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

✐

▼ö❝ ❧ö❝
❉❛♥❤ ♠ö❝ ❤➻♥❤
▲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♦♥ ❧î♣ κ = 2
✸✳✷✳✷
❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧î♣ κ = 4
❈→❝ t❛♠ ❣✐→❝ ❧î♣
= tb + tc ✳ ✳ ✳ ✳ ✳
✸✳✸✳✶
❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧î♣
=1
✸✳✸✳✷
❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❧î♣
=2
❈→❝ t❛♠ ❣✐→❝ ❧î♣

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✸

✳ ✳ ✳ ✳ ✳ ✳

✹✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✹

✺✼
✺✽


✐✐✐

❉❛♥❤ ♠ö❝ ❤➻♥❤
✶✳✶

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

✶✳✷

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

✶✳✸

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

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

✹

✹

✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

2

AB =

R · AB
OA.OB

✶✳✹

❑❤♦↔♥❣ ❝→❝❤

✶✳✺

❚➼♥❤ ❝❤➜t ❜↔♦ ❣✐→❝

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

✽

✶✳✻

❱➼ ❞ö ✈➲ ❝æ♥❣ 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

✸✳✺

❚❛♠ ❣✐→❝ ❦✐➸✉ ❙♦❞❞②

✸✳✻

✣÷í♥❣ t❤➥♥❣ ●❡r❣♦♥♥❡ s♦♥❣ s♦♥❣ ✈î✐

✸✳✼

◗✉ÿ t➼❝❤ ✤✐➸♠

✸✳✽

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

✸✳✾

❚❛♠ ❣✐→❝ ❍❡r♦♥ ❧î♣

A

t✐➳♣ t✉②➳♥ ✤➾♥❤

E = X481
Fl =

G

∩

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

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

✸✾

C

θ

ABC

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

AD

✹✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷

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

✹✺

✸✳✶✵ ❚❛♠ ❣✐→❝ ❍❡r♦♥ ❧î♣

=1✳
=2✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✻

ớ ỡ
t ữủ ởt tổ ổ ữủ
sỹ ữợ ú ù t t ừ P t
rữớ ồ Pỏ ổ t tọ ỏ
t ỡ s s t ỷ ớ tr t ừ tổ ố ợ ỳ
t tổ
ổ t ỡ ỏ t qỵ t
ổ ợ ồ rữớ ồ ồ
ồ t t tr t ỳ tự qỵ ụ
ữ t tổ t õ ồ
ổ ỷ ớ ỡ t t tợ ỳ
ữớ ổ ở ộ trủ t ồ tổ tr sốt
q tr ồ t tỹ
tr trồ ỡ

Pỏ t
ữớ t

ổ rồ


✶

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

ABC

❝â ♥❤ú♥❣ t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t✱

❜➔✐ t♦→♥ ❞ü♥❣ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞② ❧➔ tr÷í♥❣ ❤ñ♣ r✐➯♥❣ q✉❛♥ trå♥❣ ❝õ❛
❜➔✐ t♦→♥ ❆♣♦❧✐❧♦♥✐✉s✳ ❈❤❛ ✤➫ ❝õ❛ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ ✤✐➸♠ ❙♦❞❞②✱ ✤÷í♥❣
t❤➥♥❣ ❙♦❞❞②✱ t❛♠ ❣✐→❝ ❙♦❞❞②✱✳✳ ❧➔ ❋r❡❞❡r✐❝❦ ❙♦❞❞②✱ ♥❣÷í✐ ✤➣ ❞➔♥❤ ✤÷ñ❝
❣✐↔✐ 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♥❣ ✤÷ñ❝ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ ①→❝ ✤à♥❤ ❝→❝ ❜→♥ ❦➼♥❤ ❝õ❛ ❝❤ó♥❣ t❤❡♦
❝→❝ ②➳✉ tè ❝õ❛ t❛♠ ❣✐→❝ ❝❤♦ tr÷î❝❄ ❝→❝ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ ❝→❝ ✤÷í♥❣ t❤➥♥❣
❙♦❞❞② ❝â ❧✐➯♥ q✉❛♥ ❣➻ ✈î✐ ❝→❝ ✤÷í♥❣ trá♥ ✈➔ ✤÷í♥❣ t❤➥♥❣ ✤➣ ❜✐➳t ❦❤→❝❄
❚r➻♥❤ ❜➔② ❝→❝❤ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ tr➯♥ ❧➔ ❧þ ❞♦ ✤➸ tæ✐ ❝❤å♥ ✤➲ t➔✐
✧✣÷í♥❣ trá♥ ❙♦❞❞② ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥✧✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔✿
✲ ❚r➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ ❝→❝❤ ①→❝ ✤à♥❤ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ t➼♥❤ ✤÷ñ❝
❝→❝ ❜→♥ ❦➼♥❤✱ t➻♠ ✤÷ñ❝ ❝→❝ t➼♥❤ ❝❤➜t ♠î✐ ❝õ❛ ✤÷í♥❣ trá♥ ❙♦❞❞② ♥ë✐ ✈➔
✤÷í♥❣ trá♥ ❙♦❞❞② ♥❣♦↕✐✳ ❚ø ✤â ✤÷❛ r❛ ❝→❝❤ ❞ü♥❣ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❝→❝
✤÷í♥❣ trá♥✱ ✤÷í♥❣ t❤➥♥❣ ❙♦❞❞② tr♦♥❣ tå❛ ✤ë ❜❛r②❝❡♥tr✐❝✳
✲ ❳→❝ ✤à♥❤ ♠è✐ q✉❛♥ ❤➺ ❝õ❛ t❛♠ ❣✐→❝ ❙♦❞❞② ✈î✐ ❝→❝ ✤✐➸♠ ✈➔ ✤÷í♥❣
t❤➥♥❣ ✤➦❝ ❜✐➺t ❦❤→❝✳
✲ P❤➙♥ ❧♦↕✐ ✤÷ñ❝ ❝→❝ t❛♠ ❣✐→❝ ❧î♣

κ = ta + tb + tc

❦❤↔♦ s→t ❝→❝ tr÷í♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t ❝õ❛ ✷ ❧î♣ ✤â✳

✈➔ ❧î♣

= tb + tc ✱




ở t ỳ qt
ở ữủ ữỡ

ữỡ tự ờ s
ờ s ừ ỡ ữủ sỷ ử ổ ử
qt t t r P tồ ở rtr ữỡ
ỗ ử
P tr t
ồ ở rtr t t

ữỡ ữớ trỏ
ở ữỡ sỹ ữớ trỏ
ũ ở ừ õ ữỡ ồ sỡ ữỡ
tồ ở ởt tr ỳ trồ t ừ ữỡ
ỗ ử s tờ ủ ờ s tứ
ỹ ữớ trỏ
ữớ trỏ
ữớ trỏ tr tồ ở rtr
t rr

ữỡ ởt số q
ữỡ t q ữớ trỏ t
tỹ t trữớ ủ r q trồ q
tr ồ t r ữỡ
ữủ t tờ ủ t ở ỗ

t ợ
t ợ

= ta + tb + tc
= tb + tc


✸

❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❜ê s✉♥❣
❚❛ ♥❤➢❝ ❧↕✐ ✈➔ ❜ê s✉♥❣ ❤❛✐ ♥ë✐ ❞✉♥❣ ❝➛♥ ❝❤♦ ❝→❝ ❝❤÷ì♥❣ s❛✉✿ ❚❤ù ♥❤➜t✱
✤✐➸♠ q✉❛ ✈➲ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ✤➣ ✤÷ñ❝ ♥❣❤✐➯♥ ❝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 ←→ ∞✳
❚❛ ❦þ ❤✐➺✉ ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ ✤â ❧➔

✤÷í♥❣ trá♥ ♥❣❤à❝❤ ✤↔♦✳

fRO2 ✱

✤÷í♥❣ trá♥

(O, R)

✤÷ñ❝ ❣å✐ ❧➔

P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ♥➔② ❝ô♥❣ ❣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❤➥♥❣ ❦❤æ♥❣ 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 trö❝ ✤è✐ ①ù♥❣ tr♦♥❣ ♣❤➨♣ ✤è✐ ①ù♥❣✮✳ ▼å✐ ✤✐➸♠ ð tr♦♥❣
❜✐➳♥ t❤➔♥❤ ✤✐➸♠ ð ♥❣♦➔✐ ✈➔ ♥❣÷ñ❝ ❧↕✐✳

(O, R)


✺

❍➻♥❤ ✶✳✸✿ ❷♥❤ ❝õ❛ ✤÷í♥❣ trá♥ ❦❤æ♥❣ 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♥ ❦❤æ♥❣ ✤✐ q✉❛

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

✤÷í♥❣ trá♥ ❦❤æ♥❣ ✤✐ q✉❛
t❤➔♥❤ ✤÷í♥❣ trá♥ ✤ç♥❣

(I, r) ❜✐➳♥ t❤➔♥❤ ❝❤➼♥❤
t➼❝❤ p✱ ✈î✐ 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♥❤✳ ❱î✐ ♠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♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ OH ✳ ✣↔♦ ❧↕✐✱ ✈î✐ ♠å✐ N tr➯♥
❝✮ ❍↕




OH

OH

ON ổ t t ởt
N N O t t ổ t tr ự N HH N

2
ở t õ õ ố 90 r ON.ON = OH.OH = R
t H, H ừ ồ M M ữớ trỏ
ữớ OH ừ ữớ t ổ q O ữớ trỏ
q O
t ố ủ t õ ừ ữớ trỏ q O ữớ t
ổ q O

ữớ trỏ ữớ

ứ ỹ ừ ởt t s r ồ ữớ
trỏ t r t ữớ trỏ 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 trỏ

C



O

O

t số tỹ

R2 /p

H O

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

p

tữỡ ố t

tỹ t

t õ ỵ tỹ t

t

ồ

t õ ỹ

t

ũ ỹ

O

t

ợ

C

h = R2 /p.

ổ q

O

t ừ ữỡ t

ổ tự t t
t ỳ ừ trữợ

(O, R) ữớ trỏ A , B



ừ A, B t

R2 AB
AB =
OA.OB
ự

õ

OAB OB A



AB
OA
OA ã OA
R2
R2 ã AB
=
=
=
= A B =
.
AB
OB

OA ã OB
OA ã OB
OA.OB
ồ 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♥❣ ❞ö♥❣ ♥❤í t➼♥❤ ❝❤➜t ❜↔♦ ❣✐→❝✱ tù❝ ❦❤æ♥❣ t❤❛② ✤ê✐
❣â❝ ❣✐ú❛ ✷ ✤÷í♥❣ ❝♦♥❣ ✭t❤➥♥❣✱ trá♥✮ q✉❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐✳ ❈ö t❤➸

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

❤♦➦❝ ✤÷í♥❣ tò② þ✮ 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♥❣ ❤ñ♣ ✈➲ ✈à tr➼ t÷ì♥❣ ✤è✐ ❝õ❛
✭✐✳✮ ❍❛✐ ✤÷í♥❣ t❤➥♥❣ ❦❤æ♥❣ q✉❛

γ1 , γ2

O❀

✤è✐ ✈î✐ ❝ü❝ ♥❣❤à❝❤ ✤↔♦✿


✽

✭✐✐✳✮ ▼ët ✤÷í♥❣ t❤➥♥❣ q✉❛

O

✈➔ ♠ët ✤÷í♥❣ t❤➥♥❣ ❦❤æ♥❣ q✉❛

✭✐✐✐✳✮ ❍❛✐ ✤÷í♥❣ t❤➥♥❣ ❝➢t ♥❤❛✉ t↕✐

O❀

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

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

❈❤➥♥❣ ❤↕♥ t❛ ❝❤ù♥❣ ♠✐♥❤ tr÷í♥❣ ❤ñ♣

γ1 ∩ γ2 = P = O✱

❤➻♥❤ ✶✳✺✳

❍➻♥❤ ✶✳✺✿ ❚➼♥❤ ❝❤➜t ❜↔♦ ❣✐→❝
γ1 ≡ a ❜✐➳♥ t❤➔♥❤ ✤÷í♥❣ trá♥ q✉❛ O✱ t✐➳♣ t✉②➳♥
❝õ❛ ♥â t↕✐ O s♦♥❣ s♦♥❣ ✈î✐ a✱ t÷ì♥❣ tü✱ ✤÷í♥❣ t❤➥♥❣ γ2 ≡ b ❜✐➳♥ t❤➔♥❤
✤÷í♥❣ trá♥ q✉❛ O ✱ t✐➳♣ t✉②➳♥ ❝õ❛ ♥â t↕✐ O s♦♥❣ s♦♥❣ ✈î✐ 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✐ ❤ñ♣ ♥➯♥ ♠➺♥❤ ✤➲ ❤✐➸♥ ♥❤✐➯♥ tr♦♥❣ tr÷í♥❣ ❤ñ♣ γ1 , γ2 ❧➔ ❤❛✐
✤÷í♥❣ trá♥ q✉❛ O ✳ ❈❤ó þ r➡♥❣ ✈î✐ ✷ ✤÷í♥❣ trá♥ ❝➢t ♥❤❛✉ t↕✐ P t❛ ❝❤✉②➸♥
✈➲ ①➨t ✷ t✐➳♣ t✉②➳♥ t↕✐ P ✳
❚❛ t❤➜② ✤÷í♥❣ t❤➥♥❣

▼➺♥❤ ✤➲ ✤÷ñ❝ sû ❞ö♥❣ t❤÷í♥❣ ①✉②➯♥ ❦❤✐
✈î✐ ♥❤❛✉✳

γ1 , γ2

t✐➳♣ ①ó❝ ❤♦➦❝ trü❝ ❣✐❛♦


✾

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

❚❛ ❝è ✤à♥❤ t❛♠ ❣✐→❝
❑þ ❤✐➺✉

XY Z

ABC ✱

❣å✐ ♥â ❧➔ t❛♠ ❣✐→❝ ❝ì sð ✭❦❤æ♥❣ s✉② ❜✐➳♥✮✳

❧➔ ❞✐➺♥ t➼❝❤ ✤↕✐ sè ❝õ❛ t❛♠ ❣✐→❝

XY Z ✳

❚❛ ❝â ✤à♥❤ ♥❣❤➽❛

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ●✐↔ sû ABC ❧➔ t❛♠ ❣✐→❝ ❝ì sð✳ ❚å❛ ✤ë ❜❛r②❝❡♥tr✐❝ ❝õ❛
✤✐➸♠ M ✤è✐ ✈î✐ 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❛✿ ♥➳✉

❧➛♥ ❧÷ñt ❧➔

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

❧➛♥ ❧÷ñt ❝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❤✉ ✤÷ñ❝
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) ✤÷ñ❝ ❣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 ) : · · · : · · · )

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

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♥❣ tæ✐ tâ♠ t➢t ❝→❝ ❦➳t q✉↔ ❝ì ❜↔♥ ✤➣ ✤÷ñ❝ P❛✉❧ ❨✐✉ ♥➯✉ tr♦♥❣ ❬✼❪✳

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

P✳

●✐❛♦ ✤✐➸♠

❣å✐ ❧➔ ✈➳t ❝õ❛

P✳

AP , BP , CP

P

✤➳♥ ✸ ✤➾♥❤ t❛♠ ❣✐→❝ ❣å✐ ❧➔ ❝→❝ ❝❡✈✐❛♥

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

❚å❛ ✤ë ❝→❝ ✈➳t ❝â ❞↕♥❣

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

✣à♥❤ ❧þ ✶✳✶

✳ ❇❛ ✤✐➸♠ X ∈ BC, Y ∈ CA, Z ∈ AB ❧➔ ✈➳t

✭✣à♥❤ ❧þ ❈❡✬✈❛✮

❝õ❛ ♠ët ✤✐➸♠ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝❤ó♥❣ ❝â tå❛ ✤ë ❞↕♥❣

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

✭❜✮ ✣✐➸♠ ●❡r❣♦♥♥❡ ✈➔ ✤✐➸♠ ◆❛❣❡❧
❇❛ t✐➳♣ ✤✐➸♠

X, Y, Z

❝õ❛ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ✈î✐ ❝→❝ ❝↕♥❤ 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 ✳

❚✐➳♣ ✤✐➸♠ ❝õ❛ ❝→❝ ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ ✈î✐ ❝→❝ ❝↕♥❤ 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 ❧➔ ✈➼ ❞ö ✈➲
✤✐➸♠ ✤➥♥❣ ❤ñ♣ ✭❧✐➯♥ ❤ñ♣ ✤➥♥❣ ❝ü✮✳ ❍❛✐ ✤✐➸♠ P, Q ✭❦❤æ♥❣ ♥❤➜t t❤✐➳t

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

ð tr➯♥ ❝↕♥❤ t❛♠ ❣✐→❝✮ ✤÷ñ❝ ❣å✐ ❧➔ ❤❛✐ ✤✐➸♠ ✤➥♥❣ ❤ñ♣ ♥➳✉ ❝→❝ ✈➳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➩ ❦þ ❤✐➺✉ ✤✐➸♠ ✤➥♥❣ ❤ñ♣

✭❝✮ ❈æ♥❣ t❤ù❝ ❈♦♥✇❛②
σ = 2SABC
σθ = σ. cot θ✳ ❑❤✐ ✤â
❑þ ❤✐➺✉

✭❤❛✐ ❧➛♥ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝

b2 + c2 − a2
σA =

,
2

c2 + a2 − b2
σB =
,
2

ABC ✮✱

✈î✐

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 ·
·
= 2·
·
=

.
4R sin A
4R sin A.2bc
2
❱î✐

θ, ϕ

tò② þ ✤➸ ❝❤♦ 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 )) .
❱î✐ ❝→❝❤ ❜✐➸✉ ❞✐➵♥ ♥➔②✱ tê♥❣ ❝→❝ tå❛ ✤ë ❝õ❛

O

❜➡♥❣

2σ 2 ✳

❈❤ó þ✳
✲ ❚å❛ ✤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

+
−
: ... : ... .
σB σC σA

∆ABC ✱

tr♦♥❣ ❬✻❪ ❦þ ❤✐➺✉ ❧➔

✳ ❱î✐ ♠å✐ ✤✐➸♠ P

✭❈æ♥❣ t❤ù❝ ❈♦♥✇❛②✮

X20 ✳

❝õ❛ ♠➦t ♣❤➥♥❣

ABC ❦þ ❤✐➺✉ CBP = θ, BCP = ϕ t❤➻ t❛ ❝â✿

P −a2 : σC + σϕ : σB + σθ
π π
❈→❝ ❣â❝ θ, ϕ ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ − ,
✈➔ ❣â❝ θ ❞÷ì♥❣ ❤❛② ➙♠ tò② t❤❡♦
2 2
❝→❝ ❣â❝ CBP ✈➔ CBA ❦❤→❝ ❤÷î♥❣ ❤❛② ❝ò♥❣ ❤÷î♥❣✳
❈❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❬✼❪✳

❱➼ ❞ö ✶✳✷✳✺✳ ❳➨t ❤➻♥❤ ✈✉æ♥❣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 ✳


✶✹

❍➻♥❤ ✶✳✻✿ ❱➼ ❞ö ✈➲ ❝æ♥❣ 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✳

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

O a2 σA : b2 σB : c2 σC

✲P❤÷ì♥❣ tr➻♥❤ ❝→❝ ❝↕♥❤ ❇❈✱ ❈❆✱ ❆❇ ❧➛♥ ❧÷ñt ❧➔
✲❚r✉♥❣ trü❝ ❝↕♥❤ ❇❈ ❧➔ ✤÷í♥❣ t❤➥♥❣ ♥è✐ t➙♠
✈î✐ tr✉♥❣ ✤✐➸♠

I(0 : 1 : 1)

♥➯♥ ❝â ♣❤÷ì♥❣ 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) ✈î✐ 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 σC

✈î✐ ✤✐➸♠

I(a : b : c)

♥➯♥ ❝â ♣❤÷ì♥❣ 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

✭❡✮ ✣✐➸♠ ✈æ 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♥❣

✈æ

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

❱➼ ❞ö ✶✳✷✳✽✳ ❈→❝ ✤✐➸♠ ✈æ 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)✳

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

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

P (u : v : w)

s♦♥❣ s♦♥❣ ✈î✐

L : px + qy + rz = 0✱

❝â ♣❤÷ì♥❣ tr➻♥❤

q−r r−p p−q
= 0.
u
v
w
x
y
z




ữớ t
ữớ 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

ổ t tr ữớ t
t

L

õ t ừ õ ợ ữớ

L : x + y + z = 0 ữớ 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
ổ t tr ữớ t ổ õ ợ L L CA = Y (r : 0 : p)

L AB = Z(q : p : 0) t ữớ ổ õ tứ A ố L
t t t ữỡ tr ữớ t q Y ổ õ ợ AB q
Z ổ õ ợ CA õ
ữớ t ổ õ ữớ t

B A c2
r 0
p =0
x y
z



C b2 A
q p 0 = 0.
x
y
z

tự t õ ữỡ tr

A px + c2 r B p y + A rz = 0
A px + A qy + b2 q C p z = 0
ữớ ổ õ t t trỹ t

AY Z

õ tồ ở

X = ã ã ã : A p A r b 2 q + C p : A p A q + B p c 2 r

= (ã ã ã : C (p q) A (q r) : A (q r) B (r p)) .
ữớ t q

A

ổ õ ợ

AX

õ ữỡ tr

1
0
0
ã ã ã C (p q) A (q r) A (q r) + B (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) − σB (r − p))
❈❤ó þ r➡♥❣ ✤✐➸♠ ✈æ t➟♥ ❝õ❛

L

❧➔

(q − r : r − p : p − q)✳

▼➺♥❤ ✤➲ ✶✳✸✳ ◆➳✉ L ❝â ✤✐➸♠ ✈æ t➟♥ (f : g : h) t❤➻ ✤÷í♥❣ ✈✉æ♥❣ ❣â❝ ✈î✐
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♥❣✱ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ ✈î✐ ✤✐➸♠ ✈æ t➟♥

(f : g : h )

s➩ ✈✉æ♥❣ ❣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♣ ❞ö♥❣ ♣❤➨♣ ✈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 ❧➛♥ ❧÷ñt ❧➔ ♣❤÷ì♥❣ t➼❝❤ ❝õ❛ A, B, C ✤è✐ ✈î✐ ✤÷í♥❣ trá♥ C
❤❛② px + qy + rz = 0 ❧➔ trö❝ ✤➥♥❣ ♣❤÷ì♥❣ ❝õ❛ C ✈➔ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣

tr♦♥❣ ✤â✱




(ABC) ữớ trỏ õ t (x : y : z) ợ
x = a2 (A + B (r p) C (p q) y = b2 B + C (p q) A (r p)
z = c2 C + A (q r) B (r p).
2
ữủ =
a2 b2 c2 2 a2 A ã p + b2 B q + c2 C r + A (q r)2 + B (r p)2 + C (p q)2
4 2

P (x1 : y1 : z1 ) , Q (x2 : y2 : z2 ) ủ ợ

2
2
2
tỗ t số k R õ x1 x2 = ka , y1 y2 = kb ã z1 z2 = kc




t t
ABC t ỡ s sỷ P (p1 : p2 : p3 ) Q(q1 : q2 : q3 )

R(r1 : r2 : r3 ) õ tồ ở t ỹ õ t ABC õ


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 ứ õ




P Q = (q1 p1 ) OA + (q2 p2 ) OB + (q3 p3 ) OC.



O C t õ P Q = (q1 p1 ) CA + (q2 p2 ) CB ữỡ tỹ

1
P R = (r1 p1 ) CA + (r2 p2 ) CB ữủ 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 ) ợ r1 + r2 + r3 = 1 tự
(q1 r2 q2 r1 ) ợ p1 +p2 +p3 = 1 tự (r1 p2 r2 p1 ) ợ q1 +q2 +q3 = 1
tự

t ữủ


✶✾

❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② tâ♠ t➢t ✷ ♥ë✐ ❞✉♥❣✿ P❤➨♣ ♥❣❤à❝❤ ✤↔♦ ✈î✐ ✤÷í♥❣
trá♥ ♥❣❤à❝❤ ✤↔♦ ❝❤♦ tr÷î❝ ❤❛② ❝á♥ ❣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 ❞ö♥❣ ❝❤ó♥❣ ❧➔♠ ❝æ♥❣ ❝ö ✤➸ t➻♠ ❤✐➸✉
s➙✉ ✈➲ ✤÷í♥❣ trá♥ ❙♦❞❞②✱ ✤✐➸♠ ❙♦❞❞②✱ t❛♠ ❣✐→❝ ❙♦❞❞②✱✳✳✳