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 VUÔNG GÓC,

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 TOÁN CHỨNG MINH TÍNH VUÔNG GÓC,

SONG SONG TRONG HÌNH HỌC

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 Trịnh Thanh Hải

THÁI NGUYÊN - 2019

❈✉è✐ ❝ò♥❣ ❡♠ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ♥❣÷í✐ t❤➙♥ ❧➔ ♥❤ú♥❣ ♥❣÷í✐

❧✉æ♥ õ♥❣ ❤ë✱ ✤ë♥❣ ✈✐➯♥ ❡♠ ✈÷ñt 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✐♥❤ ♣❤↔✐ ♥➢♠ ✤÷ñ❝ ❝→❝ ❦✐➳♥ t❤ù❝ ♥➙♥❣ ❝❛♦✱ ✤➙② ❧➔ ❝→❝ ✤à♥❤ ❧þ✱ t➼♥❤ ❝❤➜t

❱î✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ ❝→❝ ✤à♥❤ ❧þ✱ t➼♥❤ ❝❤➜t ❝ô♥❣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣

❝❤ù♥❣ ♠✐♥❤ 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❛✉✿

• ❚➻♠ ❤✐➸✉ ❝→❝ ✤à♥❤ ❧þ✱ ❝→❝ t➼♥❤ ❝❤➜t ❧✐➯♥ q✉❛♥ ✤➳♥ ✤✐➲✉ ❦✐➺♥ ✤➸ ❤❛✐ ✤÷í♥❣t❤➥♥❣ s♦♥❣ s♦♥❣ ✭❤❛② ✈✉æ♥❣ ❣â❝✮ ✈î✐ ♥❤❛✉ ❝ô♥❣ ♥❤÷ ❝→❝ ❤➺ q✉↔ ❝â ✤÷ñ❝

tø ✈✐➺❝ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ ✭❤❛② ✈✉æ♥❣ ❣â❝✮✳

• ❙÷✉ t➛♠ ❝→❝ ❜➔✐ t♦→♥ ❧✉②➺♥ t❤✐ ✤ë✐ t✉②➸♥ ❤å❝ s✐♥❤ ❣✐ä✐✱ ❝→❝ ✤➲ t❤✐ ❤å❝s✐♥❤ ❣✐ä✐ t♦→♥ ✈➲ ❤➻♥❤ ❤å❝ ♣❤➥♥❣ ❧✐➯♥ q✉❛♥ ✤➳♥ t➼♥❤ s♦♥❣ s♦♥❣✱ t➼♥❤

✈✉æ♥❣ ❣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✐ ✈î✐ ♥❤ú♥❣

❜➔✐ t♦→♥✱ ✤➲ t❤✐ ♠➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➾ ❝â ❧í✐ ❣✐↔✐ ✈➢♥ t➢t ❤♦➦❝ ✤à♥❤

❤÷î♥❣ ❧í✐ ❣✐↔✐✳

❧þ ❚❤❛❧❡s ✤↔♦✱ ✤à♥❤ ❧þ P②t❤❛❣♦r❛s✱ ✤à♥❤ ❧þ ❈❡✈❛ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✤÷í♥❣t❤➥♥❣ ✤æ✐ ♠ët s♦♥❣ s♦♥❣ ❤♦➦❝ ✤ç♥❣ q✉②✱ ✤à♥❤ ❧þ ▼❡♥❡❧❛✉s tr♦♥❣ t❛♠ ❣✐→❝

✈➔ tù ❣✐→❝✱ ✤à♥❤ ❧þ ❈❛r♥♦t t❤✉ ✤÷ñ❝ tø ❝→❝ ✤÷í♥❣ t❤➥♥❣ ✈✉æ♥❣ ❣â❝ ♥➡♠ tr➯♥

❝→❝ ❝↕♥❤ ❝õ❛ t❛♠ ❣✐→❝ ✣ç♥❣ t❤í✐ ❝ô♥❣ ✤÷❛ r❛ ♠ët sè ❜➔✐ t➟♣ →♣ ❞ö♥❣

❝→❝ ✤à♥❤ ❧þ tr➯♥ ✤➸ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ✈✉æ♥❣ ❣â❝ ✈➔ s♦♥❣ s♦♥❣✳

❈❤÷ì♥❣ ✷✳ ❈→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ✈✉æ♥❣ ❣â❝ tr♦♥❣ ❝→❝ ✤➲t❤✐ ❍å❝ s✐♥❤ ❣✐ä✐

◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ t÷í♥❣ ♠✐♥❤ ✈✐➺❝ ✈➟♥ ❞ö♥❣ ❝→❝

✤à♥❤ ❧þ✱ t➼♥❤ ❝❤➜t ✤➸ ❝❤ù♥❣ ♠✐♥❤ ♠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➼♥❤ ✈✉æ♥❣ ❣â❝✳

❈❤÷ì♥❣ ✸✳ ❈→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ s♦♥❣ s♦♥❣ tr♦♥❣ ❝→❝ ✤➲t❤✐ ❍å❝ s✐♥❤ ❣✐ä✐

◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✸ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ t÷í♥❣ ♠✐♥❤ ✈✐➺❝ ✈➟♥

❞ö♥❣ ❝→❝ ✤à♥❤ ❧þ✱ 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✐ ❣✐❛♥ ❣✐î✐ ❤↕♥ ♥➯♥ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❧✉➟♥ ✈➠♥ t➟♣tr✉♥❣ ❝❤õ ②➳✉ ❧➔ ❝→❝ ❜➔✐ t♦→♥ t❤✉ë❝ ❍➻♥❤ ❤å❝ ♣❤➥♥❣✳

❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✻ t❤→♥❣ ✸ ♥➠♠ ✷✵✶✾

❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥

◆❣å❝ ❚❤à ❍➔

❤❛✐ ✤÷í♥❣ t❤➥♥❣ ✈✉æ♥❣ ❣â❝ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝❤ó♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❍❛✐ ✤÷í♥❣ t❤➥♥❣ xx′, yy′ ❝➢t ♥❤❛✉ ✈➔ tr♦♥❣ ❝→❝ ❣â❝ t↕♦t❤➔♥❤ ❝â ♠ët ❣â❝ ✈✉æ♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ ✈✉æ♥❣ ❣â❝ ✈➔ ✤÷ñ❝ ❦þ

M M′ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➨♣ ✤è✐ ①ù♥❣ t➙♠ I✳ P❤➨♣ ✤è✐ ①ù♥❣ t➙♠ t❤÷í♥❣ ✤÷ñ❝ ❦➼

❤✐➺✉ ❧➔ ✣I

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❈❤♦ ✤✐➸♠ O ✈➔ ❣â❝ ❧÷ñ♥❣ ❣✐→❝ α✳ P❤➨♣ ❜✐➳♥ ❤➻♥❤ ❜✐➳♥ Ot❤➔♥❤ ❝❤➼♥❤ ♥â✱ ❜✐➳♥ ♠é✐ ✤✐➸♠ M ❦❤→❝ O t❤➔♥❤ ✤✐➸♠ M′ s❛♦ ❝❤♦ OM =

OM′ ✈➔ ❣â❝ ❧÷ñ♥❣ ❣✐→❝ (OM, OM\ ′) = α ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➨♣ q✉❛② t➙♠ O ❣â❝

✭✐✐✐✮ ❇✐➳♥ t❛♠ ❣✐→❝ t❤➔♥❤ t❛♠ ❣✐→❝ ✤ç♥❣ ❞↕♥❣ ✈î✐ ♥â✱ ❜✐➳♥ ❣â❝ t❤➔♥❤ ❣â❝

❜➡♥❣ ♥â✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✳ ❈❤♦ ❤❛✐ ✤÷í♥❣ trá♥ ❦❤æ♥❣ ✤ç♥❣ t➙♠ (O1, R1); (O2, R2)✳

❚➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ M ❝â ♣❤÷ì♥❣ t➼❝❤ ✤è✐ ✈î✐ ❤❛✐ ✤÷í♥❣ trá♥ ❜➡♥❣ ♥❤❛✉ ❧➔

♠ët ✤÷í♥❣ t❤➥♥❣✳ ✣÷í♥❣ t❤➥♥❣ ♥➔② ❣å✐ ❧➔ trö❝ ✤➥♥❣ ♣❤÷ì♥❣ ❝õ❛ ❤❛✐ ✤÷í♥❣trá♥ ✤➣ ❝❤♦✳

❚➼♥❤ ❝❤➜t ✶✳✺✳ ▼ët ✤÷í♥❣ t❤➥♥❣ ✈✉æ♥❣ ❣â❝ ✈î✐ ♠ët tr♦♥❣ ❤❛✐ ✤÷í♥❣ t❤➥♥❣s♦♥❣ s♦♥❣ t❤➻ ♥â ❝ô♥❣ ✈✉æ♥❣ ❣â❝ ✈î✐ ✤÷í♥❣ t❤➥♥❣ ❦✐❛✳

❚➼♥❤ ❝❤➜t ✶✳✻✳ ❍❛✐ ✤÷í♥❣ t❤➥♥❣ ♣❤➙♥ ❜✐➺t ❝ò♥❣ s♦♥❣ s♦♥❣ ✈î✐ ♠ët ✤÷í♥❣t❤➥♥❣ t❤ù ❜❛ t❤➻ ❝❤ó♥❣ s♦♥❣ s♦♥❣ ✈î✐ ♥❤❛✉✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ✤÷í♥❣ t❤➥♥❣ xx′ ❝➢t ❝→❝ ❝↕♥❤ AB, AC ❝õ❛ t❛♠ ❣✐→❝ABC t❤❡♦ t❤ù tü t↕✐ D ✈➔ E s❛♦ ❝❤♦

❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥✿ ●✐↔ sû AD, BE, CF ✤ç♥❣ q✉②✳ ❚ø A ✈➩ ✤÷í♥❣t❤➥♥❣ s♦♥❣ s♦♥❣ ✈î✐ BC ❝➢t BE, CF t↕✐ I ✈➔ H✳ ❚❤❡♦ ✤à♥❤ ❧þ ❚❤❛❧❡s t❛ ❝â

RB

RA · P C

P B · QA

QC = 1 ✭✶✳✼✮

AM = BD

BC ⇒ AB · CD = AM · BD ✭✶✳✶✹✮

❚ø ✭✶✳✶✸✮ ✈➔ ✭✶✳✶✹✮ t❛ ✤÷ñ❝

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

⇒ AB.CD + AD.BC = AC.BD

⇒ M B2 + N C2 + P A2 = (OB2 − OM2) + (OC2 − ON2) + (OA2 − OP2)

= (OC2 − OM2) + (OA2 − ON2) + (OB2 − 0P2) = M C2 + N A2 + P B2

KA2+KB2−2KA.KB cos α+KC2+KD2−2KC.KD cos α = AB2+CD2

KA2+KD2−2KA.KD cos α+KC2+KB2−2KC.KB cos α = AD2+BC2

❝â ✤ë ❦❤â t➠♥❣ ❞➛♥ ✈➔ ✈➟♥ ❞ö♥❣ ❧✐♥❤ ❤♦↕t ❝❤ó♥❣✳ ✣ç♥❣ t❤í✐ ♠ët sè ❜➔✐ t♦→♥t→❝ ❣✐↔ ❝â ✤÷❛ r❛ ✷ ❧í✐ ❣✐↔✐ ❦❤→❝ ♥❤❛✉ ✤➸ ❤å❝ s✐♥❤ ❧ü❛ ❝❤å♥ ❝→❝ ❤÷î♥❣ t✐➳♣

❈❤ù♥❣ ♠✐♥❤✳ ❑➨♦ ❞➔✐ CK, BK ❝➢t AB, AC t↕✐ P, Q ❚❛ ❝â EM, F M ❧➛♥

❧÷ñt ❧➔ ✤÷í♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛ △BP C, △BQC ♥➯♥ BE = P E, QF = CF ✳

❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ✤à♥❤ ❧þ ❈❡✈❛ ❝❤♦ △ABC ✈î✐ ❝→❝ ✤÷í♥❣ t❤➥♥❣ ✤ç♥❣q✉② AD, BF ✈➔ CE t❛ ❝â

✣÷í♥❣ t❤➥♥❣ DE ❝➢t ❝↕♥❤ AC t↕✐ M✱ ✤÷í♥❣ t❤➥♥❣ DF ❝➢t ❝↕♥❤ AB t↕✐ N

❈❤ù♥❣ ♠✐♥❤ r➡♥❣ MN k BC

❝→❝ ✤✐➸♠ P, M, I t❤➥♥❣ ❤➔♥❣ ❤❛② P I ❧➔ ✤÷í♥❣ ❦➼♥❤ ❝õ❛ ✤÷í♥❣ trá♥ ♥❣♦↕✐t✐➳♣ t❛♠ ❣✐→❝ ABC✳ ❉♦ ✤â QD = QE ⇒ DP Q\ = QP E\✳ ❙✉② r❛ ❜❛ ✤✐➸♠

⑩♣ ❞ö♥❣ ✤à♥❤ ❧➼ ❈❡✈❛ ❝❤♦ t❛♠ ❣✐→❝ ABC ✈î✐ ❜❛ ✤÷í♥❣ BD, CE, AJ ✤ç♥❣q✉② t↕✐ I t❛ ❝â J C

AH, CK ❝ò♥❣ ✈✉æ♥❣ ❣â❝ ✈î✐ ✤÷í♥❣ t❤➥♥❣ m ♥➯♥ AH k CK s✉② r❛ AE

EK =AH

▼➦t ❦❤→❝ AD, BK ❧➔ ✤÷í♥❣ ❝❛♦ t❛♠ ❣✐→❝ ABC ♥➯♥ DAE\ = CBK\ =

90◦ − \ACB s✉② r❛ △AED ∼ △BKC (g.g)✳ ❉♦ ✤â AE

DB = 1.

❈❤ù♥❣ ♠✐♥❤✳ ❑➫ ✤÷í♥❣ ♣❤➙♥ ❣✐→❝ BH ❝õ❛ △EBC✳ ❇ð✐ ✈➻ EBC\ = ACK\

♥➯♥HBC\+HCB\ = \ACE+ECB\ = 90◦✱ tù❝ ❧➔ BH ⊥ CE ❙✉② r❛ △EBC

❇➔✐ t♦→♥ ✶✳✶✼✳ ❬✶❪ ❈❤♦ tù ❣✐→❝ ABCD, AB ❦❤æ♥❣ s♦♥❣ s♦♥❣ ✈î✐ CD

✣÷í♥❣ trá♥ (C1) q✉❛ A, B ✈➔ t✐➳♣ ①ó❝ ✈î✐ CD t↕✐ P ✱ ✤÷í♥❣ trá♥ (C2) q✉❛

C, D t✐➳♣ ①ó❝ ✈î✐ AB t↕✐ Q✱ (C1) ✈➔ (C2) ❝➢t ♥❤❛✉ t↕✐ E ✈➔ F ✳ ❈❤ù♥❣ ♠✐♥❤r➡♥❣ EF ✤✐ q✉❛ tr✉♥❣ ✤✐➸♠ ❝õ❛ P Q ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ BC k AD

❈❤ù♥❣ ♠✐♥❤✳ ●å✐ I ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ EF ✈➔ P Q✱ ✤÷í♥❣ t❤➥♥❣ P Q ❝➢t ✤÷í♥❣trá♥ (C1) ✈➔ ✤÷í♥❣ trá♥ (C2) t❤❡♦ t❤ù tü t↕✐ K ✈➔ G✳

❚❤❡♦ ❣✐↔ t❤✐➳t AB ❦❤æ♥❣ s♦♥❣ s♦♥❣ ✈î✐ CD ♥➯♥ AB ✈➔ CD ❣✐❛♦ ♥❤❛✉✱

❣å✐ ❣✐❛♦ ✤✐➸♠ ❧➔ M✳ ❚❤❡♦ ❤➺ t❤ù❝ ✤÷í♥❣ trá♥ s✉② r❛

M P2 = M A.M B ✈➔ MQ2 = M C.M D ✭✶✳✸✷✮

▼➦t ❦❤→❝ t❛ ❝â

IE.IF = IP.IK = IQ.IG ⇒ IP (IQ + QK) = IQ(IP + IG)

⇒ IP.QK = IQ.P G,s✉② r❛

❇➔✐ t♦→♥ ✶✳✷✵✳ ❬✷❪ ❈❤♦ tù ❣✐→❝ ABCD, AC = BD.AC ❝➢t BD t↕✐ P ●å✐(O1) ❧➔ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABP ✱ (O2) ❧➔ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣

❈❤ù♥❣ ♠✐♥❤✳ ●å✐ K, L t❤❡♦ t❤ù tü ❧➔ ❣✐❛♦ ✤✐➸♠ t❤ù ❤❛✐ ❝õ❛ AE, AF ✈➔(AB′C′)✳ ❱➻ A, B′, C′, L, K ✤ç♥❣ ✈✐➯♥ ✈➔ BE k B′A, CF k C′A ♥➯♥

OE2 − OF2 = (OE2 − R2) − (OF2 − R2) = EA.EC − F A.F B

❈❤÷ì♥❣ ✷

❈→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ ✈✉æ♥❣ ❣â❝ tr♦♥❣ ❝→❝ ✤➲ t❤✐ ❍å❝ s✐♥❤ ❣✐ä✐

❈❤ù♥❣ ♠✐♥❤✳ ●å✐ H = AN ∩ MD, K = BN ∩ MC, I = MN ∩ CD✳ ❚❛♠

❣✐→❝ AMD ✈✉æ♥❣ t↕✐ A, AH ❧➔ ✤÷í♥❣ ❝❛♦ ♥➯♥ AM2 = M H.M D✳

❚❛♠ ❣✐→❝ BMC ✈✉æ♥❣ t↕✐ B, BK ❧➔ ✤÷í♥❣ ❝❛♦ ♥➯♥ BM2 = M K.M C✳

▼➦t ❦❤→❝✱ AM = BM ❞♦ ✤â MH.MD = MK.MC✳ ❚❛♠ ❣✐→❝ MKH ✈➔t❛♠ ❣✐→❝ MDC ❝â KM H\ ❝❤✉♥❣ ✈➔ M H

\

ABC = ADC\ = 60◦ ❞♦ tù ❣✐→❝ ABCD ❧➔ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤✳

❳➨t t❛♠ ❣✐→❝ BCE✱ ❝â \EBC = \BEC = 60◦ ♥➯♥ △BEC ❧➔ t❛♠ ❣✐→❝ ✤➲✉✳

◆❣♦➔✐ r❛ DCE\ = 180◦ −DAE\ = 60◦ s✉② r❛ DCE\ = ADC\ ♥➯♥ ❤➻♥❤ t❤❛♥❣AECD ❧➔ ❤➻♥❤ t❤❛♥❣ ❝➙♥✳ ❑❤✐ ✤â ACD\ = EDC\ ♥➯♥ t❛♠ ❣✐→❝ ICD ❝➙♥ t↕✐

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â AE k DB ❞♦ ✤â \EAO = AOB\ = \AOE s✉② r❛ △AEO

❝➙♥ t↕✐ E✳ ●å✐ I ❧➔ tr✉♥❣ ✤✐➸♠ OA, J ❣✐❛♦ ✤✐➸♠ ❝õ❛ EF ✈➔ AD✳ ❚❛ ❝â

OA = IF

OB ⇔ IE

OA = IF

OD.

✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ ❈❤♦ tù ❣✐→❝ ❧ç✐ ABCD ❝â ❝→❝ ❝➦♣ ❝↕♥❤ ✤è✐ ❦❤æ♥❣ s♦♥❣s♦♥❣✳ AB ❝➢t CD t↕✐ E, AD ❝➢t BC t↕✐ F ✳ ❍➻♥❤ t↕♦ ❜ð✐ tù ❣✐→❝ ABCD✱

✣÷í♥❣ trá♥ (O) ✤✐ q✉❛ B ✈➔ D ❝➢t AB, AD t↕✐ E, F ❀ DE ❝➢t BF t↕✐ G; M

❧➔ tr✉♥❣ ✤✐➸♠ AG✳ ❈❤ù♥❣ ♠✐♥❤ CM ⊥ AO✳

✣➸ ❣✐↔✐ ❜➔✐ t♦→♥ ♥➔② t❛ ①➨t ❜ê ✤➲ s❛✉✳

❇ê ✤➲ ✷✳✶✳ ❈❤♦ tù ❣✐→❝ t♦➔♥ ♣❤➛♥ ABCDEF ✳ ◗✉❛ F ❦➫ ❝→❝ ✤÷í♥❣ t❤➥♥❣s♦♥❣ s♦♥❣ ✈î✐ AB, CD, AC, BD ❝➢t CD, AB, BD, AC t↕✐ M, N, P, Q t❤❡♦t❤ù tü✳ ❑❤✐ ✤â M, N, P, Q t❤➥♥❣ ❤➔♥❣ ✈➔ ✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ❝❤ó♥❣ s♦♥❣s♦♥❣ ✈î✐ EG✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â ♥❤➟♥ ①➨t ✤ì♥ ❣✐↔♥ s❛✉✿ ❝❤♦ ❤❛✐ t❛♠ ❣✐→❝ ABC, DEF ❝â

❝→❝ ✤÷í♥❣ t❤➥♥❣ AD, BE, CF ✤ç♥❣ q✉②✳ ❑❤✐ ✤â ♥➳✉ AB k DE ✈➔ BC k EFt❤➻ CD k DF

⑩♣ ❞ö♥❣ ♥❤➟♥ ①➨t tr➯♥ ❝❤♦ ❝→❝ ❝➦♣ t❛♠ ❣✐→❝ (F QN, DGE); (F MN, BEG);(F M P, AEG) s✉② r❛ ❝→❝ ✤÷í♥❣ t❤➥♥❣ MN, MP, NQ ❝ò♥❣ s♦♥❣ s♦♥❣ ✈î✐

EG✱ ❞♦ ✤â ❜è♥ ✤✐➸♠ M, N, P, Q t❤➥♥❣ ❤➔♥❣ ✈➔ ✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ❝❤ó♥❣s♦♥❣ s♦♥❣ ✈î✐ EG✳

✤✐➸♠ AB, AC✳ ❈→❝ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABE ✈➔ t❛♠ ❣✐→❝ ACD

❝➢t ♥❤❛✉ t↕✐ K ❦❤→❝ A✳ ✣÷í♥❣ t❤➥♥❣ LB ❝➢t ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ABE t↕✐ ✤✐➸♠ t❤ù ❤❛✐ M✱ ✤÷í♥❣ t❤➥♥❣ LC ❝➢t ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ ACDt↕✐ ✤✐➸♠ t❤ù ❤❛✐ N✳ ❈❤ù♥❣ ♠✐♥❤ M, N, K t❤➥♥❣ ❤➔♥❣ ✈➔ MN ⊥ OL

❚❛ ❝â IBN[ = U CV\ = 90◦ −\AU B = EIU[ = M IB\✱ ❞♦ ✤â IM = BM✳

❚÷ì♥❣ tü t❛ ❝ô♥❣ ❝â IN = NC✳ ❉♦ ✤â

IM2 − IN2 = BM2 − CN2 = SM2 − SN2

❚❤❡♦ ✤à♥❤ ❧þ ✹ ✤✐➸♠ t❛ ❝â SI ⊥ MN✳

2AQ.AC ⇔ AD.AE = AQ.AC.

▼➦t ❦❤→❝ AD.AE = AB.AC ✭❞♦ △ADB ∼ △ACE✮ ♥➯♥ AQ = AB ❤❛② B

✤è✐ ①ù♥❣ Q q✉❛ AD ❞♦ ✤â EB = EQ✳

❍ì♥ ♥ú❛✱ M RQ\ = AN M\ = ACB\ ❞♦ ✤â tù ❣✐→❝ BRQC ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣

❤❛② ER = EB = EC ✭❞♦ E ❧➔ tr✉♥❣ ✤✐➸♠ ❝✉♥❣ BC⌢ ❦❤æ♥❣ ❝❤ù❛ A ❝õ❛(O)✮✳

✤✐➸♠ ❝õ❛ AC ✈➔ BD ❜✐➳t r➡♥❣ K t❤✉ë❝ ✤♦↕♥ BE (K 6= B; K 6= E)✳ ✣÷í♥❣t❤➥♥❣ ✤✐ q✉❛ K s♦♥❣ s♦♥❣ ✈î✐ BC ❝➢t AC t↕✐ P ✳ ❈❤ù♥❣ ♠✐♥❤ KP ⊥ P M✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â P AD\ = CBD\ = P KD\ ✭✈➻ P AD\ = CBD\ ❤❛✐ ❣â❝ ♥ë✐t✐➳♣ ❝ò♥❣ ❝❤➢♥ ♠ët ❝✉♥❣✱ CBD\ = P KD\ ❤❛✐ ❣â❝ ✤ç♥❣ ✈à✮ ♥➯♥ tù ❣✐→❝AKP D ♥ë✐ t✐➳♣✳

❚ù ❣✐→❝ AKP D ♥ë✐ t✐➳♣ ♥➯♥ AP D\ = AKD\ = 90◦ ✈➔ DKP\ = \DAP

❝ô♥❣ ❧➔ ✤÷í♥❣ ❝❛♦✱ tr✉♥❣ t✉②➳♥ ❝õ❛ △ABP ✳

❳➨t tù ❣✐→❝ BHCP ✱ t❛ ❝â BHP\ = 90◦ ✭✈➻ P O ⊥ AB✮✱ BCP\ = 90◦ ✭✈➻

❦➲ ❜ò BCD\ = 90◦ ♥ë✐ t✐➳♣ ♥û❛ ✤÷í♥❣ trá♥ (O)✮✳ ❉♦ ✤â BHP\ = BCP\ s✉②r❛ tù ❣✐→❝ BHCP ♥ë✐ t✐➳♣✳

❳➨t △ACH t❛ ❝â HAC\ = Bc1 ✭❝❤➢♥ ❝✉♥❣ BKC⌢ ❝õ❛ ✤÷í♥❣ trá♥ (O)✮✳

▼➦t ❦❤→❝ Bc1 = Hc1 ✭❞♦ tù ❣✐→❝ BHCP ♥ë✐ t✐➳♣✮ ❞♦ ✤â HAC\ = Hc1✳ ◆❣♦➔✐r❛✱ t❛ ❝â Hc1 +AHC\ = 90◦ ✭✈➻ P O ⊥ AB) ♥➯♥ HAC\+AHC\ = 90◦✳ ❉♦ ✤â

△AHC ✈✉æ♥❣ t↕✐ C✳ ❍❛② AC ⊥ CH✳

❇➔✐ t♦→♥ ✷✳✶✸ ✭❈❤✉②➯♥ ❚P✳ ❍ç ❈❤➼ ▼✐♥❤ ♥➠♠ ❤å❝ ♥➠♠ ❤å❝ ✷✵✶✹ ✕ ✷✵✶✺✮✳

❚ø ♠ët ✤✐➸♠ M ♥➡♠ ♥❣♦➔✐ ✤÷í♥❣ trá♥ (O)✱ ❦➫ ❝→❝ t✐➳♣ t✉②➳♥ MA, MB ✈î✐(O) (A, B ❧➔ ❝→❝ t✐➳♣ ✤✐➸♠✮✳ ●å✐ H ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ A❇ ✈î✐ OM, I ❧➔ tr✉♥❣

✤✐➸♠ ❝õ❛ MH✳ ✣÷í♥❣ t❤➥♥❣ AI ❝➢t (O) t↕✐ ✤✐➸♠ K (K 6= A)✳ ❈❤ù♥❣ ♠✐♥❤

HK ⊥ AI✳

❈❤ù♥❣ ♠✐♥❤✳ ❑➫ ✤÷í♥❣ ❦➼♥❤ AE ❝õ❛ (O), EH ❝➢t (O) t↕✐ K′, AK′ ❝➢t EBt↕✐ D✳ ❉➵ t❤➜② H ❧➔ trü❝ t➙♠ t❛♠ ❣✐→❝ AED ♥➯♥ DH ⊥ AO s✉② r❛

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❣✐↔ t❤✐➳t BE ⊥ AC, CF ⊥ AB s✉② r❛ tù ❣✐→❝ BF EC

♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ BC ❝â t➙♠ ❧➔ M

❚÷ì♥❣ tü t❛ ❝â tù ❣✐→❝ AF HE ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ AH ❝ât➙♠ ❧➔ N✳ ❈❤♦ ✤÷í♥❣ trá♥ (N) ❝➢t NM t↕✐ K′ s✉② r❛ AK\′H = 90◦✳ ❉♦ ✤â

❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ I ❧➯♥ BC; M ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ BC; K ❧➔ ✤✐➸♠ ✤è✐ ①ù♥❣

❝õ❛ M q✉❛ AI✳ ❈❤ù♥❣ ♠✐♥❤ KD ⊥ OI

❈❤ù♥❣ ♠✐♥❤✳ ●å✐ AI ❣✐❛♦ (O) t↕✐ N ❦❤→❝ A, AI ❣✐❛♦ BC t↕✐ F, MN ❣✐❛♦(O)t↕✐ T ❦❤→❝ N, MK ❣✐❛♦ AN t↕✐ H, DK ❣✐❛♦ OI t↕✐ E✳ ❚❛ ❝â △NMH ∼

❝❤♦ AB = AP ; AC = AQ;\BAP = \CAQ = 30◦✳ ❈→❝ ✤÷í♥❣ t❤➥♥❣ BQ ✈➔

❧➔ ❝→❝ ✤÷í♥❣ ❝❛♦ ❝õ❛ t❛♠ ❣✐→❝ ABC (E ∈ AC, F ∈ AB)✳ ✣÷í♥❣ t❤➥♥❣ EF

❝➢t BC t↕✐ G✱ ✤÷í♥❣ t❤➥♥❣ AG ❝➢t (O) t↕✐ M✳ ●å✐ T ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛

BC✳ ❈❤ù♥❣ ♠✐♥❤ GH ⊥ AT

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â tù ❣✐→❝ BCEF ♥ë✐ t✐➳♣ ♥➯♥ GE.GF = GB.GC✳ ▼➦t

❦❤→❝✱ GB.GC = GM.GA tø ✤â s✉② r❛ GE.GF = GM.GA✳ ❉♦ ✤â tù ❣✐→❝

AM F E ♥ë✐ t✐➳♣✳

❍ì♥ ♥ú❛✱ tù ❣✐→❝ AEHF ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ AH ♥➯♥ M ♥➡♠tr➯♥ ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ AH ❤❛② MH ⊥ MA✳

●å✐ R ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ MH ✈î✐ (O) t❛ ❝â AM R\ = 90◦ ♥➯♥ AR ❧➔ ✤÷í♥❣

❦➼♥❤ ❝õ❛ (O) s✉② r❛ tù ❣✐→❝ HBRC ❧➔ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤✳ ❉♦ ✤â HR ❝➢t BCt↕✐ tr✉♥❣ ✤✐➸♠ ❝õ❛ BC ❤❛② M, H, T t❤➥♥❣ ❤➔♥❣✳

❱➟② H ❧➔ trü❝ t➙♠ ❝õ❛ t❛♠ ❣✐→❝ AGT ♥➯♥ GH ⊥ AT

✭✐✮ ❈❤♦ t❛♠ ❣✐→❝ ABC ✈➔ ❤❛✐ ✤÷í♥❣ Ax, Ay ✤➥♥❣ ❣✐→❝ tr♦♥❣ ❣â❝ \BAC ❦❤✐

✤â xAB[ = yAC[✳

▼å✐ t❛♠ ❣✐→❝ ✤➲✉ ❝â ✸ ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ ♣❤➙♥ ❜✐➺t✱ ♠é✐ ✤÷í♥❣ trá♥t✐➳♣ ①ó❝ ✈î✐ ♠ët ❝↕♥❤✳ ❚➙♠ ❝õ❛ ♠ët ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛

❚❤❡♦ ❣✐↔ t❤✐➳t✱ BE k AD, DE k O1A ♥➯♥

\

O1AD = BED\ ⇒ABC\ = 90◦ −BED.\

❙✉② r❛ \ABC = \ACB ❞♦ ✤â △ABC ❝➙♥ t↕✐ A ♥➯♥ AB = AC✳

❳➨t tù ❣✐→❝ O1DCO2 ❝â CAO\2 = DAO\1 ✭❣â❝ ✤è✐ ✤➾♥❤✮✳ ▼➔ t❛♠ ❣✐→❝ADO1 ✈➔ t❛♠ ❣✐→❝ ACO2 ❝➙♥ ♥➯♥ AO\1D = AO\2C ⇒ CO\1D = DO\2C ❞♦

✤â tù ❣✐→❝ O1DCO2 ♥ë✐ t✐➳♣✳ ▼➦t ❦❤→❝ AO\2O1 = 90◦ −BED\ ✈➔ O\1AD =

❚❤❡♦ ❣✐↔ t❤✐➳t✱ EA = EB ♥➯♥ P E k KB s✉② r❛ P E ⊥ BC.

❇➔✐ t♦→♥ ✷✳✷✺ ✭❇r❛③t❦♦s ♥➠♠ ✷✵✶✹✱ ❬✷❪✮✳ ❈❤♦ ❤➻♥❤ t❤❛♥❣ ABCD ♥ë✐ t✐➳♣

✤÷í♥❣ trá♥ (O) ✤÷í♥❣ ❦➼♥❤ AB, E ❧➔ ❣✐❛♦ ✤✐➸♠ ❤❛✐ ✤÷í♥❣ ❝❤➨♦ AC ✈➔ BD✳

✣÷í♥❣ trá♥ t➙♠ B ❜→♥ ❦➼♥❤ BE ❝➢t (O) t↕✐ P ✈➔ Q ✭P ❝ò♥❣ ♣❤➼❛ ✈î✐ C ❜í

AB✮✳ ✣÷í♥❣ t❤➥♥❣ q✉❛ E ✈✉æ♥❣ ❣â❝ ✈î✐ BD ❝➢t CD t↕✐ M✳ ❈❤ù♥❣ ♠✐♥❤r➡♥❣ P M ⊥ DQ

❈❤ù♥❣ ♠✐♥❤✳ ●å✐ N ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ ✤÷í♥❣ t❤➥♥❣ ME ✈➔ AB✳ ❚❤❡♦ ❣✐↔t❤✐➳t AB k CD✱ ✤÷í♥❣ trá♥ t➙♠ B ❜→♥ ❦➼♥❤ BE ❝➢t ✤÷í♥❣ trá♥ (O) t↕✐ P

BM ⊥ M O✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû t❛♠ ❣✐→❝ ABC ♥❤å♥ ✭tr÷í♥❣ ❤ñ♣ t❛♠ ❣✐→❝ ABC tò ❣✐↔✐t÷ì♥❣ tü✮✳ ●å✐ T ✈➔ R ❧➛♥ ❧÷ñt ❧➔ t➙♠ ❝→❝ ✤÷í♥❣ trá♥ (ABC) ✈➔ (KBN)t÷ì♥❣ ù♥❣✳ ❱➻ BN K\ = \BAC ♥➯♥ △BNK ∼ △BAC✳ ❉♦ ✤â t❛ ❝â

\

CBR = ABT[ = 90◦ −ACB\

AE⊥DF ⇔ AD2 − AF2 = ED2 − EF2 ✭✷✳✶✽✮AB⊥BF ⇔ AB2 − AF2 = −BF2 ✭✷✳✶✾✮

AD⊥DE ⇔ AD2 − AE2 = −DE2 ✭✷✳✷✵✮

▲➜② ✭✷✳✶✽✮ ✕ ✭✷✳✶✾✮ ✰ ✭✷✳✷✵✮ t❤❡♦ ✈➳ ✭✈î✐ ❝❤ó þ AD = AB✮ t❛ ✤÷ñ❝

AB2 − AE2 = F B2 − F E2 ⇔ AF ⊥BE

❈❤÷ì♥❣ ✸

❈→❝ ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ s♦♥❣ s♦♥❣ tr♦♥❣ ❝→❝ ✤➲ t❤✐ ❍å❝ s✐♥❤ ❣✐ä✐

D, E, F✳ ●å✐ K ❧➔ ❤➻♥❤ ❝❤✐➳✉ ✈✉æ♥❣ ❣â❝ ❝õ❛ B tr➯♥ ✤÷í♥❣ t❤➥♥❣ DE, M ❧➔tr✉♥❣ ✤✐➸♠ ❝õ❛ ✤♦↕♥ t❤➥♥❣ DF ✳ ●å✐ L ❧➔ ❤➻♥❤ ❝❤✐➳✉ ✈✉æ♥❣ ❣â❝ ❝õ❛ C tr➯♥

OF ⊥ AM✳ ❱➻ AN ⊥ AM ⇒ OF k AN✳

❚❤❡♦ ❣✐↔ t❤✐➳t O ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ AB ♥➯♥ F O ❧➔ ✤÷í♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛t❛♠ ❣✐→❝ ABP ❉♦ ✤â F ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ BP ✳ ❚❛ ❝â △BMQ ✈✉æ♥❣ ð

M ✈➔ E ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ BQ ⇒ ME = 1

2BQ = EB ❞♦ ✤â △OME =

△OBE (c.c.c)✳ ❙✉② r❛ EM O\ = EBO\ = 90◦ ⇒ EM ⊥ M N✳ ❈❤ù♥❣ ♠✐♥❤t÷ì♥❣ tü t❤➻ F N ⊥ MN✳ ❚ø ✤â s✉② r❛ ME k NF

AOE = 60◦ ⇒\OEA = \OAE = 60◦ ⇒\OAC = \OCA = 30◦

❉♦ t❛♠ ❣✐→❝ COB ❝➙♥ t↕✐ O ♥➯♥ OCB\ = OBC\ = 75◦ ⇒ \ACB = 45◦✳

❳➨t t❛♠ ❣✐→❝ AHC ✈✉æ♥❣ t↕✐ H✱ ❝â ACH\ = 45◦ ⇒ HAC\ = 45◦✳ ▼➦t

❑❤✐ ✤â O\′N C = O\′CN ✭✈➻ △OBC ❝➙♥ t↕✐ O′✮ ✈➔ OBC\ = OCB\ ✭✈➻

△OBC ❝➙♥ t↕✐ O✮✳ ❙✉② r❛ OBC\ = O\′N C✱ ❤❛✐ ❣â❝ ♥➔② ❧↕✐ ✤ç♥❣ ✈à✱ ❞♦ ✤â

❚ø ✭✸✳✺✮ ✈➔ ✭✸✳✻✮ s✉② r❛ M OD\ = M OE\✳ ❑➳t ❤ñ♣ ✈î✐ OD = OE ❝❤ó♥❣ t❛s✉② r❛ MD = ME✳ ❱➟②✱ MO ❧➔ tr✉♥❣ trü❝ ✤♦↕♥ CE ♥➯♥ MO ⊥ DE✳

▼➦t ❦❤→❝✱ ACBE ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ♥➯♥ BCE\ = \BAE✳ ❑➳t ❤ñ♣ ✈î✐ ✭✷✳✺✮

✈➔ ❞♦ H ∈ BA s✉② r❛ BHM\ = \BAE tø ✤â ❞➝♥ ✤➳♥ HM k AE✳

❇➔✐ t♦→♥ ✸✳✾ ✭❈❤å♥ ✤ë✐ t✉②➸♥ ❑✐➯♥ ●✐❛♥❣ ♥➠♠ ❤å❝ ✷✵✶✷ ✕ ✷✵✶✸✮✳ ❈❤♦

❤➻♥❤ t❤❛♥❣ ABCD ❝â ✤→② ❧î♥ ❧➔ CD✳ ◗✉❛ A ✈➩ AK k BC (K ∈ CD) ✈➔q✉❛ B ❦➫ BI k AD (I ∈ CD❀ BI ❝➢t AC t↕✐ F, AK ❝➢t BD t↕✐ E✳ ❈❤ù♥❣

♠✐♥❤ KD = CI ✈➔ EF k AB.✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❣✐↔ t❤✐➳t AK k BC, BI k AD ✈➔ ABCD ❧➔ ❤➻♥❤ t❤❛♥❣

♥➯♥ AB k CD✱ s✉② r❛ ❝→❝ tù ❣✐→❝ ABCK, ABID ❧➔ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤✳ ❉♦ ✤â

DI = CK ✭❝ò♥❣ ❜➡♥❣ AB✮✱ s✉② r❛ DI + IK = CK + IK✳ ❱➟② DK = CI

P Q k BC.

❇➔✐ t♦→♥ ✸✳✶✸ ✭❚r✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣ ❈❤✉②➯♥ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐

♥➠♠ ❤å❝ ✷✵✶✻ ✲ ✷✵✶✻✮✳ ❈❤♦ t❛♠ ❣✐→❝ ABC✳ ✣÷í♥❣ trá♥ (I) ♥ë✐ t✐➳♣ t❛♠

❣✐→❝ ABC t✐➳♣ ①ó❝ ✈î✐ 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❤➥♥❣

❝õ❛ ✤÷í♥❣ t❤➥♥❣ BD ✈î✐ ✤÷í♥❣ trá♥ (C)✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ MN k AE✳

2s✤ P M⌢ ✮✳ ❍ì♥ ♥ú❛✱ P M Q\ + ICM\ + IBM\ = 180◦ ♥➯♥ P M Q\ + P IG[ =

180◦✳ ❉♦ ✤â tù ❣✐→❝ MP IQ ♥ë✐ t✐➳♣ ♥➯♥ t❛ ❝â M QP\ = M IK\ ✭❝ò♥❣ ❜➡♥❣1

2s✤ P M⌢ ✮✳ ▼➔ M IK\ = M IC\ ✭❝ò♥❣ ❜➡♥❣ KBM\✮ ♥➯♥ M QP\ = M CI\✳ ❙✉②r❛ P Q k BC