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 TỐN 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: 8 46 01 13

LUẬN VĂN THẠC SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC
PGS.TS. Trịnh Thanh Hải

THÁI NGUYÊN - 2019

ử ử

tự


ỵ ✤➲ ✈➲ t➼♥❤ ✈✉æ♥❣ ❣â❝✱ s♦♥❣ s♦♥❣ tr♦♥❣ ❤➻♥❤
❤å❝ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✶✳✶

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✶✳✷

❈→❝ t➼♥❤ ❝❤➜t ✈➲ t➼♥❤ ✈✉æ♥❣ ❣â❝✱ s♦♥❣ s♦♥❣ tr♦♥❣ ❤➻♥❤
❤å❝ ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ỵ t➼♥❤ s♦♥❣ s♦♥❣ ✈➔ ✈✉æ♥❣ ❣â❝

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❤❛♠ ❦❤↔♦
✽✶




ớ ỡ
rữợ t ỷ ớ ỡ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ♥❤➜t tỵ✐ P●❙✳❚❙✳

❚rà♥❤ ❚❤❛♥❤ ❍↔✐✱ ữớ t ợ ỏ t t ổ t t
tứ ỳ t ỗ tớ ✤÷❛ r❛ ♥❤ú♥❣ ❧í✐ ❦❤✉②➯♥ ❜ê ➼❝❤
❣✐ó♣ ❡♠ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳
❊♠ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❝→❝ t❤➛② ❝æ✱ t➟♣ t❤➸ ❝→♥ ❜ë ❦❤♦❛ ❚♦→♥
✲ ❚✐♥✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱
ỗ r t ữợ ✈➔ ●✐→♦ ❞ư❝ t❤÷í♥❣ ①✉②➯♥ t➾♥❤
◗✉↔♥❣ ◆✐♥❤✱ ❝ị♥❣ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❧ỵ♣ ❝❛♦ ❤å❝ ❚♦→♥ ❑✶✶❉✱ ✤➣ ❦❤ỉ♥❣ ❝❤➾
tr❛♥❣ ❜à ❝❤♦ ❡♠ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❜ê ➼❝❤ ♠➔ ❝á♥ ❧✉ỉ♥ ❣✐ó♣ ✤ï✱ t↕♦ ✤✐➲✉ ❦✐➺♥
t❤✉➟♥ ❧đ✐ 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❝ ❧✉ỉ♥ ❧➔ ❝→❝ ❜➔✐ t➟♣ t❤ó ✈à ♥❤÷♥❣ t❤÷í♥❣
r➜t ❦❤â✳ ✣➦❝ ❜✐➺t ❧➔ ♥❤ú♥❣ ❜➔✐ t♦→♥✱ ✤➲ t❤✐ ❞➔♥❤ ❝❤♦ ❤å❝ s✐♥❤ ❣✐ä✐ t❤➻ ❤å❝
s✐♥❤ ♣❤↔✐ ♥➢♠ ✤÷đ❝ ❝→❝ ❦✐➳♥ tự ỵ t t
❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝❤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➯♥ ♥❣➔♥❤ P❤÷ì♥❣ ♣❤→♣
t♦→♥ sì

ợ ố t ỵ t t ❝ơ♥❣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣
❝❤ù♥❣ ♠✐♥❤ t➼♥❤ s♦♥❣ s♦♥❣✱ t➼♥❤ ✈✉ỉ♥❣ ❣â❝ q✉❛ ♠ët sè ❜➔✐ t♦→♥✱ ✤➲ t❤✐ ❤å❝
s✐♥❤ ❣✐ä✐ ✤➸ ❧➔♠ t➔✐ ❧✐➺✉ ❝❤♦ ✈✐➺❝ ❣✐↔♥❣ ❞↕② ❝õ❛ ❜↔♥ t❤➙♥ ✈➔ ❧➔♠ t➔✐ ❧✐➺✉
t❤❛♠ ❦❤↔♦ ❝❤♦ ❤å❝ s✐♥❤ tü ❤å❝✱ tỉ✐ ❝❤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♦♥❣ ✭❤❛② ✈✉ỉ♥❣ õ ợ ụ ữ q õ ữủ
tứ ữớ 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 ữ r ớ tữớ ố ợ ỳ
t♦→♥✱ ✤➲ t❤✐ ♠➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➾ ❝â ớ tt
ữợ ớ

ã

ố ợ ởt t ố ữ r ♥❤✐➲✉ ❧í✐ ❣✐↔✐ ✤➸ ♠✐♥❤ ❤å❛ t➼♥❤
❧✐♥❤ ❤♦↕t tr♦♥❣ ✈✐➺❝ ử t t ỵ ự
t♦→♥ ✈➲ t➼♥❤ s♦♥❣ s♦♥❣✱ t➼♥❤ ✈✉ỉ♥❣ ❣â❝✳

❱ỵ✐ ♠ư❝ t✐➯✉ ♥❣❤✐➯♥ ❝ù✉ ♥❤÷ ✈➟②✱ ❜è ❝ư❝ ❝õ❛ ❧✉➟♥ ✈➠♥ ❜❛♦ ỗ ữỡ

ữỡ tự

ở ữỡ tố t t ỵ ✈➔ ♣❤÷ì♥❣
♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❜➔✐ t♦→♥ ✈➲ t➼♥❤ ✈✉ỉ♥❣ ❣â❝ ✭✤÷í♥❣ t❤➢♥❣✱ ❣â❝✮ ✈➔ t➼♥❤
s♦♥❣ s♦♥❣ tr♦♥❣ ❤➻♥❤ ❤å❝ 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 ữớ t ổ õ tr
ừ t

...

ỗ tớ ụ ữ r ởt số t ử

ỵ tr ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ✈✉ỉ♥❣ ❣â❝ ✈➔ s♦♥❣ s♦♥❣✳

❈❤÷ì♥❣ ✷✳ ❈→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ✈✉æ♥❣ ❣â❝ tr♦♥❣ ❝→❝ ✤➲
t❤✐ ❍å❝ s✐♥❤ ❣✐ä✐
◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ tữớ ử
ỵ t t

...

ự ♠✐♥❤ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤

✈✉æ♥❣ ❣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➻♥❤ ởt tữớ
ử ỵ t➼♥❤ ❝❤➜t

. . . ✤➸ ❝❤ù♥❣ ♠✐♥❤ ♠ët 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 ❍➔




ữỡ

tự

ỵ ✈➲ t➼♥❤ ✈✉æ♥❣ ❣â❝✱ s♦♥❣ s♦♥❣
tr♦♥❣ ❤➻♥❤ ❤å❝ ♣❤➥♥❣

✶✳✶✳✶ ❑✐➳♥ tự

rữợ t ú 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 ổ ❣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✐ ♥❤❛✉✳




t
õ

A1

ứ ữợ ú t❛ ①→❝ ✤à♥❤ ❝→❝ ❝➦♣ ❣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 ❝➦♣ õ 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

ồ 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♥❣ trư❝ 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,α) ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ trữợ ởt O sốtỹ 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❤➔♥❤ ữớ t s s trũ ợ õ

t t t✐❛✱ ❜✐➳♥ ✤♦↕♥ t❤➥♥❣ t❤➔♥❤ ✤♦↕♥ t❤➥♥❣✳
✭✐✐✐✮ ❇✐➳♥ t❛♠ ❣✐→❝ t t ỗ ợ õ õ t ❣â❝
❜➡♥❣ ♥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✐ ữỡ t

ừ M ố ợ ữớ trỏ (O; R) PM/(O)
t q ừ ỵ s ✤➙② t❤÷í♥❣ ✤÷đ❝ ❞ị♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝
❜➔✐ t♦→♥ tr♦♥❣ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ✈➲ t➼♥❤ s♦♥❣ s♦♥❣ ✈➔ ✈✉æ♥❣ ❣â❝✱ ú t s
ọ q ự

ỵ

t❛♠ ❣✐→❝ ABC

✭❍➺ t❤ù❝ ❧÷đ♥❣ tr♦♥❣ t❛♠ ❣✐→❝ ✈✉ỉ♥❣✮

✈✉ỉ♥❣ t↕✐ A✱ ✤÷í♥❣ ❝❛♦ AH ✱ t❛ ❝â

AB 2 = BC.BH, AC 2 = BC.HC, AH 2 = BH.CH, BC.AH = AC.AH,
1
1
1
=
+
.
AH 2
AB 2 AC 2

ỵ M ♥❣♦➔✐ ✤÷í♥❣ trá♥ (O) t❛ ✈➩ ✤÷đ❝ t✐➳♣ t✉②➳♥ M T

tợ ữớ trỏ õ PM/(O) = M A.M B = M T 2✳


✼

✣à♥❤ ♥❣❤➽❛ ✶✳✾✳ ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ ❧➔ tù ❣✐→❝ ❝â ❜è♥ ✤➾♥❤ ❝ị♥❣

♥➡♠ tr➯♥ ✤÷í♥❣ trá♥✳ ✣÷í♥❣ trá♥ ✤â ✤÷đ❝ ❣å✐ ❧➔ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ tù
❣✐→❝✳

◆❤➟♥ ①➨t ✶✳✸✳

❚ù ❣✐→❝ ♥ë✐ t✐➳♣ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿

✭✐✮ ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ ❝â tê♥❣ ❤❛✐ ❣â❝ ✤è✐ ❜➡♥❣

180◦ .

✭✐✐✮ ❚ù ❣✐→❝ ❝â ❤❛✐ ✤➾♥❤ ❦➲ ❝ò♥❣ ♥❤➻♥ ①✉è♥❣ ởt ỏ ữợ ởt õ
t ở t✐➳♣✳
✭✐✐✐✮ ❚ù ❣✐→❝ ❝â ✹ ✤➾♥❤ ❝→❝❤ ✤➲✉ ♠ët ✤✐➸♠ trữợ t ở t
õ t ởt ❝õ❛ ♠ët tù ❣✐→❝ ❜➡♥❣ ❣â❝ tr♦♥❣ ✤è✐ ❞✐➺♥ ✈ỵ✐
✤➾♥❤ õ ừ tự t ở t

ỵ ❚ù ❣✐→❝ ABCD ❝â ❤❛✐ ❝↕♥❤ ✤è✐ AB, CD ❝➢t ♥❤❛✉ t↕✐ M ✳ ✣✐➲✉

❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ tù ❣✐→❝ ABCD ♥ë✐ t✐➳♣ ✤÷đ❝ ✤÷í♥❣ trá♥ ❧➔ M A.M B =

M C.M D.

ỵ ự ABCD õ ❤❛✐ ✤÷í♥❣ ❝❤➨♦ AC, BD ❝➢t ♥❤❛✉ t↕✐
N ✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔
N A.N C = N B.N D.

✤õ ✤➸ tự ở t ữủ ữớ trỏ

ỵ ✶✳✺✳ ❈❤♦ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ AB, M T ♣❤➙♥ ❜✐➺t ❝➢t ♥❤❛✉ t↕✐ M (M

❦❤ỉ♥❣ trị♥❣ A, B, T ✮✳ ❑❤✐ ✤â ♥➳✉ M A.M B
t✐➳♣ t❛♠ ❣✐→❝ ABT t✐➳♣ ú ợ M T t T

= MT 2

t ữớ trá♥ ♥❣♦↕✐

ữớ trỏ ổ ỗ t (O1, R1); (O2, R2)✳
❚➟♣ ❤đ♣ ❝→❝ ✤✐➸♠ M ❝â ♣❤÷ì♥❣ t ố ợ ữớ trỏ
ởt ữớ t❤➥♥❣✳ ✣÷í♥❣ t❤➥♥❣ ♥➔② ❣å✐ ❧➔ trư❝ ✤➥♥❣ ♣❤÷ì♥❣ ❝õ❛ ❤❛✐ ✤÷í♥❣
trá♥ ✤➣ ❝❤♦✳

◆❤➟♥ ①➨t ✶✳✹✳

❚rư❝ ✤➥♥❣ ♣❤÷ì♥❣ ❝õ❛ ❤❛✐ ✤÷í♥❣ trá♥ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿

✭✐✮ ❚rư❝ ✤➥♥❣ ♣❤÷ì♥❣ ừ ữớ trỏ ổ õ ợ ữớ ố t
◆➳✉ ❤❛✐ ✤÷í♥❣ trá♥ ❝➢t ♥❤❛✉ t↕✐

A

✈➔

B

t❤➻

AB

❝❤➼♥❤ ❧➔ trư❝ ✤➥♥❣

♣❤÷ì♥❣✳
✭✐✐✐✮
q

M

M

õ ũ ữỡ t ợ ữớ trỏ t ữớ t

ổ õ ợ ữớ ố t trử ✤➥♥❣ ♣❤÷ì♥❣✳

M, N
M N ❧➔

✭✐✈✮ ◆➳✉ ❤❛✐ ✤✐➸♠
✤÷í♥❣ t❤➥♥❣

❝â ❝ị♥❣ ữỡ t ố ợ ữớ trỏ t
trử ữỡ

õ ũ ữỡ t ợ ữớ trá♥ t❤➻ ❝❤ó♥❣ t❤➥♥❣
❤➔♥❣✳
✭✈✐✮ ◆➳✉

(O1 ), (O2 )

O1 O2

❝➢t ♥❤❛✉ t

trử ữỡ

A

t ữớ t q

A

ổ õ ợ


✾

✶✳✶✳✷ ❈→❝ t➼♥❤ ❝❤➜t ✈➲ t➼♥❤ ✈✉æ♥❣ ❣â❝✱ s♦♥❣ s♦♥❣ tr♦♥❣ ❤➻♥❤ ❤å❝
♣❤➥♥❣
❚➼♥❤ ❝❤➜t ✶✳✶✳
a
O
❈â ♠ët ✈➔ ❝❤➾ ♠ët ✤÷í♥❣ t

õ ợ ữớ t

t

a

q

ổ

trữợ

c

✤÷í♥❣ t❤➥♥❣

❝➢t ❤❛✐ ✤÷í♥❣ t❤➥♥❣

a, b

✈➔ tr♦♥❣ ❝→❝

❣â❝ t↕♦ t❤➔♥❤ ❝â ♠ët ❝➦♣ ❣â❝ s♦ ❧❡ tr♦♥❣ ❜➡♥❣ ♥❤❛✉ t❤➻
✭✐✮ ❍❛✐ ❣â❝ s♦ ❧❡ tr♦♥❣ ❝á♥ ❧↕✐ ❜➡♥❣ ♥❤❛✉❀
✭✐✐✮ ❍❛✐ õ ỗ

t

ởt ữớ t ❝➢t ❤❛✐ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ t❤➻

✭✐✮ ❍❛✐ ❣â❝ s♦ tr
õ ỗ
❍❛✐ ❣â❝ tr♦♥❣ ❝ị♥❣ ♣❤➼❛ ❜ị ♥❤❛✉✳

❚➼♥❤ ❝❤➜t ✶✳✹✳

❍❛✐ ✤÷í♥❣ t t ũ ổ õ ợ ởt ữớ

t tự t ú s s ợ

t

ởt ữớ t ổ õ ợ ởt tr ữớ t

s s t õ ụ ổ õ ợ ữớ t

t

ữớ t t ũ s s ợ ởt ữớ

t tự t ú s s ợ

ỵ t s s ổ õ tr
ồ
ỵ
ởt ữớ t t
ỵ s tr t

ừ ởt t ❣✐→❝ ✈➔ s♦♥❣ s♦♥❣ ✈ỵ✐ ❝↕♥❤ ❝á♥ ❧↕✐ t❤➻ ♥â ✤à♥❤ r❛ tr➯♥
❤❛✐ ❝↕♥❤ ❝á♥ ❧↕✐ ♥❤ú♥❣ ✤♦↕♥ t❤➥♥❣ t➾ ❧➺✳
❈❤ù♥❣ ♠✐♥❤✳ ❳➨t t❛♠ ❣✐→❝ ABC ✈➔ ❣✐↔ sû ✤÷í♥❣ t❤➥♥❣ xx BC ✱ ❝➢t ❝↕♥❤
AB

✈➔

AC

t÷ì♥❣ ù♥❣ t↕✐

D

✈➔

E✳

❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤

AE
AD
=
.
DB
EC

✭✶✳✶✮


✶✵

❱➻

DE

❚r♦♥❣

BC ✱ ♥➯♥ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝ DEB
ABE ❦➫ ✤÷í♥❣ ❝❛♦ EF. ❑❤✐ ✤â
SADE
SBDE

❜➡♥❣ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝

DEC ✳

1

AD.EF
AD
= 2
=
.
1
BD
BD.EF
2

✭✶✳✷✮

SADE
AE
=
.
SCDE
EC

✭✶✳✸✮

❚÷ì♥❣ tü t❛ ❝â

❚ø ✭✶✳✷✮ ✈➔ ✭✶✳✸✮ s✉② r❛ ❤➺ t❤ù❝

ỵ

ởt ữớ t t ừ

ỵ s

ởt t r tr ❤❛✐ ❝↕♥❤ ➜② ♥❤ú♥❣ ✤♦↕♥ t❤➥♥❣ t÷ì♥❣ ù♥❣ t➾ ❧➺
t❤➻ ữớ t õ s s ợ ỏ ừ t❛♠ ❣✐→❝✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ✤÷í♥❣ t❤➥♥❣ xx ❝➢t ❝→❝ ❝↕♥❤ AB, AC ❝õ❛ t❛♠ ❣✐→❝
ABC

t❤❡♦ t❤ù tü t↕✐

D

✈➔

E

s❛♦ ❝❤♦

AB
AC
=
.
DB
EC
❚❛ ự


D

DE

BC.

ữớ t s s ợ

BC

t

AC

t

E

ỵ s t t õ

AD
AE
AE
AE
AE
AE
=

=

+1=
+1
DB
EC
EC

EC
EC
EC
AE + E C
AE + EC
AC
AC

=

=
EC
EC
EC
EC


E C = EC

ỵ

tự

EE

õ

DE

BC.

r ởt t ổ ữỡ

ỵ Ptrs

ở ❤✉②➲♥ ❜➡♥❣ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ✤ë ❞➔✐ ❤❛✐ ❝↕♥❤ ❣â❝ ✈✉æ♥❣✳


✶✶

❈❤ù♥❣ ♠✐♥❤✳ ❚r➯♥ BC

❧➜② ❤❛✐ ✤✐➸♠

M, N

t❤ä❛ ♠➣♥

BM = BN = AB.

❑❤✐ ✤â✱

❉♦

1
1
BN A = BAN = 90◦ − ABC, N AC = 90◦ − BAN = ABC,
2
2
1

AM B = ABC.
2
✤â✱
M CA ∼ ACN (g.g) ♥➯♥ t❛ ❝â
MC
CA
AB + BC
AC
=
⇒
=
.
AC
CN
AC
BC − AB

❉♦ ✈➟②

BC 2 = AB 2 + AC 2 .

ỵ

ữỡ ở ởt

ỵ Ptrs

ừ t tờ ữỡ ở ❞➔✐ ❝õ❛ ❤❛✐ ❝↕♥❤ ❦✐❛✱ t❤➻ ❣â❝ ♥➡♠
❣✐ú❛ ❤❛✐ ❝↕♥❤ ❝õ❛ t❛♠ ❣✐→❝ ✤â ❜➡♥❣ ❣â❝ ✈✉æ♥❣✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû
t❤➥♥❣ ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐

ABC ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ t❛♠ ❣✐→❝ ✈✉ỉ♥❣✱ tø B ữớ
AC t AC t D ỵ P②t❤❛❣♦r❛s t❛ ❝â
BC 2 = DB 2 + DC 2 .

❚❤❡♦ ❣✐↔ t❤✐➳t

BC 2 = AB 2 + AC 2 .

❙✉② r❛

AB 2 − DB 2 = DC 2 − AC 2 ⇒ AD2 = AD(DC + AC)
❉♦ ✤â

AD = DC + AC

ỵ

t

t ABC D, E, F

ỵ

ữủt tr BC, AC, AB. ự AD, BE, CF ỗ q✉② ❤♦➦❝ ✤æ✐
♠ët s♦♥❣ s♦♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
DB EC F A
·

·
= −1.
DC EA F B

✭✶✳✹✮


✶✷

❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥✿ ●✐↔ sû AD, BE, CF ỗ q ứ A ữớ
t s s ợ

BC

t

BE, CF

t

I



H

ỵ s t õ

DB
IA EC

BC F A
AH
=
;
=
;
=
.
DC
HA EA
IA F B
BC
DB EC F A
Ã
Ã
= 1.
õ
DC EA F B
ợ trữớ ủ AD
BE CF ử ỵ s

t ụ ❝â ❦➳t

q✉↔

DB EC F A
·
·
= −1.
DC EA F B

✣✐➲✉ ❦✐➺♥ ✤õ✿ ●✐↔ sû t❛ ❝â
●å✐

H, I

G

✈➔

F

❧➛♥ ❧÷đt ❧➔

DB EC F A
·
·
= −1.
DC EA F B
❣✐❛♦ ✤✐➸♠ ❝õ❛ AD ❝➢t BE ✱ GC

✭✶✳✺✮
❝➢t

AB ✳

❈→❝ ✤✐➸♠

♥❤÷ tr♦♥❣ ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ r

ứ s r

ỵ

DB EC F A
Ã
Ã
= 1.
DC EA F B
F F

ỵ ▼❡♥❡❧❛✉s tr♦♥❣ t❛♠ ❣✐→❝✮

✭✶✳✻✮

✳ ❈❤♦ t❛♠ ❣✐→❝ ABC ✱ tr➯♥

❝→❝ ✤÷í♥❣ t❤➥♥❣ ❝❤ù❛ ❝→❝ ❝↕♥❤ BC, CA, AB ❧➜② ❝→❝ ✤✐➸♠ P, Q, R t÷ì♥❣ ù♥❣
s❛♦ ❝❤♦ ♠é✐ ✤✐➸♠ ❦❤ỉ♥❣ trị♥❣ ✈ỵ✐ ✤➾♥❤ t❛♠ ❣✐→❝✳ ❑❤✐ ✤â✱ ❜❛ ✤✐➸♠ P, Q, R
t❤➥♥❣ ❤➔♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
RB P C QA
·
·
= 1.
RA P B QC

✭✶✳✼✮


✶✸

❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥✿

●✐↔ sû ❜❛ ✤✐➸♠

❦➫ ✤÷í♥❣ t❤➥♥❣ s s ợ
s t õ



AL

BC

t ữớ

P, Q, R t A
t (d) t L ỵ

LA
QA
CP Ã QA
=
⇔
,
PC
PC
QC

✭✶✳✽✮

RB
PB
RB LA
=
⇔
·
=1
RA
LA
RA P B

✭✶✳✾✮

ð ✭✶✳✽✮ ✈➔♦ ✭✶✳✾✮ t❛ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

✣✐➲✉ ❦✐➺♥ ✤õ✿ ●✐↔ sû t❛ ❝â

RB P C QA
·
·
= 1.
RA P B QC
●å✐

Q

❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛

PR

✈➔ ❝↕♥❤

AC.

❑❤✐ ✤â t❤❡♦ ✤✐➲✉ ❦✐➺♥ ❝➛♥ t❛ ❝â

RB Q A P C
·
·
= 1.
RA Q C P B
❚ø ✭✶✳✼✮ ✈➔ ✭✶✳✶✵✮ t❛ s r

ỵ

QA
QA
=

QC
QC





QQ.

tự ABCD

ỵ ▼❡♥❡❧❛✉s tr♦♥❣ tù ❣✐→❝✮

♠ët ✤÷í♥❣ t❤➥♥❣ (d) ❝➢t AB, BC, CD, DA ❧➛♥ ❧÷đt ð M, N, P, Q. ❑❤✐ ✤â t❛
❝â
M A N B P C QD
·
·
·
= 1.
M B N C P D QA

✭✶✳✶✶✮


✶✹

❈❤ù♥❣ ♠✐♥❤✳ ❚r➯♥ ✤÷í♥❣ t❤➥♥❣ (d) ❧➜② ❤❛✐ ✤✐➸♠ I, J
CD.

s

AI

BJ

ỵ s t õ

MA
JA N B

JB OD
PD
=
,
=
,
=
.
MB
JB N C
P C OA
IA
❉♦ ✤â

M A N B P C QD
IA JB P C P D
·
·
·
=
·
·
·
= 1.
M B N C P D QA
JB P C P D IA

ỵ

ự ỗ ABCD ở t ởt ữớ

ỵ Pt

trỏ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tê♥❣ ❝õ❛ t➼❝❤ ❝→❝ ❝➦♣ ❝↕♥❤ ✤è✐ ❞✐➺♥ ❜➡♥❣ t➼❝❤ ❤❛✐ ✤÷í♥❣
❝❤➨♦✱ ♥❣❤➽❛ ❧➔
AB.CD + AD.BC = AC.BD

✭✶✳✶✷✮

❈❤ù♥❣ ♠✐♥❤✳ ▲➜② M t❤✉ë❝ ✤÷í♥❣ ❝❤➨♦ AC s❛♦ ❝❤♦ ABD = M BC. ❑❤✐ ✤â✱
ABD ✈➔ M BC ❝â ABD = M BC, ADB = M CB ✳
M BC (g.g). ❉♦ ✤â t❛ ❝â

①➨t

▼➦t ❦❤→❝✱

AD
MC
=
⇒ AD · BC = BD · M C.
BD
BC
BA
BM
=
✈➔ ABM = DBC ♥➯♥
ABM ∼
BD
BC

AB
BD
=
⇒ AB · CD = AM · BD
AM
BC

◆➯♥

ABD ∼

✭✶✳✶✸✮

DBC

❚ø ✭✶✳✶✸✮ ✈➔ ✭✶✳✶✹✮ t❛ ✤÷đ❝

AD.BC + AB.CD = BD.M C + AM.BD = AC.BD
⇒ AB.CD + AD.BC = AC.BD.

s✉② r❛

✭✶✳✶✹✮




ỵ

ỵ rt

t ABC õ M, N, P t❤❡♦ t❤ù

tü ♥➡♠ tr➯♥ ❝→❝ ❝↕♥❤ BC, CA, AB. ❱➩ ❝→❝ ✤÷í♥❣ t❤➥♥❣ d1, d2, d3 ✈✉ỉ♥❣ ❣â❝
✈ỵ✐ BC, CA, AB t❤❡♦ t❤ù tü t↕✐ M, N, P. ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✤✐➲✉ ❦✐➺♥ ❝➛♥
✈➔ ✤õ ✤➸ M, N, P ỗ q t õ tự
M B 2 + N C 2 + P A2 = M C 2 + N A 2 + P B 2

✭✶✳✶✺✮

❈❤ù♥❣ ồ O ỗ q ừ d1, d2, d3 ử
ỵ Ptrs t õ

M B 2 = OB 2 − OM 2 , N C 2 = OC 2 − ON 2 , P A2 = OA2 − OP 2
⇒ M B 2 + N C 2 + P A2 = (OB 2 − OM 2 ) + (OC 2 − ON 2 ) + (OA2 − OP 2 )
= (OC 2 − OM 2 ) + (OA2 − ON 2 ) + (OB 2 − 0P 2 ) = M C 2 + N A2 + P B 2 .

✣✐➲✉ ❦✐➺♥ ✤õ✿ ●✐↔ sû ❝â ❤➺ t❤ù❝ ✭✶✳✶✺✮✳ ●å✐ O ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ d2, d3✳ ❱➩

OM ⊥ BC (M ∈ BC)✳

❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ð ✤✐➲✉ ❦✐➺♥ ❝➛♥✱ t❛ ❝â

M B 2 + N C 2 + P A2 = M C 2 + N A 2 + P B 2 ⇒ M B 2 = M B 2
⇒ MB = M B M M


d1 , d2 , d3

ỗ q t

ỵ

O

ố A, B, C, D t tr

ỵ

t õ AB CD ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ AC 2 − AD2 = BC 2 − BD2.
❈❤ù♥❣ ♠✐♥❤✳ ●å✐ H, K ❧➛♥ ❧÷đt ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ A, B ❧➯♥ ✤÷í♥❣ t❤➥♥❣
CD✳

◆➳✉

AB ⊥ CD

t

HK

t ỵ Ptrs t õ

AC 2 AD2 = HC 2 − HD2 = BC 2 − BD2 .
◆❣÷đ❝ ❧↕✐✱ ♥➳✉

AC 2 − AD2 = BC 2 − BD2

t❤➻ t❛ ❝â

a = AC 2 −AD2 = HC 2 −HD2 = HC 2 −(CD ± HC)2 ⇒ HC = ±

a + CD2
2CD

a + CD2
✣ê✐ ✈❛✐ trá H ❝❤♦ K t❛ ❝ô♥❣ ❝â KC = ±
. ❉♦ ✤â HC = KC. ▲↕✐
2CD
✤ê✐ ✈❛✐ trá C ❜ð✐ D t❛ ❝ô♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ HD = KD. ◆❤÷ ✈➟② H ≡ K ✱
s✉② r❛ AB ⊥ CD.




ỵ

ự õ ữớ ổ õ

ự ỗ ABCD

õ ữớ AC BD ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ AB + CD2 = AD2 + BC 2.
❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥✿ ●✐↔ sû AC ⊥ BD K ừ AC


BD

2

ỵ P②t❤❛❣♦r❛s t❛ ❝â

AB 2 + CD2 = KA2 + KB 2 + KC 2 + KD2
= KA2 + KD2 + KB 2 + KC 2 = AD2 + BC 2 .

✣✐➲✉ ❦✐➺♥ ✤õ✿ ●✐↔ sû t❛ ❝â
AB 2 + CD2 = AD2 + BC 2 .
✣➦t

α = AKB.

❑❤✐ ✤â t❛ ❜✐➸✉ ❞✐➵♥

KA2 +KB 2 −2KA.KB. cos α+KC 2 +KD2 −2KC.KD. cos α = AB 2 +CD2
KA2 +KD2 −2KA.KD. cos α+KC 2 +KB 2 −2KC.KB. cos α = AD2 +BC 2 .
❱➟②

s✉② r❛

(KA.KB + KC.KD − KA.KD − KA.KC). cos α = 0.

= AC BD.
2

ỵ

ỵ ❉❡s❛r❣✉❡s✮

✳ ❈❤♦ ❤❛✐ t❛♠ ❣✐→❝ ABC ✈➔ A1B1C1✳

●å✐ M ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ AB ✈➔ A1B1✱ N ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ AC ✈➔ A1C1✱ P

❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ BC ✈➔ B1C1✳ ❑❤✐ ✤â ❜❛ ✤✐➸♠ M, N, P t❤➥♥❣ ❤➔♥❣
AA1, BB1, CC1 ỗ q

ự AA1, BB1, CC1 ỗ q t O t ❝❤ù♥❣
♠✐♥❤
✤✐➸♠

M, N, P t❤➥♥❣
N, A1 , C1 t❛ ❝â

❤➔♥❣✳ ⑩♣ ử ỵ s

N A C1 C A 1 O
Ã
Ã
= 1.
N C C1 O A1 A

OAC

ợ



ự tữỡ tü t❛ ❝â

P C B1 B C1 O
M B A1 A B1 O
·
·

= 1,
·
·
= 1.
P B B1 O C1 C
M A A1 O B1 B

✭✶✳✶✼✮


✶✼
❚ø ✭✶✳✶✻✮ ✈➔ ✭✶✳✶✼✮ t❛ ❝â

NA P C MB
·
·
= 1,
NC P B MA
õ ử ỵ s t❛♠ ❣✐→❝

ABC ✱ t❛ ❝â ❜❛ ✤✐➸♠ M, N, P

t❤➥♥❣ ❤➔♥❣✳

❈❤✐➲✉ t❤✉➟♥✿ ❈❤♦ ❜❛ ✤✐➸♠ M, N, P t❤➥♥❣ ❤➔♥❣✱ t ự AA1, BB1, CC1

ỗ q t t
t

M BB1

N CC1

õ

M N, BC, B1 C1

ỗ q

P

O ✤✐➸♠ ❝õ❛ BB1 ✈➔ CC1 ✳ ❍ì♥ ♥ú❛ A ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ M B
✈➔ N C, A1 ❧➔ ❣✐❛♦ ❝õ❛ M B1 ✈➔ N C1 ✳ ❉♦ ✤â t❤❡♦ ❝❤ù♥❣ ♠✐♥❤ tr➯♥✱ t❛ ❝â
O, A, A1 t❤➥♥❣ ❤➔♥❣✱ ❤❛② AA1 , BB1 , CC1 ỗ q
õ

ởt số ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ ✈✉æ♥❣ ❣â❝✱
s♦♥❣ s♦♥❣ tr♦♥❣ ồ
ử ỵ t t t➼♥❤ s♦♥❣ s♦♥❣ ✈➔ ✈✉ỉ♥❣ ❣â❝ ✤➣ ✤÷đ❝
tr➻♥❤ ❜➔② tr♦♥❣ ♠ö❝ ✶✳✶✳✸✱ ♥ë✐ ❞✉♥❣ ♠ö❝ t✐➳♣ t❤❡♦ s➩ tr➻♥❤ ❜➔② ❝→❝ ❜➔✐ t♦→♥
❝â ✤ë ❦❤â t➠♥❣ ❞➛♥ ✈➔ ✈➟♥ ❞ö♥❣ t ú ỗ tớ ởt số t
t ❝â ✤÷❛ r❛ ✷ ❧í✐ ❣✐↔✐ ❦❤→❝ ♥❤❛✉ ✤➸ ❤å❝ s ỹ ồ ữợ t
ũ ủ t♦→♥ tr♦♥❣ ♠ư❝ ✶✳✷ ✤÷đ❝ t❤❛♠ ❦❤↔♦ ❝❤õ ②➳✉ tø ❬✶❪ ✈➔
❬✷❪✳

❇➔✐ t♦→♥ ✶✳✶

✳ ❈❤♦ tù ❣✐→❝ ABCD ♥ë✐ t✐➳♣

✭✣à♥❤ ỵ rt

ữớ trỏ (O) õ ữớ AC ✈➔ BD ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ ♥❤❛✉ t↕✐ M ✳
❈❤ù♥❣ ♠✐♥❤ r ữớ t ố tr ởt ợ M ✈✉ỉ♥❣ ❣â❝
❝↕♥❤ ✤è✐ ❞✐➺♥ ✈➔ ♥❣÷đ❝ ❧↕✐✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû I
❣✐↔ t❤✐➳t
✤➾♥❤✮✳

AC ⊥ BD

AB ✳ M I ❝➢t CD
ABM = AM I ✳ ❚❛ ❝â AM I = CM E

❧➔ tr✉♥❣ ✤✐➸♠ ❝↕♥❤

♥➯♥

t↕✐

E.

❚❤❡♦

✭❤❛✐ ❣â❝ ✤è✐


✶✽

▼➦t ❦❤→❝ tù ❣✐→❝

ABCD

♥ë✐ t✐➳♣ ♥➯♥

ABM = M CE

s✉② r❛

M AB + M BA = EM C + ECM ⇒ AM B = M EC ⇒ M EC = 90◦ .
❉♦ ✤â

M I ⊥ CD.

❇➔✐ t♦→♥ ✶✳✷✳ ❈❤♦ ❤➻♥❤ t❤❛♥❣ ABCD ✈ỵ✐ AB
CD✳

●å✐ I ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ AM
❈❤ù♥❣ ♠✐♥❤ r➡♥❣ IK AB.

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â

AIB ∼

t❛ ❝â

▼➦t ❦❤→❝✱

M D = M C, AB

❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛
✈➔ BC, K ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ BM ✈➔ AC ✳

M ID

AB

IM
MD
=
.
IA
AB
M C ✭t❤❡♦

KM
MC
=
KB
AB
ỵ s s r

t



IK

CD, M

M D, AIB = M ID)✱

❞♦ ✤â

t❤✐➳t✮✳ ❉♦ ✤â

IM
KM
=
.
IA
KB

AB.

tr➯♥ tr✉♥❣ t✉②➳♥ AM ❧➜② ✤✐➸♠ K ❜➜t ❦➻ ❦❤→❝
A, M. ◗✉❛ M ữủt ữớ t s s ợ KB, KC ❣✐❛♦ AC, AB
t↕✐ F, E ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ EF BC.
ABC,

❈❤ù♥❣ ♠✐♥❤✳ ❑➨♦ ❞➔✐ CK, BK
❧÷đt ❧➔ ✤÷í♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛

AB, AC t↕✐ P, Q. ❚❛ ❝â EM, F M ❧➛♥
BP C, BQC ♥➯♥ BE = P E, QF = CF ✳

❝➢t

õ

AP
AK
AQ
AP + P E
AQ + QF
=
=

=
PE
KM
QF
PE
QF



ỵ s ✤↔♦ s✉②

❇➔✐ t♦→♥ ✶✳✹✳ ❈❤♦

❧➔ ❤❛✐ ✤✐➸♠ ♥➡♠ tr➯♥
r➡♥❣ EF BC ✳

AF

AE
=
.
EB
FC
r❛ EF
BC.

ABC. ●å✐ D ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ BC, E ✈➔ F ❧➛♥ ❧÷đt
AB, AC s❛♦ ❝❤♦ AD, BF, CE ỗ q ự

ự ử ỵ
q

AD, BF





BD = CD

õ

EF

CE

ABC

ợ ữớ t ỗ

t õ

AE BD CF
.
.
= 1.
EB DC F A
AE CF
EA
FA
.
= 1 s✉② r❛
=
✳
EB F A
EB
FC

ỵ s t

BC.

t ồ I ❧➔ t➙♠ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ t❛♠ ❣✐→❝ ABC. ✣÷í♥❣

t❤➥♥❣ AI, BI, CI ❝➢t ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABC t↕✐ D, E ✈➔ F.
✣÷í♥❣ t❤➥♥❣ DE ❝➢t ❝↕♥❤ AC t↕✐ M ✱ ✤÷í♥❣ t❤➥♥❣ DF ❝➢t ❝↕♥❤ AB t↕✐ N.

❈❤ù♥❣ ♠✐♥❤ r➡♥❣ M N BC.


✷✵

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❣✐↔ t❤✐➳t BE

❧➔ ♣❤➙♥ ❣✐→❝ ❝õ❛ ❣â❝

EBC ⇒ ADE = EDC ✳
AM
AD
AN
AD
❳➨t
ADC ❝â
=
, t÷ì♥❣ tü
=
.
MC
DC
NB
DB
♣❤➙♥ ❣✐→❝ ❝õ❛ ❣â❝ BAC s✉② r❛ DB = DC
⇒

ABC

♥➯♥

ABE =

❚❤❡♦ ❣✐↔ t❤✐➳t

AD

❧➔

AD
AD
AM
AN
=
⇒
=
.
DB
DC
MC
NB

❚❤❡♦ ỵ s s r

MN

BC.

t t ❣✐→❝ ABC ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ (O) ✈➔ ♥❣♦↕✐ t✐➳♣

✤÷í♥❣ trá♥ (I). H ❧➔ trü❝ t➙♠ ✈➔ D ❧➔ t✐➳♣ ✤✐➸♠ ❝õ❛ (I) ✈ỵ✐ ❝↕♥❤ BC ✳ ●✐↔
sû OI BC ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ AO HD.

❈❤ù♥❣ ♠✐♥❤✳ ●å✐ K

❧➔ ✤✐➸♠ ✤è✐ ①ù♥❣ ❝õ❛

D

q✉❛

I

✈➔

M

❧➔ tr✉♥❣ ✤✐➸♠

BC, AK ❝➢t ❝↕♥❤ BC t↕✐ E t❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣
♥➯♥ M D = M E ✱ s✉② r❛ IM ❧➔ ✤÷í♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛
KDE ♥➯♥ IM AE.
❚❤❡♦ ❣✐↔ t❤✐➳t OI
BC ♥➯♥ tù ❣✐→❝ OIM D ❧➔ ❤➻♥❤ ❝❤ú ♥❤➟t✱ ❞♦ ✤â
IO = M E ✳ ❱➟②✱ tù ❣✐→❝ IOM E ❧➔ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤✱ s✉② r❛ IM OE ✳ ❚ø
✤â s✉② r❛ O t❤✉ë❝ ✤÷í♥❣ t❤➥♥❣ AE ✈➔ KD = 2OM ✳
❱➻ H ❧➔ trü❝ t➙♠ ♥➯♥ AH ⊥ BC ⇒ AH
KD, O ❧➔ t➙♠ ✤÷í♥❣ trá♥
♥❣♦↕✐ t✐➳♣
ABC ♥➯♥ AH = 2OM ⇒ AH = KD✳ ❱➟② tù ❣✐→❝ AKDH ❧➔

❤➻♥❤ ❜➻♥❤ ❤➔♥❤✳ ❉♦ ✤â HD
AK ⇒ HD AO.

❝↕♥❤

❇➔✐ t♦→♥ ✶✳✼✳ ❬✶❪ ❈❤♦ tù ❣✐→❝ ABCD ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ t❤ä❛ ♠➣♥ AB =

BC, CD = DA✳ M ❧➔
❝➢t DB t↕✐ Q✳ ❈❤ù♥❣

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦

✤✐➸♠ ♥➡♠ tr➯♥ ❝✉♥❣ CD,
♠✐♥❤ r➡♥❣ P Q AC

MB

❝➢t CD t↕✐ P ✈➔ M A

AB = BC, CD = DA
∆CBD (c.g.c) ⇒ DAB = DCB
❣✐↔✐ t❤✐➳t

s✉② r❛

⇒ DAB + DCB = 180◦ ⇒ DAB = DCB = 90◦

∆ABD =

✷✶

❙✉② r❛

BD

❧➔ ✤÷í♥❣ ❦➼♥❤ ❝õ❛ ✤÷í♥❣ trá♥ ❞♦ ✤â

AC ⊥ BD.

1
1
DQM = (s✤ M D +s✤ AB), DP M = (s✤ M D +s✤ BC), AB = BC.
2
2
DQM = DP M ✳ ❱➟② tù ❣✐→❝ DQP M ♥ë✐ t✐➳♣ ✈ỵ✐ BD ❧➔ ✤÷í♥❣ ❦➼♥❤✳
◦
◦
✤â DM B = 90 ✱ s✉② r❛ DQP = 90 ✳ ❱➟②✱ P Q ⊥ BD ⇒ P Q
AC.

❉♦ ✤â
❉♦

❇➔✐ t♦→♥ ✶✳✽✳ ❬✶❪ ❈❤♦ t❛♠ ❣✐→❝ ♥❤å♥ ABC ✱ ✤÷í♥❣ ❝❛♦ AD, BE, CF ✈➔ M

❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ BC ✳ ✣÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ AEF ❝➢t AM t↕✐
N, BN ❝➢t AD t↕✐ P, CF ❝➢t AM t↕✐ Q✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ P Q BC ✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦

AC, CF
BF HD

✈✉æ♥❣ ❣â❝

❣✐↔ t❤✐➳t

AB ✳

●å✐

AD
H ❧➔

BC, BE ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐
❣✐→❝ ABC. ❙✉② r❛ tù ❣✐→❝

✈✉ỉ♥❣ ❣â❝ ✈ỵ✐
trü❝ t➙♠ t❛♠

♥ë✐ t✐➳♣✱ ❞♦ ✤â

AF.AB = AH.AD
AEF ❝➢t AM t↕✐ N, AEH = AF H = 90◦ ✳
AN H = 90◦ ⇒ HN M + HDM = 180◦ s✉②

✣÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝
❚❛ ❝â✱

AH

❧➔ ✤÷í♥❣ ❦➼♥❤ ♥➯♥