â MN AE õ õ ỗ ❇➔✐ t♦→♥ ✸✳✶✺ ✭✣➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ t➾♥❤ ◗✉↔♥❣ ◆❣➣✐ ♥➠♠ ✷✵✶✵✮ ✳ ❈❤♦ t❛♠ ❣✐→❝ ABC ❝➙♥ t↕✐ A ữớ trỏ (O) t ú ợ AB, AC t B, C ✳ ❚r➯♥ ❝✉♥❣ BC ♥➡♠ tr♦♥❣ t❛♠ ❣✐→❝ ABC ❧➜② ♠ët ✤✐➸♠ M (M = B, C)✳ ●å✐ I, K, H ❧➛♥ ❧÷đt ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ M tr➯♥ BC, AB, CA ✈➔ P ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ M B ✈ỵ✐ IK, Q ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ M C ✈ỵ✐ IH ✳ ❈❤ù♥❣ ♠✐♥❤ P Q BC ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ tù ❣✐→❝ BIM K ✈➔ CIM H ♥ë✐ t✐➳♣ ♥➯♥ KIM = KBM , HIM = HCM , P IG = KIM + HIM = KBM + HCM ▼➦t ❦❤→❝ KBM = ICM ✭❝ò♥❣ ❜➡♥❣ s✤ BM ✮✱ HCM = IBM ✭❝ò♥❣ ❜➡♥❣ ◦ s✤ P M ✮✳ ❍ì♥ ♥ú❛✱ P M Q + ICM + IBM = 180 ♥➯♥ P M Q + P IG = 180◦ ✳ ❉♦ ✤â tù ❣✐→❝ M P IQ ♥ë✐ t✐➳♣ ♥➯♥ t❛ ❝â M QP = M IK ✭❝ò♥❣ ❜➡♥❣ s✤ P M ✮✳ ▼➔ M IK = M IC ✭❝ò♥❣ ❜➡♥❣ KBM ✮ ♥➯♥ M QP = M CI ✳ ❙✉② r❛ P Q BC ✼✸ ❇➔✐ t♦→♥ ✸✳✶✻ ✭✣➲ t❤✐ s ọ t ỗ ✷✵✶✺✮ ✳ ❈❤♦ ✤÷í♥❣ trá♥ (C1) t➙♠ I ✳ ▲➜② ✤✐➸♠ O tr➯♥ (C1)✱ ❞ü♥❣ ✤÷í♥❣ trá♥ (C2) t➙♠ O s❛♦ ❝❤♦ (C2) ❝➢t (C1) t↕✐ C ✈➔ D✳ ❚✐➳♣ t✉②➳♥ ✈ỵ✐ (C2) t↕✐ C ❝➢t (C1 ) t↕✐ A ✈➔ t✐➳♣ t✉②➳♥ ✈ỵ✐ (C1 ) t↕✐ C ❝➢t (C2 ) t↕✐ B ✳ ✣÷í♥❣ t❤➥♥❣ AB ❝➢t (C1 ) t↕✐ F (F = A) ✈➔ ❝➢t (C2 ) t↕✐ E (E = B)✳ ✣÷í♥❣ t❤➥♥❣ CE ❝➢t (C1 ) t↕✐ G (G = C)✱ ✤÷í♥❣ t❤➥♥❣ CF ❝➢t ✤÷í♥❣ t❤➥♥❣ GD t↕✐ H ✳ ❈❤ù♥❣ ♠✐♥❤ CG F D ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ tù ❣✐→❝ OCAD ♥ë✐ t✐➳♣ ♥➯♥ OAD = 90◦✳ ❉♦ ✤â AC, AD ❧➔ ❝→❝ t✐➳♣ t✉②➳♥ ❝õ❛ (C2 )✳ ❙✉② r❛ AC = AD✳ ❚❛ ❝â BCF = CAE ⇒ ACE = BCF ⇒ CEF = CF E CBF = ACE ✼✹ ▼➦t ❦❤→❝ AEG = CEF = CF E = ACD ⇒ AEG = AGE = ADC = ACD AEG = ADC ❉♦ ✤â CAD = GAE ⇒ CGD = GCF ❚ù ❣✐→❝ CF DG ❇➔✐ t♦→♥ ✸✳✶✼ ♥ë✐ t✐➳♣ ❞♦ ✤â HF D = CGD = GCF ✳ ❱➟② FD ✭✣➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ▲♦♥❣ ❆♥ ❇↔♥❣ ❇ ♥➠♠ ✷✵✶✻✮ GC ✳ ❈❤♦ t❛♠ ❣✐→❝ ABC ❝â ❜❛ ❣â❝ ♥❤å♥ (AB < AC)✱ ❞ü♥❣ ✈➲ ♣❤➼❛ ♥❣♦➔✐ t❛♠ ❣✐→❝ ABC ❝→❝ t❛♠ ❣✐→❝ ABD ✈✉æ♥❣ ❝➙♥ t↕✐ A✱ t❛♠ ❣✐→❝ ACE ✈✉æ♥❣ ❝➙♥ t↕✐ A✳ ●å✐ I ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ BE ✈➔ CD✳ ●å✐ M, N ❧➛♥ ❧÷đt ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ BC ✈➔ DE ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ AI M N ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â Q(A,900 ) (D) = B ⇒ Q(A,900 ) (DC) = BE ⇒ Q(A,900 ) (C) = E ●å✐ F, K t❤♦✐ ✭✈➻ CE ✳ ❑❤✐ ✤â tù ❣✐→❝ M F N K M F = F N = N K = KM = CD✮ s✉② r❛ M N ⊥ F K ❚❛♠ ❣✐→❝ BD✳ ❧➛♥ ❧÷đt ❧➔ tr✉♥❣ ✤✐➸♠ BAD ❉♦ ✤â F ✈✉æ♥❣ t↕✐ BD CD = BE CD⊥BE A ✈➔ t❛♠ ✈➔ ❣✐→❝ t❤✉ë❝ tr✉♥❣ trü❝ ❝↕♥❤ BID ✈✉æ♥❣ t↕✐ I ♥➯♥ ❧➔ ❤➻♥❤ ✭✸✳✶✻✮ FA = FI = AI ✳ CAE ✈✉æ♥❣ t↕✐ A ✈➔ t❛♠ ❣✐→❝ CIE ✈✉æ♥❣ t↕✐ I ♥➯♥ KA = KI = CE ❉♦ ✤â K t❤✉ë❝ tr✉♥❣ trü❝ ❝↕♥❤ AI ✳ ❱➟② F K t❤✉ë❝ tr✉♥❣ trü❝ ❝↕♥❤ AI ♥➯♥ AI ⊥ F K ✭✸✳✶✼✮ ❚❛♠ ❣✐→❝ ❚ø ✭✸✳✶✻✮ ✈➔ ✭✸✳✶✼✮ s✉② r❛ AI M N ✼✺ ❇➔✐ t♦→♥ ✸✳✶✽ ✭✣➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ❚➾♥❤ ✣➢❝ ▲➢❝ ♥➠♠ ❤å❝ ✷✵✶✻ ✕ ✷✵✶✼✮ ✳ ❈❤♦ ✤÷í♥❣ trá♥ (O)✳ ❉➙② AB ❝è ✤à♥❤ ❦❤ỉ♥❣ ♣❤↔✐ ✤÷í♥❣ ❦➼♥❤✳ ●å✐ I ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ ✤♦↕♥ AB ✳ ❚r➯♥ ❝✉♥❣ ♥❤ä A❇ ❧➜② ❤❛✐ ✤✐➸♠ C, E s❛♦ ❝❤♦ ❣â❝ CIA ✈➔ EIB ❧➔ ❣â❝ ♥❤å♥✳ CI ❝➢t ✤÷í♥❣ trá♥ (O) t↕✐ ✤✐➸♠ D ❦❤→❝ C EI ❝➢t ✤÷í♥❣ trá♥ (O) t↕✐ ✤✐➸♠ F E t t ợ ữớ trỏ (O) t↕✐ C ✈➔ D ❝➢t ♥❤❛✉ t↕✐ M ✱ t t ợ ữớ trỏ (O) t E F ❝➢t ♥❤❛✉ t↕✐ N ✳ ◆è✐ OM ❝➢t CD t↕✐ P ✈➔ ON ❝➢t EF t↕✐ Q✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ M N AB ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â OC = OD M C = M D ✭✈➻ trü❝ ❝õ❛ CD ✱ ❞♦ ✤â ✭✈➻ ❝ò♥❣ ❧➔ ❜→♥ ❦➼♥❤✮✱ M C, M D ❧➔ ✷ t✐➳♣ t✉②➳♥ ❝➢t ♥❤❛✉✮ ♥➯♥ OM ❧➔ tr✉♥❣ OM ⊥ DP ❳➨t ODM ❝â ODM = 90◦ ✭✈➻ M D ❧➔ t✐➳♣ t✉②➳♥ ❝õ❛ (O) t↕✐ D✮✱ OM ⊥ DP s✉② r❛ OD2 = OP.OM ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ✤÷đ❝ OF = OQ.ON ✈➔ OD = OF ✳ ❉♦ ✤â OP.OM = OQ.ON ⇒ ❳➨t OP Q ✈➔ ON M ❝â O ❝❤✉♥❣✱ OP ON = OQ OM OP ON = OQ OM ♥➯♥ OP Q ∼ ON M ♥➯♥ OP Q = ON M ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ tù ❣✐→❝ OP IQ ✭✸✳✶✽✮ OM ⊥ DP, ON ⊥ F Q ♥➯♥ OP I = OQI = 90◦ ✳ ❱➟② ♥ë✐ t✐➳♣ ♥➯♥ QOI = QP I ❚ø ✭✸✳✶✽✮ ✈➔ ✭✸✳✶✾✮ s✉② r❛ ✭✸✳✶✾✮ ✼✻ ❉♦ ✤â MN✱ ON T ✈✉æ♥❣ t↕✐ ♠➦t ❦❤→❝ ❇➔✐ t♦→♥ ✸✳✶✾ T ✭T OI ⊥ AB ✭✈➻ ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ OI IA = IB = AB ✮✳ ✈➔ ❱➟② M N ✮✳ MN ❙✉② r❛ OI ⊥ BC ✳ ❈❤♦ t❛♠ ❣✐→❝ ✭❱á♥❣ r ợ ABC, ữớ ❝❛♦ BD ✈➔ CE P ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ D tr➯♥ AB, ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ P tr➯♥ BC ✳ P Q ❝➢t BD t↕✐ K ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ EK Q ❧➔ AC ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ H ❧➔ trü❝ t➙♠ t❛♠ ❣✐→❝ ABC ✳ Q ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ P tr➯♥ BC ✈➔ AH ⊥ BC ❞♦ ✤â AH P Q✳ DP ⊥ AB ♥➯♥ DP CE ✳ ❚❤❡♦ ✤à♥❤ ❧➼ ❚❤❛❧❡s t❛ BH BE = ❚❛ ❝â ✤➥♥❣ t❤ù❝ BD BP BK BK BH BP BE BE = = = BD BH BD BA BP BA ❚❛ ❝â✱ ❚❤❡♦ ✤à♥❤ ❧➼ ❚❤❛❧❡s ✤↔♦ t❛ ❝â ❇➔✐ t♦→♥ ✸✳✷✵ EK AC ❝â BK BP = BH BA ✈➔ ✳ ❈❤♦ t❛♠ ❣✐→❝ ♥❤å♥ ABC ❝â ✭■▼❖ ❙❤♦rt❧✐st ♥➠♠ ✷✵✶✷✱ ❬✶❪✮ ❝→❝ ✤÷í♥❣ ❝❛♦ AD, BE, CF ●å✐ M, N ❧➔ t➙♠ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ t❛♠ ❣✐→❝ BF D, CDE ✈➔ P, Q ❧➔ t➙♠ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABM, ACN ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ M N P Q ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ AD, BE, CF ❧➔ ❝→❝ ✤÷í♥❣ ❝❛♦ ♥➯♥ F BD = DEC ✈➔ BF D = ECD ❞♦ ✤â BF D ∼ ECD✳ ❚❤❡♦ ❣✐↔ t❤✐➳t ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❤❛✐ t❛♠ ❣✐→❝ ✤â ♥➯♥ FD DM = , M DN = F DC CD DN ❉♦ ✤â DM N ∼ DF C ✱ s✉② r❛ DM N = DF C = DAC ✳ M, N ❧➔ t➙♠ ✼✼ ▼➦t ❦❤→❝✱ BM N = BM D + DM N = 90◦ + BF D + DF C 1 = 90◦ + ACB + 90◦ − ACB = 180◦ − ACB 2 ◦ = 180 − N CB ❉♦ ✈➟②✱ tù ❣✐→❝ BM N C ♥ë✐ t✐➳♣✳ K ❧➔ t➙♠ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ AEF ✱ t÷ì♥❣ tü s✉② r❛ ❝→❝ tù ❣✐→❝ BM KA, AKN C ❧➔ ❝→❝ tù ❣✐→❝ ♥ë✐ t✐➳♣✱ ❞♦ ✤â AK ❧➔ ❞➙② ❝❤✉♥❣ ❝õ❛ ❤❛✐ ✤÷í♥❣ trá♥ (BM KA) ✈➔ (AKN C)✳ ❚❤❡♦ ❣✐↔ t❤✐➳t P, Q ❧➔ t➙♠ ❝õ❛ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABM, ACN s✉② r❛ P, Q ❝❤➼♥❤ ❧➔ t➙♠ ❝õ❛ ❤❛✐ ✤÷í♥❣ trá♥ (BM KA) ✈➔ (AKN C) ⇒ AK ⊥ P Q ▼➦t ❦❤→❝✱ ❜❛ ✤✐➸♠ M, N, K ❧➔ t➙♠ ❝õ❛ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ❝→❝ t❛♠ ❣✐→❝ BDF, CDE, AEF ♥➯♥ M, N, K ♥➡♠ tr➯♥ BI, CI, AI ✱ s✉② r❛ ●å✐ ❉♦ ✤â IKN = N CA = ACB IM N = N CB = ACB ✳ ❱➟② IKN = IM N ✳ ❚÷ì♥❣ tü✱ t❛ ❝â IM K = IN K, IKM = IN M ❙✉② r❛ I ❧➔ trü❝ t➙♠ ❝õ❛ ❇➔✐ t♦→♥ ✸✳✷✶ ABC ✳ KM N ⇒⇒ KI⊥M N ⇒ P Q ✭■▼❖ ♥➠♠ ✷✵✶✽✮ M N ✳ ❈❤♦ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ♥❤å♥ ❈→❝ ✤✐➸♠ D, E ❧➛♥ ❧÷đt ♥➡♠ tr➯♥ AB ✈➔ AC s❛♦ ❝❤♦ AD = AE ✳ ❚r✉♥❣ trü❝ BD ✈➔ CE ❝➢t ❝✉♥❣ ♥❤ä AB ✈➔ AC ❝õ❛ τ t↕✐ F ✈➔ G✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ DE ✈➔ F G s♦♥❣ s♦♥❣ ❤♦➦❝ trò♥❣ ♥❤❛✉✳ ✼✽ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â F BD = F DB = ADI ✱ ♠➔ AIF = F BA ✭❣â❝ ♥ë✐ t✐➳♣ F A✮ ❤❛② AID = F BD✳ ❙✉② r❛ ADI = AID✳ ❱➟② t❛♠ ❣✐→❝ A ♥➯♥ AD = AI ✳ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â AE = AH ✳ ❝ò♥❣ ❝❤➢♥ ❝✉♥❣ AID ❝➙♥ t↕✐ ❚❛ ❝â AD = AE; AD = AI; AE = AH ⇒ AE = AH = AD = AI ❙✉② r❛ tù ❣✐→❝ HDEI ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ t➙♠ HIF = HGF ✣÷í♥❣ trá♥ t➙♠ A ❜→♥ ❦➼♥❤ AD ❤❛② A✱ ❜→♥ ❦➼♥❤ HID = HGF GF ✳ ❚❛ ❝â ✭✸✳✷✵✮ ❝â HID = HED ❚ø ✭✸✳✷✵✮ ✈➔ ✭✸✳✷✶✮ s✉② r❛ AD✳ HED = HGF ⇒ ED ✭✸✳✷✶✮ s♦♥❣ s♦♥❣ ❤♦➦❝ trò♥❣ ✼✾ ❑➳t ❧✉➟♥ ❱ỵ✐ ♠ư❝ t✐➯✉ tü ❝❤✉➞♥ ❜à ❝❤♦ ❜↔♥ t❤➙♥ ♠ët ❝❤✉②➯♥ ✤➲ ✤➸ ♣❤ö❝ ✈ö ❜↔♥ t❤➙♥ tr♦♥❣ ❞↕② ❤å❝ t♦→♥ ð ❚r✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣ ✈➔ ①✉➜t ♣❤→t tø t❤ü❝ t✐➵♥ ❝❤♦ t❤➜② ❣➛♥ ✤➙② ❤å❝ s✐♥❤ r➜t sñ ❤➻♥❤ ❤å❝✱ ♥❤➜t ❧➔ ❝→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ ♥➯♥ tæ✐ ✤➣ ❝❤å♥ ❝❤õ ✤➲✿ ✏❇➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ s♦♥❣ s♦♥❣ ✈➔ ✈✉æ♥❣ ❣â❝ tr♦♥❣ ❤➻♥❤ ❤å❝✳✑ ▲✉➟♥ ✈➠♥ ✤➣ ❤♦➔♥ t❤➔♥❤ ❝→❝ ♥❤✐➺♠ ✈ö s❛✉✿ ✶✳ ❚➻♠ ỵ t t q ✤✐➲✉ ❦✐➺♥ ✤➸ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ ✭❤❛② ✈✉ỉ♥❣ õ ợ ụ ữ q õ ữủ tø ✈✐➺❝ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ ✭❤❛② ✈✉ỉ♥❣ ❣â❝✮✳ ✷✳ ❙÷✉ t➛♠ ❝→❝ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ s♦♥❣ s♦♥❣ ❤❛② ✈✉æ♥❣ ❣â❝✳ ❚r➯♥ ❝ì sð ✤â✱ ❧✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ s❛✉✿ ✶ ●✐ỵ✐ t❤✐➺✉ ♠ët sè ✤à♥❤ ỵ t t q ởt tt ❝❤ù♥❣ ♠✐♥❤ t❤÷í♥❣ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ s♦♥❣ s♦♥❣ ❤❛② ✈✉æ♥❣ ❣â❝✳ ✷ ❚r➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝â ❝❤å♥ ❧å❝ ❝→❝ ✤➲ t❤✐ t✉②➸♥ s✐♥❤ ✈➔♦ ❧ỵ♣ ✶✵ trữớ tr ữợ q ✣➲ t❤✐ ❝❤å♥ ✤ë✐ t✉②➸♥❀ ✣➲ t❤✐ ❍å❝ s✐♥❤ ❣✐ä✐ tr ữợ qố t q ❧í✐ ❣✐↔✐ ❝õ❛ ❝→❝ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ t♦→♥ ✈➲ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ s♦♥❣ s♦♥❣✱ t➼♥❤ ✈✉æ♥❣ ❣â❝✱ tr♦♥❣ ✤â ❝â ♥❤✐➲✉ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ❝õ❛ ♥❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙② ✤➸ ♠✐♥❤ ❤å❛ ❝❤♦ ♥❣÷í✐ ✤å❝ t❤➜② ❞↕♥❣ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ s♦♥❣ s♦♥❣✱ ✈✉ỉ♥❣ ❣â❝ t✉② ❧➔ ♥❤ú♥❣ ❜➔✐ t♦→♥ ❝ì ❜↔♥ ❝õ❛ ❤➻♥❤ ❤å❝ ♥❤÷♥❣ ✈➝♥ ❧➔ ♠ët ❝❤✉②➯♥ ✤➲ ✏♥â♥❣✑ tr♦♥❣ ✈✐➺❝ ❝❤å♥ ❤å❝ s✐♥❤ ❣✐ä✐✳ ✸ ✣è✐ ✈ỵ✐ ♠ët ✈➔✐ ❜➔✐ t♦→♥✱ ❧✉➟♥ ✈➠♥ ✤➣ ❝è ❣➢♥❣ ✤÷❛ r❛ ❧í✐ ❣✐↔✐ ❝❤✐ t✐➳t ❤ì♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✭✤➸ ❤å❝ s✐♥❤ ❚r✉♥❣ ❤å❝ ❝ì sð ❞➵ ❞➔♥❣ ❤ì♥ ❦❤✐ ✤å❝ ❝→❝ ❧í✐ ❣✐↔✐✮✳ ✣è✐ ✈ỵ✐ ♠ët ✈➔✐ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐✱ ❧✉➟♥ ✈➠♥ ❝ơ♥❣ ✤➣ ❝è ❣➢♥❣ ✤÷❛ r❛ ♥❤✐➲✉ ❧í✐ ❣✐↔✐ ✤➸ ♠✐♥❤ ❤å❛ t➼♥❤ ❧✐♥❤ ❤♦↕t tr t t ỵ ❝❤ù♥❣ ♠✐♥❤ ❜➔✐ t♦→♥ ✈➲ t➼♥❤ s♦♥❣ s♦♥❣✱ t➼♥❤ ✈✉æ♥❣ õ ữợ ự trữợ t t ự s s♦♥❣✱ ✈✉ỉ♥❣ ❣â❝ tr♦♥❣ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ✈➔ ✤➙② ❝ơ♥❣ ❧➔ ♥❤✐➺♠ ✈ö t✐➳♣ t❤❡♦ ❝õ❛ ❤å❝ ✈✐➯♥ ✤➸ ❝â ♥❤ú♥❣ ❝❤✉②➯♥ ✤➲ ♣❤ư❝ ✈ư ❝ỉ♥❣ ✈✐➺❝ ❣✐↔♥❣ ❞↕② ❚♦→♥ ❝õ❛ ♠➻♥❤ ð ❚r÷í♥❣✳ ✽✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❘➧♥ ❧✉②➺♥ ♣❤→t tr✐➸♥ t÷ ❞✉② t❤ỉ♥❣ q✉❛ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❤➻♥❤ ❤å❝ ♣❤➥♥❣✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ö❝ ❱✐➺t ◆❛♠✳ ❬✷❪ ◆❣✉②➵♥ ❇→ ✣❛♥❣ ✭✷✵✶✽✮✱ P❤→t tr✐➸♥ ❦❤↔ ♥➠♥❣ ❣✐↔✐ t♦→♥ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ❞➔♥❤ ❝❤♦ ❜➟❝ t sữ P ỗ ❈❤➼ ▼✐♥❤✳ ❬✶❪ ◆❣✉②➵♥ ❇→ ✣❛♥❣ ✭✷✵✶✽✮✱ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❱✐❦t♦r Pr❛s♦❧♦✈ ✭✷✵✵✻✮✱ Pr♦❜❧❡♠s ✐♥ ♣❧❛♥❡ ❛♥❞ s♦❧✐❞ ●❡♦♠❡tr②✳ ❚r❛♥s✲ ❧❛t❡❞ ❛♥❞ ❡❞✐t❡❞ ❜② ❉✐♠✐tr② ▲❡✐t❡s✱ ▼♦s❝♦✇ t❡①t❜♦♦❦s✳ ❖♥ t❤❡ ◆♦t✐♦♥ ♦❢ ❖r✐❡♥t❡❞ ❆♥❣❧❡s ✐♥ P❧❛♥❡ ❊❧❡♠❡♥t❛r② ●❡♦♠❡tr② ❛♥❞ ❙♦♠❡ ♦❢ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ▼▼ ❘❡s❡❛r❝❤ Pr❡♣r✐♥ts✱ ✶✕✶✷ ❬✹❪ ❲✉ ❲✳❚ ✭✷✵✵✺✮✱ ❑▲▼▼✱ ❆▼❙❙✱ ❆❝❛❞❡♠✐❛ ❙✐♥✐❝❛ ❱♦❧✳ ✷✹✳ ... HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC  - NGỌC THỊ HÀ BÀI TOÁN CHỨNG MINH TÍNH VNG GĨC, SONG SONG TRONG HÌNH HỌC Chun ngành: Phương pháp Toán sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC... TRONG HÌNH HỌC Chun ngành: Phương pháp Toán sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC PGS.TS Trịnh Thanh Hải THÁI NGUYÊN - 2019 ✐ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝➠♥