Đường tròn soddy và các vấn đề liên quan

ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC
---------------------------

NGÔ TRỌNG THÀNH

ĐƯỜNG TRÒN SODDY
VÀ CÁC VẤN ĐỀ LIÊN QUAN

LUẬN VĂN THẠC SĨ TOÁN HỌC

THÁI NGUYÊN - 2019


ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC
---------------------------

NGÔ TRỌNG THÀNH

ĐƯỜNG TRÒN SODDY
VÀ CÁC VẤN ĐỀ LIÊN QUAN
Chuyên 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. Nguyễn Việt Hải

THÁI NGUYÊN - 2019

✐

▼ư❝ ❧ư❝
❉❛♥❤ ♠ư❝ ❤➻♥❤
▲í✐ ❝↔♠ ì♥
▼ð ✤➛✉
✶ ❑✐➳♥ t❤ù❝ ❜ê s✉♥❣
✶✳✶

✶✳✷

✐✐✐
✐✈
✶
✸

P❤➨♣ ♥❣❤à❝❤ ✤↔♦ tr♦♥❣ ♠➦t ♣❤➥♥❣

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✶✳✶

✣à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✶✳✷

❈æ♥❣ t❤ù❝ ❦❤♦↔♥❣ ❝→❝❤✱ t➼♥❤ ❝❤➜t ❜↔♦ ❣✐→❝

✳ ✳ ✳ ✳ ✳

✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✷✳✶

✣à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✷✳✷

▼ët sè ❦➳t q✉↔ tr♦♥❣ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

❚å❛ ✤ë ❜❛r②❝❡♥tr✐❝ t❤✉➛♥ ♥❤➜t

✷ ❈→❝ ✤÷í♥❣ trá♥ ❙♦❞❞②

✷✵

✷✳✶

✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝❤ ❞ü♥❣ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

✷✳✷

❇→♥ ❦➼♥❤ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞②

✷✸

✷✳✸

✷✳✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✷✳✶

❇→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ❙♦❞❞② ♥ë✐

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

✷✳✷✳✷

❇→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ❙♦❞❞② ♥❣♦↕✐

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✹

✣÷í♥❣ trá♥ ❙♦❞❞② tr♦♥❣ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✺

✷✳✸✳✶

❈→❝ ✤✐➸♠ ❙♦❞❞② ✈➔ ✤÷í♥❣ t❤➥♥❣ ❙♦❞❞②

✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✺

✷✳✸✳✷

P❤÷ì♥❣ tr➻♥❤ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞②

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

❚❛♠ ❣✐→❝ ❙♦❞❞② ✈➔ t❛♠ ❣✐→❝ ❊✉❧❡r✲●❡r❣♦♥♥❡✲❙♦❞❞②

✳ ✳ ✳ ✳

✸ ▼ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥
✸✳✶

❚❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞②

✷✾

✸✺
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺


✐✐

✸✳✷

✸✳✸

✸✳✶✳✶

▼ët sè ❤➺ t❤ù❝ ❤➻♥❤ ❤å❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✳✶✳✷

❚❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ✈➔ ❝→❝ t➼♥❤ ❝❤➜t

✸✳✶✳✸

❚❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ❝↕♥❤ ♥❣✉②➯♥

✸✳✶✳✹

❉ü♥❣ t❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ❜✐➳t ♠ët ❝↕♥❤

κ = ta + tb + tc ✳ ✳ ✳
✸✳✷✳✶
❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ κ = 2
✸✳✷✳✷
❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣ κ = 4
❈→❝ t❛♠ ❣✐→❝ ❧ỵ♣
= tb + tc ✳ ✳ ✳ ✳ ✳
✸✳✸✳✶
❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣
=1
✸✳✸✳✷
❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣
=2
❈→❝ t❛♠ ❣✐→❝ ❧ỵ♣

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✸

✳ ✳ ✳ ✳ ✳ ✳

✹✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✹

✺✼
✺✽


✐✐✐

❉❛♥❤ ♠ö❝ ❤➻♥❤
✶✳✶

❷♥❤ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ✤✐➸♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷

❛✮ ❷♥❤ ✤÷í♥❣ t❤➥♥❣ ❦❤ỉ♥❣ q✉❛ ❝ü❝❀ ❜✮ ❷♥❤ ✤÷í♥❣ trá♥ ❝â
t➙♠ ❧➔ ❝ü❝

✶✳✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❷♥❤ ❝õ❛ ✤÷í♥❣ trá♥ ❦❤ỉ♥❣ q✉❛ ❝ü❝ ♥❣❤à❝❤ ✤↔♦

✹

✹

✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

2

AB =

R · AB
OA.OB

✶✳✹

❑❤♦↔♥❣ ❝→❝❤

✶✳✺

❚➼♥❤ ❝❤➜t ❜↔♦ ❣✐→❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✻

❱➼ ❞ư ✈➲ ❝ỉ♥❣ t❤ù❝ ❈♦♥✇❛② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✷✳✶

✣÷í♥❣ trá♥ ❙♦❞❞② ♥ë✐ ✈➔ ✤÷í♥❣ trá♥ ❙♦❞❞② ♥❣♦↕✐

✳ ✳ ✳ ✳ ✳

✷✶

✷✳✷

❈→❝❤ ❞ü♥❣ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✷

✷✳✸

❚å❛ ✤ë ❜❛r②❝❡♥tr✐❝ ❝õ❛ ❝→❝ ✤✐➸♠ ❙♦❞❞② ✈➔ ✤÷í♥❣ t❤➥♥❣ ❙♦❞❞② ✷✻

✷✳✹

❚➙♠ ❙♦❞❞② ♥ë✐✱ ♥❣♦↕✐ ✈➔ ✤✐➸♠ ❊♣♣st❡✐♥

✳ ✳ ✳ ✳ ✳

✸✵

✷✳✺

❈→❝ ✤÷í♥❣ t❤➥♥❣ ❊✉❧❡r ✈➔ ●❡r❣♦♥♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✶

✷✳✻

❚❛♠ ❣✐→❝ ❊✉❧❡r✲●❡r❣♦♥♥❡✲❙♦❞❞② ✈✉æ♥❣ t↕✐

S

✳ ✳

✸✷

✷✳✼

▼ët sè ✤✐➸♠ tr➯♥ ❝↕♥❤ t❛♠ ❣✐→❝ ❊✉❧❡r✲●❡r❣♦♥♥❡✲❙♦❞❞②

✳ ✳

✸✸

✸✳✶

AD✲❝❡✈✐❛♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✻

✸✳✷

❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝❡✈✐❛♥ t✐➳♣ t✉②➳♥ ✤➾♥❤ ❆

✸✳✸

❈→❝ ❤➺ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥

✸✳✹

P Q ⊥ AD







ữớ t r s s ợ



t ✤✐➸♠

✸✳✽

❉ü♥❣ t❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞② ❜✐➳t ♠ët ❝↕♥❤

✸✳✾

❚❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣

A

t✐➳♣ t✉②➳♥ ✤➾♥❤

E = X481
Fl =

G

∩

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✾

C

θ

ABC

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

AD

✹✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺

✸✳✶✵ ❚❛♠ ❣✐→❝ ❍❡r♦♥ ❧ỵ♣

=1✳
=2✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✻

✐✈

▲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♥ ỏ t qỵ t
ổ ợ ồ rữớ ồ ❦❤♦❛ ❤å❝ ✲ ✣↕✐
❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ t ỳ tự qỵ ụ
ữ t ❝❤♦ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳
❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ỡ 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 ▲✉➟♥ ✈➠♥

◆❣ỉ ❚rå♥❣ ❚❤➔♥❤


✶

▼ð ✤➛✉
✶✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥
❈→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ❝õ❛ t❛♠ ❣✐→❝

ABC

❝â ♥❤ú♥❣ t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t✱

❜➔✐ t♦→♥ ❞ü♥❣ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ❧➔ tr÷í♥❣ ❤đ♣ r✐➯♥❣ q✉❛♥ trå♥❣ ❝õ❛
❜➔✐ t♦→♥ ❆♣♦❧✐❧♦♥✐✉s✳ ❈❤❛ ✤➫ ❝õ❛ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ ✤✐➸♠ ❙♦❞❞②✱ ✤÷í♥❣
t❤➥♥❣ ❙♦❞❞②✱ t❛♠ ❣✐→❝ ❙♦❞❞②✱✳✳ ❧➔ ❋r❡❞❡r✐❝❦ ❙♦❞❞②✱ ♥❣÷í✐ ✤➣ ❞➔♥❤ ✤÷đ❝
❣✐↔✐ t❤÷ð♥❣ ◆♦❜❡❧ ✈➲ ❍â❛ ❤å❝✳ P❤→t tr✐➸♥ ❝→❝ ❦❤→✐ ♥✐➺♠ ♥➔② tr♦♥❣ ♥❤ú♥❣
♥➠♠ ❣➛♥ ✤➙②✱ ♥❤✐➲✉ t→❝ ❣✐↔ ✭◆✳ ❉❡r❣✐❛❞❡s ♥➠♠ ✷✵✵✼✱ ▼✳ ❏❛❝❦s♦♥ ♥➠♠ ✷✵✶✸✱
▼✳ ❏❛❝❦s♦♥ ✈➔ ❚❛❦❤❛❡✈ ♥➠♠ ✷✵✶✺✱ ✷✵✶✻ ✮ ✤➣ ❝æ♥❣ ❜è ❝→❝ ♣❤→t ❤✐➺♥ ❤➻♥❤
❤å❝ s➙✉ s➢❝ s✐♥❤ r❛ tø ✤÷í♥❣ trá♥ ❙♦❞❞②✳ ❇➔✐ t♦→♥ ✤➦t r❛ ❧➔ ❧➔♠ t❤➳ ♥➔♦
❞ü♥❣ ✤÷đ❝ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ ①→❝ ✤à♥❤ ❝→❝ ❜→♥ ❦➼♥❤ ❝õ❛ ❝❤ó♥❣ t❤❡♦
❝→❝ ②➳✉ tè ❝õ❛ t❛♠ ❣✐→❝ ❝❤♦ trữợ ữớ trỏ ữớ t
õ q ợ ữớ trỏ ữớ t ❜✐➳t ❦❤→❝❄
❚r➻♥❤ ❜➔② ❝→❝❤ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ tr➯♥ ỵ tổ ồ t
ữớ trỏ ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥✧✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔✿
✲ ❚r➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ ❝→❝❤ ①→❝ ✤à♥❤ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ t➼♥❤ ✤÷đ❝
❝→❝ ❜→♥ ❦➼♥❤✱ t➻♠ ✤÷đ❝ ❝→❝ t t ợ ừ ữớ trỏ ở
ữớ trỏ ❙♦❞❞② ♥❣♦↕✐✳ ❚ø ✤â ✤÷❛ r❛ ❝→❝❤ ❞ü♥❣ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❝→❝
✤÷í♥❣ trá♥✱ ✤÷í♥❣ t❤➥♥❣ ❙♦❞❞② tr♦♥❣ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝✳
✲ ❳→❝ ✤à♥❤ ♠è✐ q✉❛♥ ❤➺ ❝õ❛ t❛♠ ❣✐→❝ ❙♦❞❞② ✈ỵ✐ ❝→❝ ✤✐➸♠ ✈➔ ✤÷í♥❣
t❤➥♥❣ ✤➦❝ ❜✐➺t ❦❤→❝✳
✲ P❤➙♥ ❧♦↕✐ ✤÷đ❝ ❝→❝ t❛♠ ❣✐→❝ ❧ỵ♣

κ = ta + tb + tc

❦❤↔♦ st trữớ ủ t ừ ợ õ

❧ỵ♣

= tb + tc ✱


✷

✷✳ ◆ë✐ ❞✉♥❣ ✤➲ t➔✐✱ ♥❤ú♥❣ ✈➜♥ ✤➲ ❝➛♥ ❣✐↔✐ q✉②➳t
◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ✸ ❝❤÷ì♥❣✿

❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❜ê s✉♥❣
◆❤➢❝ ❧↕✐ ✈➔ ❜ê s✉♥❣ ❤❛✐ ❝❤õ ✤➲ ❝ì ❜↔♥ ✤÷đ❝ sû ❞ư♥❣ ❧➔♠ ❝ỉ♥❣ ❝ư
❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ ✤➦t r❛✿ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ✈➔ tå❛ ✤ë rtr ữỡ
ỗ ử
P tr t ♣❤➥♥❣
✶✳✷✳ ❚å❛ ✤ë ❜❛r②❝❡♥tr✐❝ t❤✉➛♥ ♥❤➜t

❈❤÷ì♥❣ ✷✳ ❈→❝ ✤÷í♥❣ trá♥ ❙♦❞❞②
◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ✤➲ ❝➟♣ ✤➳♥ sü ①→❝ ✤à♥❤ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞②
❝ị♥❣ ❝→❝ ❜ë ♣❤➟♥ ❝õ❛ ♥â ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤➻♥❤ ❤å❝ sì ❝➜♣ ✈➔ ♣❤÷ì♥❣
♣❤→♣ tå❛ ✤ë✳ ✣➙② ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ trå♥❣ t➙♠ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ữỡ
ỗ ử s tờ ủ ờ s tø ❝→❝ ❜➔✐ ❜→♦ ❬✶❪✱ ❬✸❪✱ ❬✼❪✮✿
✷✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝❤ ❞ü♥❣ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞②
✷✳✷✳ ❇→♥ ❦➼♥❤ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞②
✷✳✸✳ ✣÷í♥❣ trá♥ ❙♦❞❞② tr♦♥❣ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝
✷✳✹✳ ❚❛♠ ❣✐→❝ ❙♦❞❞② ✈➔ t❛♠ ❣✐→❝ ❊✉❧❡r✲●❡r❣♦♥♥❡✲❙♦❞❞②

❈❤÷ì♥❣ ✸✳ ▼ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥
❈❤÷ì♥❣ ✸ ①➨t ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ t❛♠ ❣✐→❝
❙♦❞❞②✱ t❤ü❝ ❝❤➜t ❧➔ ❝→❝ tr÷í♥❣ ❤đ♣ r✐➯♥❣ q✉❛♥ trå♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝
❦❤→✐ ♥✐➺♠ ❦❤→❝ tr♦♥❣ ❤➻♥❤ ❤å❝✱ ❝❤➥♥❣ ❤↕♥ t❛♠ ❣✐→❝ ❍❡r♦♥✳ ❈❤÷ì♥❣ ♥➔②
✤÷đ❝ t❤❛♠ ❦❤↔♦ ✈➔ tê♥❣ ❤đ♣ t❤❡♦ ❝→❝ ❜➔✐ ❜→♦ ở ỗ

t❛♠ ❣✐→❝ ❧ỵ♣
✸✳✸✳ ❈→❝ t❛♠ ❣✐→❝ ❧ỵ♣

κ = ta + tb + tc
= tb + tc ✳


✸

❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❜ê s✉♥❣
❚❛ ♥❤➢❝ ❧↕✐ ✈➔ ❜ê s✉♥❣ ❤❛✐ ♥ë✐ ❞✉♥❣ ❝➛♥ ❝❤♦ ❝→❝ ❝❤÷ì♥❣ s❛✉✿ ❚❤ù ♥❤➜t✱
✤✐➸♠ q✉❛ ✈➲ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ●✐→♦ tr➻♥❤ ❤➻♥❤
❤å❝ sì ❝➜♣❀ ❚❤ù ❤❛✐✱ ❜ê s✉♥❣ t❤➯♠ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝ ✭❞↕♥❣ ❤➻♥❤ ❤å❝ ❣✐↔✐
t➼❝❤✮✱ ♣❤→t tr✐➸♥ tø ❦❤→✐ ♥✐➺♠ t➙♠ t✛ ❝ü q✉❡♥ t❤✉ë❝✳

✶✳✶ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ tr♦♥❣ ♠➦t ♣❤➥♥❣
❚❛ ♥❤➢❝ ❧↕✐ ♠ët sè ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤
✤↔♦ q✉❛ ✤÷í♥❣ trá♥ ❤❛② ❝á♥ ❣å✐ ❧➔

♣❤➨♣ ✤è✐ ①ù♥❣ q✉❛ ✤÷í♥❣ trá♥

tr➯♥ ♠➦t

♣❤➥♥❣ ❊✉❝❧✐❞❡✳ ❈→❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝â t❤➸ t➻♠ t❤➜② tr♦♥❣ ❝→❝ ❣✐→♦
tr➻♥❤ ❍➻♥❤ ❤å❝ sì ❝➜♣ ❤✐➺♥ ❤➔♥❤✳

✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❚r➯♥ ♠➦t ♣❤➥♥❣ ❝❤♦ ✤÷í♥❣ trá♥ t➙♠ O✱ ❜→♥ ❦➼♥❤ R✳

P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝ü❝ O✱ ♣❤÷ì♥❣ t➼❝❤ k = R2 ❧➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ tr➯♥ ♠➦t
♣❤➥♥❣✱ ❜✐➳♥ P → P s❛♦ ❝❤♦ ♥➳✉ P = O t❤➻ OP.OP = R2 ❀ ♥➳✉ P ≡ O
t❤➻ P
ỵ õ

ữớ trá♥ ♥❣❤à❝❤ ✤↔♦✳

fRO2 ✱

✤÷í♥❣ trá♥

(O, R)

✤÷đ❝ ❣å✐ ❧➔

P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ♥➔② ❝ơ♥❣ ❣å✐ ❧➔ ♣❤➨♣ ✤è✐ ①ù♥❣

q✉❛ ✤÷í♥❣ trá♥✳
❉➵ t❤➜② ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝â t➼♥❤ ❝❤➜t ✤è✐ ❤ñ♣✱ tù❝ ❧➔

fRO2

2

= Id✳

❚ø

✹

❍➻♥❤ ✶✳✶✿ ❷♥❤ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ✤✐➸♠
✤à♥❤ ♥❣❤➽❛ t❛ s✉② r❛ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦✿

❍➻♥❤ ✶✳✷✿ ❛✮ ❷♥❤ ✤÷í♥❣ t❤➥♥❣ ❦❤ỉ♥❣ q✉❛ ❝ü❝❀ ❜✮ ❷♥❤ ✤÷í♥❣ trá♥ ❝â t➙♠ ❧➔ ❝ü❝
❛✮ ◗✉❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦

fRO2 ✱

✤÷í♥❣ trá♥ ♥❣❤à❝❤ ✤↔♦

(O, R)

❜✐➳♥ t❤➔♥❤

❝❤➼♥❤ ♥â✱ ♥â✐ ❝→❝❤ ❦❤→❝✱ ✤÷í♥❣ trá♥ ♥❣❤à❝❤ ✤↔♦ ❧➔ ❤➻♥❤ ❦➨♣ t✉②➺t ✤è✐
✭t÷ì♥❣ tü trư❝ ✤è✐ ①ù♥❣ tr♦♥❣ ♣❤➨♣ ✤è✐ ①ù♥❣✮✳ ▼å✐ ✤✐➸♠ ð tr♦♥❣
❜✐➳♥ t❤➔♥❤ ✤✐➸♠ ð ♥❣♦➔✐ ✈➔ ♥❣÷đ❝ ❧↕✐✳

(O, R)


✺

❍➻♥❤ ✶✳✸✿ ❷♥❤ ❝õ❛ ✤÷í♥❣ trá♥ ❦❤ỉ♥❣ q✉❛ ❝ü❝ ♥❣❤à❝❤ ✤↔♦
❜✮ ◗✉❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦

fRO2 ✱ ♠å✐ ✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ❖ ❜✐➳♥ t❤➔♥❤ ❝❤➼♥❤

♥â ✭❤➻♥❤ ❦➨♣ t÷ì♥❣ ✤è✐✮✳
❝✮ ◗✉❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦
✤÷í♥❣ trá♥ ✤✐ q✉❛

fRO2 ✱ ♠å✐ ✤÷í♥❣ t❤➥♥❣ ❦❤æ♥❣ ✤✐ q✉❛ O

O✳

❞✮ ◗✉❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦
t❤➥♥❣ ❦❤æ♥❣ ✤✐ q✉❛

fRO2 ✱

♠å✐ ✤÷í♥❣ trá♥ ✤✐ q✉❛

fRO2 ✱

❜✐➳♥ t❤➔♥❤ ✤÷í♥❣

♠å✐ ✤÷í♥❣ trá♥ ❦❤ỉ♥❣ ✤✐ q✉❛

O❀ ♠å✐ ✤÷í♥❣ trá♥ t➙♠ O✱
2
t➙♠ O ✱ ❜→♥ ❦➼♥❤ R /r ✳

✤÷í♥❣ trá♥ ❦❤ỉ♥❣ ✤✐ q✉❛
t❤➔♥❤ ữớ trỏ ỗ

(I, r) t
t p ợ p = PO/(I,r) ✳

❢✮ ✣÷í♥❣ trá♥

❈❤ù♥❣ ♠✐♥❤✳

O

O✳

❡✮ ◗✉❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦

♣❤÷ì♥❣

❜✐➳♥ t❤➔♥❤

O

❜✐➳♥ t❤➔♥❤

❜→♥ ❦➼♥❤

♥â q✉❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦

f

r

❜✐➳♥

❝ü❝

O✱

❛✮✱ ❜✮ ❤✐➸♥ ♥❤✐➯♥✳

OH ⊥ ∆✱ ❣å✐ H ❧➔ ↔♥❤ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ H t❤➻ H ❝è ✤à♥❤✳ ❱ỵ✐ ♠å✐
M ∈ ∆✱ M ❧➔ ↔♥❤ ❝õ❛ M t❤➻ OM.OM = OH.OH ♥➯♥ ✹ ✤✐➸♠ H ✱ H ✱ M ✱
M t❤✉ë❝ ♠ët ✤÷í♥❣ trá♥✳ ❚❛ ❧↕✐ ❝â M HH = 90◦ ✱ s✉② r❛ M M H = 90◦ ✱
tù❝ ❧➔ M t❤✉ë❝ ✤÷í♥❣ trỏ ữớ OH ợ ồ N tr➯♥
❝✮ ❍↕


✻

OH

∆ ⊥ OH

ON ❧✉æ♥ ❝➢t ∆ t↕✐ ♠ët
✤✐➸♠ N ✭♥➳✉ N ≡ O t❤➻ t❛ ❧➜② ✤✐➸♠ ✈æ t➟♥ tr➯♥ ∆✮✳ ❚ù ❣✐→❝ N HH N
◦
2
♥ë✐ t✐➳♣ ✈➻ ❝â ✷ ❣â❝ ✤è✐ ❞✐➺♥ ❜➡♥❣ 90 ✳ ❙✉② r❛ ON.ON = OH.OH = R
t❤❡♦ ❝→❝❤ ①→❝ ✤à♥❤ H, H ✳ ❷♥❤ ❝õ❛ ♠å✐ M ∈ ∆ ❧➔ M ∈ δ ✲✤÷í♥❣ trá♥
✤÷í♥❣ ❦➼♥❤ OH ✳ ❱➟② ↔♥❤ ❝õ❛ ✤÷í♥❣ t❤➥♥❣ ∆ ❦❤ỉ♥❣ q✉❛ O ❧➔ ✤÷í♥❣ trá♥
δ ✤✐ q✉❛ O✳
❞✮ ❉♦ t➼♥❤ ✤è✐ ❤đ♣ t❛ ❝â ♥❣❛② ↔♥❤ ❝õ❛ ✤÷í♥❣ trá♥ ✤✐ q✉❛ O ❧➔ ✤÷í♥❣ t❤➥♥❣
❦❤ỉ♥❣ ✤✐ q✉❛ O ✱ ❤➻♥❤ ✶✳✷❛✮✳

✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤

✳ ❱➻

♥➯♥

❡✮ ❚ø ❝→❝❤ ❞ü♥❣ ↔♥❤ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ♠ët ✤✐➸♠ t❛ s✉② r❛ ♥❣❛② ♠å✐ ✤÷í♥❣
trá♥ t➙♠ ❖✱ ❜→♥ ❦➼♥❤ r ❜✐➳♥ t❤➔♥❤ ✤÷í♥❣ trá♥ ỗ t
ồ

C

ữớ trỏ t

O
C

ữỡ t ừ ỹ
ữỡ t

p

s

C

ố ợ

ổ q ỹ

C

O

C

ữớ trá♥

C

❧➔

O✱

O✱

t✛ sè ✈à tü ❜➡♥❣

R2 /p

H O ✮✿

fRO2 (C) = fRO2 ◦ fpO (C) = HhO (C),
HhO (C)

p

✭❤➻♥❤ ❦➨♣ t÷ì♥❣ ✤è✐✮✳ ❱➻ t➼❝❤ ❤❛✐

❧➔ ♣❤➨♣ ✈à tü t➙♠

♥➯♥ t❛ ❝â ỵ tỹ t

t

ồ

t ❝â ♥❣❛② ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝ü❝

t❤➔♥❤ ❝❤➼♥❤

♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❝ò♥❣ ❝ü❝

O✳

✭❞➵ t❤➜②

✈ỵ✐

C

h = R2 /p.

❦❤ỉ♥❣ ✤✐ q✉❛

O✮✳

❢✮ ❍✐➸♥ ♥❤✐➯♥ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♣❤÷ì♥❣ t➼❝❤✳

✶✳✶✳✷ ❈ỉ♥❣ t❤ù❝ ❦❤♦↔♥❣ ❝→❝❤✱ t➼♥❤ ❝❤➜t ❜↔♦ ❣✐→❝
❚❛ ✤✐ t➻♠ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❤❛✐ ↔♥❤ ừ trữợ

(O, R) ❧➔ ✤÷í♥❣ trá♥ ♥❣❤à❝❤ ✤↔♦✱ A , B

❧➔ ↔♥❤ ♥❣❤à❝❤

✤↔♦ ❝õ❛ A, B t❤➻

R2 AB
AB =
OA.OB
❈❤ù♥❣ ♠✐♥❤✳

❚❛ ❝â

∆OAB ∼ ∆OB A

♥➯♥

AB
OA
OA · OA
R2
R2 · AB
=
=
=
=⇒ A B =
.
AB
OB

OA · OB
OA · OB
OA.OB
▼✐♥❤ ❤å❛ tr➯♥ ❤➻♥❤ ✶✳✹✳

✭✶✳✶✮


✼

❍➻♥❤ ✶✳✹✿ ❑❤♦↔♥❣ ❝→❝❤ A B

=

R2 · AB
OA.OB

❍➺ q✉↔ ✶✳✶✳✶✳ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ❜↔♦ t♦➔♥ t✛ sè ❦➨♣ ❝õ❛ ✹ ✤✐➸♠
CA DA
:
✳ ❚❤❛②
CB DB
R2 CB
R2 DA
R2 DB
R2 CA
❀C B =
❀DA =
❀DB =
✱

CA =
OC · OA
OC · OB
OD · OA
OD.OB
CA DA
t❛ ❝â
:
= (A, B, C, D)✳
CB DB
❱➟② (A , B , C , D ) = (A, B, C, D).
❈❤ù♥❣ ♠✐♥❤✳

❚✛ sè ❦➨♣ ❝õ❛ ✹ ✤✐➸♠

(A , B , C , D ) =

P❤➨♣ ♥❣❤à❝❤ ✤↔♦ trð ♥➯♥ ✤➦❝ s➢❝ ♥❤í ❝→❝ ✤➦❝ tr÷♥❣ ❝â t❤➸ ❜✐➳♥ ✤÷í♥❣
trá♥ t❤➔♥❤ ✤÷í♥❣ t❤➥♥❣ ✈➔ ✤÷í♥❣ t❤➥♥❣ t❤➔♥❤ ✤÷í♥❣ trá♥✳ ◆❤÷♥❣ ♥â t❤ü❝
sü ❤✐➺✉ q✉↔ tr♦♥❣ ù♥❣ ❞ư♥❣ ♥❤í t➼♥❤ ❝❤➜t ❜↔♦ ❣✐→❝✱ tù❝ ❦❤ỉ♥❣ t❤❛② ✤ê✐
❣â❝ ❣✐ú❛ ✷ ✤÷í♥❣ ❝♦♥❣ ✭t❤➥♥❣✱ trá♥✮ q✉❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐✳ ❈ö t❤➸

▼➺♥❤ ✤➲ ✶✳✷✳ ●✐↔ sû γ1, γ2 ❧➔ ❤❛✐ ✤÷í♥❣ ❝♦♥❣ ✭✤÷í♥❣ t❤➥♥❣ ✱ ✤÷í♥❣ trá♥

❤♦➦❝ ữớ tũ ỵ tr t fRO2 : γ1 → γ1 , γ2 → γ2 ✳
❑❤✐ ✤â ∠ (γ1 , γ2 ) = ∠ (γ1 , γ2 )✳
❈❤ù♥❣ ♠✐♥❤✳

❚❛ ❝❤➾ ①➨t ❝→❝ ✤÷í♥❣ ❝♦♥❣

γ1 , γ2

❧➔ ✤÷í♥❣ t❤➥♥❣ ❤♦➦❝ ✤÷í♥❣

trá♥✳ ❉♦ t➼♥❤ ❝❤➜t ↔♥❤ ❝õ❛ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ t❛ ♣❤↔✐ ❝❤✐❛ t❤➔♥❤ ♥❤✐➲✉
tr÷í♥❣ ❤đ♣ ✈➲ ✈à tr➼ t÷ì♥❣ ✤è✐ ❝õ❛
✭✐✳✮ ❍❛✐ ✤÷í♥❣ t❤➥♥❣ ❦❤ỉ♥❣ q✉❛

γ1 , γ2

O❀

✤è✐ ✈ỵ✐ ❝ü❝ ♥❣❤à❝❤ ✤↔♦✿


✽

✭✐✐✳✮ ▼ët ✤÷í♥❣ t❤➥♥❣ q✉❛

O

✈➔ ♠ët ✤÷í♥❣ t❤➥♥❣ ❦❤ỉ♥❣ q✉❛

✭✐✐✐✳✮ ❍❛✐ ✤÷í♥❣ t❤➥♥❣ ❝➢t ♥❤❛✉ t↕✐

O❀

O ✈➔ ❝→❝ tr÷í♥❣ ❤đ♣ t÷ì♥❣ tü ❦❤✐ γ1 , γ2

❝ị♥❣ ❧➔ ✤÷í♥❣ trá♥ ❤♦➦❝ ♠ët ✤÷í♥❣ t❤➥♥❣✱ ♠ët ✤÷í♥❣ trá♥✳

❈❤➥♥❣ ❤↕♥ t❛ ❝❤ù♥❣ ♠✐♥❤ tr÷í♥❣ ❤đ♣

γ1 ∩ γ2 = P = O✱

❤➻♥❤ ✶✳✺✳

❍➻♥❤ ✶✳✺✿ ❚➼♥❤ ❝❤➜t ❜↔♦ ❣✐→❝
γ1 ≡ a ❜✐➳♥ t❤➔♥❤ ✤÷í♥❣ trá♥ q✉❛ O✱ t✐➳♣ t✉②➳♥
❝õ❛ ♥â t↕✐ O s♦♥❣ s♦♥❣ ✈ỵ✐ a✱ t÷ì♥❣ tü✱ ✤÷í♥❣ t❤➥♥❣ γ2 ≡ b ❜✐➳♥ t❤➔♥❤
✤÷í♥❣ trá♥ q✉❛ O ✱ t✐➳♣ t✉②➳♥ ❝õ❛ ♥â t↕✐ O s♦♥❣ s♦♥❣ ✈ỵ✐ b✳ ❱➻ θ ❧➔ ✶ tr♦♥❣
❝→❝ ❣â❝ ❣✐ú❛ ✷ t✐➳♣ t✉②➳♥ t↕✐ O ♥➯♥ ♥â ❧➔ ♠ët tr♦♥❣ ❤❛✐ ❣â❝ ❝õ❛ γ1 ✈➔ γ2 ✳
◆❤÷♥❣ ❝→❝ ✤÷í♥❣ trá♥ ♥➔② ❦❤æ♥❣ ❝❤➾ ❝➢t ♥❤❛✉ t↕✐ O ♠➔ ❝á♥ ❝➢t ♥❤❛✉ t↕✐
P ✳ ❉♦ ✤â✱ ❣â❝ θ ❝ô♥❣ ❧➔ ❣â❝ ❣✐ú❛ ✷ ✤÷í♥❣ trá♥ t↕✐ P ✳
❉♦ t➼♥❤ ✤è✐ ❤đ♣ ♥➯♥ ♠➺♥❤ ✤➲ ❤✐➸♥ ♥❤✐➯♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ γ1 , 2
ữớ trỏ q O ú ỵ r ợ ữớ trỏ t t P t ❝❤✉②➸♥
✈➲ ①➨t ✷ t✐➳♣ t✉②➳♥ t↕✐ P ✳
❚❛ t❤➜② ✤÷í♥❣ t

ữủ sỷ ử tữớ
ợ

1 , γ2

t✐➳♣ ①ó❝ ❤♦➦❝ trü❝ ❣✐❛♦


✾

✶✳✷ ❚å❛ ✤ë ❜❛r②❝❡♥tr✐❝ t❤✉➛♥ ♥❤➜t
✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ t

ố t
ỵ

XY Z

ABC

ồ õ t❛♠ ❣✐→❝ ❝ì sð ✭❦❤ỉ♥❣ s✉② ❜✐➳♥✮✳

❧➔ ❞✐➺♥ t➼❝❤ ✤↕✐ sè ❝õ❛ t❛♠ ❣✐→❝

XY Z ✳

❚❛ ❝â ✤à♥❤ ♥❣❤➽❛

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ●✐↔ sû ABC ❧➔ t❛♠ ❣✐→❝ ❝ì sð✳ ❚å❛ ✤ë ❜❛r②❝❡♥tr✐❝ ❝õ❛
✤✐➸♠ M ✤è✐ ✈ỵ✐ t❛♠ ❣✐→❝ ABC ❧➔ ❜ë ❜❛ sè (x : y : z) s❛♦ ❝❤♦

x : y : z = M BC : M CA : M AB
M = (x : y : z) t❤➻ ❝ô♥❣ ❝â
M = (kx : ky : kz), k = 0✳ ❈❤♦ ∆ABC ❣å✐ G, I, O, H, Oa
❚ø ✤à♥❤ ♥❣❤➽❛ t❛ s✉② r❛✿ ♥➳✉

❧➛♥ ❧÷đt ❧➔

trå♥❣ t➙♠✱ t➙♠ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣✱ t➙♠ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣✱ trü❝ t➙♠✱
t➙♠ ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ tr♦♥❣ ❣â❝

A

tr♦♥❣ t❛♠ ❣✐→❝ ✤â✳ ❑❤✐ ✤â t❛ ❝â✿

❱➼ ❞ö ✶✳✷✳✶✳ ❚❛ ❝â tå❛ ✤ë ❜❛r②❝❡♥tr✐❝ ❝õ❛ G, I, O, H, Oa
❛✳

G = (1 : 1 : 1)

❜✳

I = (a : b : c)

❝✳

O = (sin 2A : sin 2B : sin 2C) =

✈➻

✈➻

✿

SGBC = SGCA = SGAB ✳

SIBC = 21 ra, SICA = 12 rb, SIAB = 21 rc✳

= a2 b2 + c2 − a2 : b2 c2 + a2 − b2 : c2 a2 + b2 − c2
✣â ❧➔ ✈➻

✳

SOBC : SOCA : SOAB =:

1
1
1
= R2 sin 2A : R2 sin 2B : R2 sin 2C
2
2
2
= sin A cos A : sin B cos B : sin C cos C
b2 + c2 − a2 c2 + a2 − b2 b2 + a2 − c2
:b
:c
=a
2bc
2ac
2ba
2
2
2
2
2
2
2
2
= a b + c − a : b c + a − b : c2 a2 + b2 − c2 .
−S(Oa BC) : S(Oa CA) : S(Oa AB) = −a : b : c✳

❞✳

Oa = (−a : b : c)

❡✳

H = (tan A : tan B : tan C) =

✈➻

b2

1
: ... : ...
+ c 2 − a2

✳


✶✵

❢✳ ❈→❝ ✤✐➸♠ tr➯♥

CA, AB
❑❤✐

BC

❧➛♥ ❧÷đt ❝â tå❛ ✤ë

M = (x : y : z)

t✉②➺t ✤è✐ ❝õ❛

(0 : y : z)✳ ❚÷ì♥❣
(x : 0 : z), (x : y : 0)✳

❝â tå❛ ✤ë ❞↕♥❣

x + y + z = 0 t❛ t❤✉ ✤÷đ❝
x
y
z
:
:
x+y+z x+y+z x+y+z

M✿

♠➔

tü ❝→❝ ✤✐➸♠ tr➯♥

tå❛ ✤ë ❜❛r②❝❡♥tr✐❝
✱ ♥➳✉

x+y+z = 1

(x : y : z) ✤÷đ❝ ❣å✐ ❧➔ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝ ❝❤✉➞♥ ❝õ❛ M ✳ ◆➳✉
P (u : v : w), Q(u : v : w ) t❤ä❛ ♠➣♥ u + v + w = u + v + w
X ❝❤✐❛ P Q t❤❡♦ t✛ sè P X : XQ = p : q ❝â tå❛ ✤ë ❧➔
t❤➻

t❤➻ ✤✐➸♠

(qu + pu : qv + pv : qw + pw ) .

❱➼ ❞ö ✶✳✷✳✷✳ ❚➻♠ tå❛ ✤ë ❝→❝ ✤✐➸♠ T, T ✱ t➙♠ ✈à tü tr♦♥❣ ✈➔ ♥❣♦➔✐ ❝õ❛

✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ ✈➔ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ t❛♠ ❣✐→❝ ABC ✳
▲í✐ ❣✐↔✐✳

❚❛ ❝â T, T ❝❤✐❛ ✤✐➲✉ ❤á❛ ✤♦↕♥ t❤➥♥❣ OI ✱ ✈➔ ❞➵ t❤➜② t✛ sè
R abc S
abcs
=
: =
✳ ✭❙ ❧➔ ❞✐➺♥ t➼❝❤✱ s ❧➔ ♥û❛ ❝❤✉ ✈✐ t❛♠ ❣✐→❝ ❆❇❈✮
r
4S s
4S 2
2
❱➻ O = a
b2 + c2 − a2 : . . . : . . . = (s.a2 (b2 + c2 − a2 ) : · · · : · · · )

✈ỵ✐ tê♥❣ ❝→❝ tå❛ ✤ë ❜➡♥❣
❞ư♥❣ ❝→❝❤ t➼♥❤ tr➯♥ ✈ỵ✐

4S 2 ✈➔ I = (a : b : c) = 8S 2 a : 8S 2 b : 8S 2 c
OT
R
=

t❛ ❝â tå❛ ✤ë ❝õ❛ T ❧➔
TI
r

✳ ⑩♣

4S 2 · sa2 b2 + c2 − a2 + abcs.8S 2 a : . . . : . . .
❘ót ❣å♥ ❜✐➸✉ t❤ù❝✿

4S 2 · sa2 b2 + c2 − a2 + abcs.8S 2 a =
= 4sS 2 a2 b2 + c2 − a2 + 2bc
= 4sS 2 a2 (b + c)2 − a2
= 4sS 2 a2 (b + c + a)(b + c − a)

❱➟② t➙♠ ✈à tü tr♦♥❣

T = a2 (b + c − a) : b2 (a + c − b) : c2 (a + b − c)

❚÷ì♥❣ tü t➙♠ ✈à tü ♥❣♦➔✐✿

T = (a2 (a + b − c)(c + a − b) : b2 (b + c − a)(a + b − c) :
c2 (c + a − b)(b + c − a).
❈ô♥❣ ❝â t❤➸ ✈✐➳t

T =

b2
c2
a2
:

:
b+c−a c+a−b a+b−c

✳

✳


✶✶

❚r♦♥❣ ❬✻❪✱

T ≡ X55 , T ≡ X56 ✳

❱➼ ❞ö ✶✳✷✳✸✳ ❚å❛ ✤ë ❜❛r②❝❡♥tr✐❝ ❝õ❛ t➙♠ ❊✉❧❡r
O9 = a cos(B − C) : b cos(C − A) : c cos(A − B) .
❈❤ù♥❣ ♠✐♥❤✳
✤✐➸♠

✣â ❧➔ ❞♦ t❛ ❝â t✛ sè

OO9 : O9 G = 3 : −1✳

❚r♦♥❣ ❬✺❪✱

O9

❧➔

X5 ✳

✶✳✷✳✷ ▼ët sè ❦➳t q✉↔ tr♦♥❣ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝
❈❤ó♥❣ tỉ✐ tâ♠ t➢t ❝→❝ ❦➳t q✉↔ ❝ì ❜↔♥ ✤➣ ✤÷đ❝ P❛✉❧ ❨✐✉ ♥➯✉ tr♦♥❣ ❬✼❪✳

✭❛✮ ❈→❝ ❝❡✈✐❛♥ ✈➔ ✈➳t
❇❛ ✤÷í♥❣ t❤➥♥❣ ♥è✐ tø ✤✐➸♠
❝õ❛

P✳

●✐❛♦ ✤✐➸♠

❣å✐ ❧➔ ✈➳t ❝õ❛

P✳

AP , BP , CP

P

✤➳♥ ✸ ✤➾♥❤ t❛♠ ❣✐→❝ ❣å✐ ❧➔ ❝→❝ ❝❡✈✐❛♥

❝õ❛ ❝→❝ ❝❡✈✐❛♥ ♥➔② ✈ỵ✐ ❝→❝ ❝↕♥❤ t❛♠ ❣✐→❝

❚å❛ ✤ë ❝→❝ ✈➳t ❝â ❞↕♥❣

AP = (0 : y : z) BP = (x : 0 : z) CP = (x : y : 0)

✣à♥❤ ỵ

X BC, Y CA, Z AB t

ỵ

ừ ởt ✈➔ ❝❤➾ ❦❤✐ ❝❤ó♥❣ ❝â tå❛ ✤ë ❞↕♥❣

X = (0 : y : z),
Y = (x : 0 : z),
Z = (x : y : 0),

✭❜✮ ✣✐➸♠ ●❡r❣♦♥♥❡ ✈➔ ✤✐➸♠
t

X, Y, Z

ừ ữớ trỏ ở t ợ ❝→❝ ❝↕♥❤ t❛♠ ❣✐→❝ ❝â

tå❛ ✤ë

1
1
:
,
s−b s−c
1
1
:0
:
,
s−a

s−c
1
1
:
:0 .
s−a s−b

X = 0
X = (0
: s − c : s − b),
Y = (s − c : 0
: s − a),
Z = (s − b : s − a : 0).

❤❛②

Y

=

Z =

:


✶✷

◆❤÷ ✈➟②✱

AX, BY, CZ

❝➢t ♥❤❛✉ t↕✐ ✤✐➸♠ ❝â tå❛ ✤ë

1
1
1
:
:
.
s−a s−b s−c
✣â ❧➔ ✤✐➸♠ ●❡r❣♦♥♥❡

Ge

❝õ❛

∆ABC ✱

tr♦♥❣ ❬✻❪ ♥â ♠❛♥❣ ♥❤➣♥

X7

ừ ữớ trỏ t ợ ❝↕♥❤ t❛♠ ❣✐→❝✿

X = (0
: s − b : s − c),
Y = (s − a : 0
: s − c),
Z = (s − a : s − b : 0).
(s − a : s − b : s − c)✱ ❝â

❣å✐ ❧➔ ✤✐➸♠ ◆❛❣❡❧ Na ❝õ❛ ∆ABC ✳ ❍❛✐ ✤✐➸♠ Ge ✈➔ Na ❧➔ ✈➼ ❞ư ✈➲
✤✐➸♠ ✤➥♥❣ ❤đ♣ ✭❧✐➯♥ ❤đ♣ ✤➥♥❣ ❝ü✮✳ ❍❛✐ ✤✐➸♠ P, Q ✭❦❤ỉ♥❣ ♥❤➜t t❤✐➳t

✣â ❧➔ ✈➳t tr➯♥ ♠é✐ ❝↕♥❤ ❝õ❛ ✤✐➸♠ ❝â tå❛ ✤ë
t➯♥
❤❛✐

ð tr➯♥ ❝↕♥❤ t❛♠ ❣✐→❝✮ ✤÷đ❝ ❣å✐ ❧➔ ❤❛✐ ✤✐➸♠ ✤➥♥❣ ❤đ♣ ♥➳✉ ❝→❝ ✈➳t t÷ì♥❣
ù♥❣ ❝õ❛ ❝❤ó♥❣ ✤è✐ ①ù♥❣ ♥❤❛✉ q✉❛ tr✉♥❣ ✤✐➸♠ ❝↕♥❤ t÷ì♥❣ ù♥❣✳ ◆❤÷ ✈➟②✱

BAP = AQ C, CBP = BQ A✱ ACP = CQ B ✳
∗
❝õ❛ P ❧➔ P ✳ ❚❛ ❝â
1 1 1
P (x : y : z) ⇔ P ∗
: :
✳
x y z

❚❛ s ỵ ủ

ổ tự
= 2SABC
= . cot õ
ỵ

t➼❝❤ t❛♠ ❣✐→❝

b2 + c2 − a2
σA =

,
2

c2 + a2 − b2
σB =
,
2

ABC ✮✱

✈ỵ✐

θ ∈ R,

✤➦t

a2 + b 2 − c 2
σC =
2

❈❤➥♥❣ ❤↕♥✿

abc cos A
abc b2 + c2 − a2
b2 + c2 − a2
σA = 2SABC · cot A = 2 Ã
Ã
= 2Ã
Ã
=

.
4R sin A
4R sin A.2bc
2
ợ

,

tũ ỵ ✤➸ ❝❤♦ t✐➺♥ ❦❤✐ tr➻♥❤ ❜➔② t❛ ✤➦t

σθϕ = σθ .σϕ ✳

❚➼♥❤ ❝❤➜t ✶✳✷✳✶✳ ❚❛ ❝â ❤❛✐ t➼♥❤ ❝❤➜t ❝õ❛ σθ
• σB + σC = a2 , σC + σA = b2 , σA + σB = c2 ✳


✶✸

• σAB + σBC + σCA = σ 2 ✳
❈❤ù♥❣ ♠✐♥❤✳

✣➥♥❣ t❤ù❝ ✤➛✉ ❤✐➸♥ ♥❤✐➯♥✳ ✣➸ ❝â ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐✱ t❛ ♥❤➟♥

A + B + C = 1800 ♥➯♥ cot(A + B + C) ❧➔ ∞✳ ▼➝✉ sè ❝õ❛ ♥â ❜➡♥❣
cot A cot B + cot B cot C + cot C cot A − 1 = 0✳ ❚ø ✤â✱
σAB + σBC + σCA = σ 2 · (cot A cot B + cot B cot C + cot C cot A) = σ 2 ✳
①➨t✿ ✈➻

❱➼ ❞ö ✶✳✷✳✹✳ ❚å❛ ✤ë trü❝ t➙♠ H ✈➔ t➙♠ ♥❣♦↕✐ t✐➳♣ O t❤❡♦ σθ
1

1
1
:
:
σA σB σC
H ❜➡♥❣ σ 2 ✳

✲ ❚rü❝ t➙♠ ❍ ❝â tå❛ ✤ë
♥❣❛② tê♥❣ ❝→❝ tå❛ ✤ë ❝õ❛

❤❛②

(σBC : σCA : σAB )✳

❚❛ ❝â

✲ ❚➙♠ ♥❣♦↕✐ t✐➳♣ ❝â tå❛ ✤ë

a2 σA : b2 σB : c2 σC = (σA (σB + σC ) : σB (σC + σA ) : σC (σB + σA )) .
❱ỵ✐ ❝→❝❤ ❜✐➸✉ ❞✐➵♥ tờ tồ ở ừ

O



2 2

ú ỵ
ồ ✤ë ✤✐➸♠ t➙♠ ❊✉❧❡r ❜✐➸✉ ❞✐➵♥ t❤❡♦

σA , σB , σC

❧➔

O9 = σ 2 + σBC : σ 2 + σCA : σ 2 + σAB .
✲ ❚å❛ ✤ë ✤✐➸♠ ✤è✐ ①ù♥❣ ❝õ❛ trü❝ t➙♠ q✉❛ t➙♠ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣✱ tù❝
❧➔ ✤✐➸♠

L

❝❤✐❛ ✤♦↕♥ t❤➥♥❣

HO

t❤❡♦ t✛ sè

L = (σCA + σAB − σBC : . . . : . . .) =
✣â ❧➔ ✤✐➸♠ ❝â t➯♥ ❞❡ ▲♦♥❣❝❤❛♠♣s ❝õ❛

❚➼♥❤ ❝❤➜t

2
HL
=

LO
1
1
1
1

+

: ... : ... .
B C A

ABC

tr ỵ ❤✐➺✉ ❧➔

✳ ❱ỵ✐ ♠å✐ ✤✐➸♠ P

✭❈ỉ♥❣ t❤ù❝ ❈♦♥✇❛②✮

X20 ✳

❝õ❛ ♠➦t

ABC ỵ CBP = , BCP = t t❛ ❝â✿

P −a2 : σC + σϕ : σB + σθ
π π
❈→❝ ❣â❝ θ, ϕ ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ − ,
✈➔ ❣â❝ θ ❞÷ì♥❣ ❤❛② ➙♠ tị② t❤❡♦
2 2
❝→❝ ❣â❝ CBP CBA ữợ ũ ữợ
ự tr

❞ư ✶✳✷✳✺✳ ❳➨t ❤➻♥❤ ✈✉ỉ♥❣π BCX1X2 ❞ü♥❣
r❛ ♥❣♦➔✐ t❛♠ ❣✐→❝ ABC ✱ ❤➻♥❤
π

✶✳✻✳ ❚❛ ❝â ❝→❝ ❣â❝ CBX1 =

4

, BCX 1 =

❚÷ì♥❣ tü✱ X2 = −a2 : σC + σ : σB ✳

2

♥➯♥ X1 = −a2 : σC : σB + σ ✳


✶✹

❍➻♥❤ ✶✳✻✿ ❱➼ ❞ư ✈➲ ❝ỉ♥❣ t❤ù❝ ❈♦♥✇❛②

✭❞✮ P❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ t❤➥♥❣
✣÷í♥❣ t❤➥♥❣ ♥è✐ ✷ ✤✐➸♠ (x1 : y1 : z1 ), (x2 : y2 : z2 ) ❧➔

x y z
x1 y1 z1 = 0
x2 y2 z2
❤❛②

(y1 z2 − y2 z1 ) x + (z1 x2 − z2 x1 ) y + (x1 y2 − x2 y2 ) z = 0✳

❱➼ ❞ư ✶✳✷✳✻✳ ▼ët sè tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✿
x = 0, y = 0, z = 0✳

O a2 σA : b2 σB : c2 σC

✲P❤÷ì♥❣ tr➻♥❤ ❝→❝ ❝↕♥❤ ❇❈✱ ❈❆✱ ❆❇ ❧➛♥ ❧÷đt ❧➔
✲❚r✉♥❣ trü❝ ❝↕♥❤ ❇❈ ❧➔ ✤÷í♥❣ t❤➥♥❣ ố t
ợ tr

I(0 : 1 : 1)

õ ữỡ tr➻♥❤

b2 σB − c2 σC x − a2 σA y + a2 σA z = 0.
❱➻

b2 σB − c2 σC = . . . = σA (σB − σC ) = −σA b2 − c2 . ♥➯♥ ✈✐➳t ❧↕✐ t❤➔♥❤
b2 − c2 x + a2 (y − z) = 0.
✲ ✣÷í♥❣ t❤➥♥❣ ❊✉❧❡r ❧➔ ✤÷í♥❣ t❤➥♥❣ ♥è✐ trå♥❣ t➙♠

t➙♠

H (σBC : σCA : σAB )

G(1 : 1 : 1) ✈ỵ✐ trü❝

♥➯♥ ❝â ♣❤÷ì♥❣ tr➻♥❤

(σAB − σCA ) x + (σBC − σAB ) y + (σCA − σBC ) z = 0.


✶✺

❈â t❤➸ ✈✐➳t t➢t

σA (σB − σC ) x = 0.
✲ ✣÷í♥❣ t❤➥♥❣

OI

♥è✐ ✤✐➸♠

O a2 σA : b2 σB : c2 C

ợ

I(a : b : c)

õ ữỡ tr

0=


b2 σB · c − c2 σC · b x =

bσB − cσC = . . . = −2(b − c)s(s − a)✱
bc(b − c)s(s − a)x = 0

❤❛②

bc (bσB − cσC ) x.

♣❤÷ì♥❣ tr➻♥❤ trð t❤➔♥❤

(b − c)(s − a)
x = 0.
a

✭❡✮ ✣✐➸♠ ✈ỉ t➟♥ ✈➔ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣
✣✐➸♠

(x0 : y0 : z0 )

❧➔ ✤✐➸♠ ✈æ t➟♥ ♥➳✉ ♥â ❦❤æ♥❣ ♣❤↔✐ ✤✐➸♠ ❝â tå❛ ✤ë

x0 + y0 + z0 = 0✳ ❚❛ t❤➜② t➜t ❝↔ ❝→❝ ✤✐➸♠
t❤➥♥❣ L∞ ✱ ❝â ♣❤÷ì♥❣ tr➻♥❤ x + y + z = 0✳

❜❛r②❝❡♥tr✐❝ t✉②➺t ✤è✐✱ tù❝ ❧➔
t➟♥ ♥➡♠ tr➯♥ ♠ët ✤÷í♥❣

✈ỉ

❱➼ ❞ư ✶✳✷✳✼✳ ❈→❝ ✤✐➸♠ ✈ỉ t➟♥ tr➯♥ ❝→❝ ✤÷í♥❣ t❤➥♥❣ BC, CA, AB ❝õ❛ t❛♠
❣✐→❝ ❝ì sð ABC ❧➛♥ ❧÷ìt ❧➔ (0 : −1 : 1), (1 : 0 : −1), (−1 : 1 : 0)✳

❱➼ ❞ư ✶✳✷✳✽✳ ❈→❝ ✤✐➸♠ ✈ỉ t➟♥ tr➯♥ ✤÷í♥❣ ❝❛♦ ✤✐ q✉❛ A ❧➔

(0 : σC : σB ) − a2 (1 : 0 : 0) = (−a2 : σC : σB )✳ ❚ê♥❣ q✉→t✱ ✤✐➸♠ ✈æ t➟♥
tr➯♥ ✤÷í♥❣ t❤➥♥❣ px + qy + rz = 0 ❧➔ (q − r : r − p : p − q)✳

❱➼ ❞ư ✶✳✷✳✾✳ ✣✐➸♠ ✈ỉ t➟♥ tr➯♥ ✤÷í♥❣ t❤➥♥❣ ❊✉❧❡r✿
3 (σBC : σCA : σAB ) − σσ(1 : 1 : 1) = (3σBC − σσ : 3σCA − σσ : 3σAB − σσ) .

❈→❝ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ ❝â ❝ị♥❣ ✤✐➸♠ ✈ỉ t➟♥✳ ✣÷í♥❣ t❤➥♥❣ q✉❛

P (u : v : w)

s♦♥❣ s♦♥❣ ✈ỵ✐

L : px + qy + rz = 0✱

❝â ♣❤÷ì♥❣ tr➻♥❤

q−r r−p p−q
= 0.
u
v
w
x
y
z


✶✻

✭❢✮ ●✐❛♦ ❤❛✐ ✤÷í♥❣ t❤➥♥❣
●✐❛♦ ❤❛✐ ✤÷í♥❣ t❤➥♥❣

q1 r1
q2 r2

:

r1 p1

:

r2 p2

p1 x + q1 y + r1 z = 0, p2 x + q2 y + r2 z = 0 ❧➔ ✤✐➸♠

p 1 q1

= (q1 r2 − q2 r1 : r1 p2 − r2 p1 : p1 q2 − p2 q1 )✳

p 2 q2

✣✐➸♠ ✈ỉ t➟♥ tr➯♥ ✤÷í♥❣ t❤➥♥❣
t❤➥♥❣

L

❝â t ừ õ ợ ữớ

L : x + y + z = 0✳ ❈→❝ ✤÷í♥❣ t❤➥♥❣ pi x + qi y + ri z = 0, i = 1, 2, 3

ỗ q

p1 q1 r1
p2 q2 r2 = 0
p3 q3 r3
L : px+qy+rz = 0✳ ❚❛ ①→❝ ✤à♥❤
✤✐➸♠ ✈ỉ t➟♥ tr➯♥ ✤÷í♥❣ t❤➥♥❣ ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ L✿ L ∩ CA = Y (−r : 0 : p)❀

L ∩ AB = Z(q : −p : 0) ✳ ✣➸ t➻♠ ✤÷í♥❣ ✈✉ỉ♥❣ ❣â❝ tø A ①✉è♥❣ L✱ ✤➛✉
t✐➯♥ t❛ t➻♠ ♣❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ t❤➥♥❣ q✉❛ Y ✱ ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ AB ✈➔ q✉❛
Z ✱ ✈✉ỉ♥❣ ❣â❝ ợ CA õ
ữớ t ổ õ ữớ t

B σA −c2
−r 0
p =0
x y
z

✈➔

σC −b2 σA
q −p 0 = 0.
x
y
z

❚➼♥❤ ✤à♥❤ t❤ù❝ t❛ ❝â ✷ ♣❤÷ì♥❣ tr➻♥❤

σA px + c2 r − σB p y + σA rz = 0
σA px + σA qy + b2 q − σC p z = 0
❍❛✐ ✤÷í♥❣ ✈✉ỉ♥❣ ❣â❝ ♥➔② ❝➢t ♥❤❛✉ t↕✐ trü❝ t➙♠

∆AY Z ✱

❝â tå❛ ✤ë

X = · · · : σA p σ A r − b 2 q + σC p : σA p σA q + σB p − c 2 r

= (· · · : σC (p − q) − σA (q − r) : σA (q − r) − σB (r − p)) .
✣÷í♥❣ t q

A

ổ õ ợ

AX

õ ữỡ tr

1
0
0
à à à C (p − q) − σA (q − r) −σA (q − r) + σB (r − p) = 0
x
y
z


✶✼

❤❛②

− (σA (q − r) − σB (r − p)) y + (σC (p − q) − σA (q − r)) z = 0✳

◆â ❝â

✤✐➸♠ ✈æ t➟♥ ❧➔

(σB (r − p) − σC (p − q) : σC (p − q) − σA (q − r) : σA (q − r) B (r p))
ú ỵ r ổ t➟♥ ❝õ❛

L

❧➔

(q − r : r − p : p − q)✳

▼➺♥❤ ✤➲ ✶✳✸✳ ◆➳✉ L ❝â ✤✐➸♠ ✈æ t➟♥ (f : g : h) t ữớ ổ õ ợ
L ❝â ✤✐➸♠ ✈æ t➟♥

(f : g : h ) = (σB · g − σC · h : σC · h − σA · f : σA · f − σB · g)
(f : g : h)
σA f f + σB gg + σC hh = 0✳

▼ët ❝→❝❤ t÷ì♥❣ ✤÷ì♥❣✱ ữớ t ợ ổ t

(f : g : h )

s➩ ✈✉ỉ♥❣ ❣â❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐

✈➔

✭❣✮ P❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ trá♥
✲ P❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ❝ì sð

ABC

❧➔

a2 yz + b2 zx + c2 xy = 0.
✲ P❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ trá♥ ❊✉❧❡r ❝õ❛

∆ABC ✿

⑩♣ ❞ư♥❣ ♣❤➨♣ ✈à tü t➙♠

1
❜✐➳♥ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❤➔♥❤ ✤÷í♥❣ trá♥ ❊✉❧❡r✳ ◆➳✉
2
P (x : y : z) ❧➔ ✤✐➸♠ tr➯♥ ✤÷í♥❣ trá♥ ❊✉❧❡r t❤➻ ✤✐➸♠ Q = 3G − 2P =
= (x + y + z)(1 : 1 : 1) − 2(x : y : z) = (y + z − x : z + x − y : x + y − z)
G✱

t✛ sè

−

t❤✉ë❝ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣✱ tù❝ ❧➔

a2 (z+x−y)(x+y−z)+b2 (x+y−z)(y+z−x)+c2 (y+z−x)(z+x−y) = 0.
❘ót ❣å♥ ❧↕✐ t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ trá♥ ❊✉❧❡r✿

σA x2 − a2 yz + σB y 2 − b2 xz + σC z 2 − c2 xy = 0.
P❤÷ì♥❣ tr➻♥❤ tê♥❣ q✉→t ❝õ❛ ✤÷í♥❣ trá♥

C

❧➔

a2 yz + b2 zx + c2 xy + (x + y + z)(px + qy + rz) = 0,
p, q, r ❧➛♥ ❧÷đt ❧➔ ♣❤÷ì♥❣ t➼❝❤ ❝õ❛ A, B, C ố ợ ữớ trỏ C
px + qy + rz = 0 ❧➔ trư❝ ✤➥♥❣ ♣❤÷ì♥❣ ❝õ❛ C ✈➔ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣

tr♦♥❣ ✤â✱


✶✽

(ABC)✳ ✣÷í♥❣ trá♥ ♥➔② ❝â t➙♠ ❧➔ ✤✐➸♠ (x : y : z) ✈ỵ✐
x = a2 (σA + σB (r − p) − σC (p − q)❀ y = b2 σB + σC (p − q) − σA (r − p)❀
z = c2 σC + σA (q − r) − σB (r − p).
2
❇→♥ ❦➼♥❤ ρ ✤÷đ❝ ❝❤♦ ❜ð✐ ρ =
a2 b2 c2 − 2 a2 σA · p + b2 σB q + c2 σC r + σA (q − r)2 + σB (r − p)2 + σC (p − q)2
4σ 2

P (x1 : y1 : z1 ) , Q (x2 : y2 : z2 ) ❧✐➯♥ ❤ñ♣ ✤➥♥❣ ợ

2
2
2
tỗ t số k R ❝â x1 x2 = ka , y1 y2 = kb · z1 z2 = kc ✳

❍❛✐ ✤✐➸♠
❦❤✐ ✈➔ ❝❤➾

✭❤✮ ❉✐➺♥ t➼❝❤ t❛♠ ❣✐→❝
ABC ❧➔ t❛♠ ❣✐→❝ ❝ì sð✳ ●✐↔ sû P (p1 : p2 : p3 )✱ Q(q1 : q2 : q3 )✱

R(r1 : r2 : r3 ) ❝â tå❛ ✤ë t✛ ❝ü ❝❤✉➞♥ ❤â❛ t❤❡♦ ABC ✳ ❑❤✐ ✤â
▲➜②

p1 q1 r1
P QR = p2 q2 r2 .ABC.
p3 q3 r3

✭✶✳✷✮

−→
−→
−−→
−→
O, OP = p1 OA + p2 OB + p3 OC ❀
−→
−→
−−→
−→
OQ = q1 OA + q2 OB + q3 OC ✳ ❚ø ✤â✱
−→
−→
−−→
−→
P Q = (q1 − p1 ) OA + (q2 − p2 ) OB + (q3 − p3 ) OC.
−→
−→
−−→
▲➜② O ≡ C t❛ ❝â✿ P Q = (q1 − p1 ) CA + (q2 − p2 ) CB ✳ ❚÷ì♥❣ tü✱
−→
−→

−−→
1 −→ −→
P R = (r1 − p1 ) CA + (r2 − p2 ) CB ✳ ❚❛ ♥❤➟♥ ✤÷đ❝✿ P QR = P Q ∧ P R =
2
−→ −−→ 1
−−→ −→
1
(q1 − p1 ) (r2 − q2 ) CA ∧ CB + (q2 − p2 ) (r1 − p1 ) CB ∧ CA✳
2
2
1 −→ −−→
1 −−→ −→
❱➻ ABC = CA ∧ CB = − CB ∧ CA t t ữủ
2
2
ự

ợ ồ

P QR = ((q1 − p1 ) (r2 − p2 ) − (q2 − p2 ) (r1 − p1 )) ABC =
= [(p1 q2 − p2 q1 ) + (q1 r2 − q2 r1 ) + (r1 p2 − r2 p1 )] ABC.
(p1 q2 − p2 q1 ) ✈ỵ✐ r1 + r2 + r3 = 1✱ ❜✐➸✉ t❤ù❝
(q1 r2 − q2 r1 ) ✈ỵ✐ p1 +p2 +p3 = 1✱ ❜✐➸✉ t❤ù❝ (r1 p2 − r2 p1 ) ✈ỵ✐ q1 +q2 +q3 = 1
❙❛✉ ❦❤✐ ♥❤➙♥ ❜✐➸✉ t❤ù❝

t❛ ♥❤➟♥ ✤÷đ❝ ✭✶✳✷✮✳


✶✾

❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② tâ♠ t➢t ✷ ♥ë✐ ❞✉♥❣✿ P❤➨♣ ợ ữớ
trỏ trữợ ỏ ❣å✐ ❧➔ ♣❤➨♣ ✤è✐ ①ù♥❣ q✉❛ ✤÷í♥❣ trá♥
✈➔ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝ ♠➔ ♠ët sè t→❝ ❣✐↔ ✤➦t t➯♥ ❧➔ tå❛ ✤ë t✛ ❝ü ❤♦➦❝ tå❛
✤ë ❞✐➺♥ t➼❝❤✳ ❚r♦♥❣ ❝❤÷ì♥❣ ✷ t❛ s➩ sû ❞ư♥❣ ❝❤ó♥❣ ❧➔♠ ❝ỉ♥❣ ❝ư ✤➸ t➻♠ ❤✐➸✉
s➙✉ ✈➲ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ ✤✐➸♠ ❙♦❞❞②✱ t❛♠ ❣✐→❝ ❙♦❞❞②✱✳✳✳