Một số kết quả mới của tứ giác nội tiếp

✐ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖ ♦✵♦ ✖✖✖✖✖ ❇Ò■ ❱❿◆ ❍Ò◆● ▼❐❚ ❙➮ ❑➌❚ ◗❯❷ ▼❰■ ❈Õ❆ ❚Ù ●■⑩❈ ◆❐■ ❚■➌P ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ t♦→♥ ❝➜♣ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ P●❙✳❚❙✳ ❚r➛♥ ❱✐➺t ❈÷í♥❣ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✷✶ ✐✐ ▲❮■ ❈❆▼ ✣❖❆◆ ❈→❝ ❦➳t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ ❝ỉ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ tỉ✐✱ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ữợ sỹ ữợ P r t ữớ ❚ỉ✐ ①✐♥ ❝❤à✉ tr→❝❤ ♥❤✐➺♠ ✈➲ ♥❤ú♥❣ ❧í✐ ❝❛♠ ✤♦❛♥ ❝õ❛ ♠➻♥❤✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✳✳✳ t❤→♥❣ ✳✳✳ ♥➠♠ ✷✵✷✶ ❚→❝ ❣✐↔ ❇ò✐ ❱➠♥ ❍ò♥❣ ✐✐✐ ▲❮■ ❈❷▼ ❒◆ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ữợ sỹ ữợ t➟♥ t➻♥❤ ❝õ❛ P●❙✳❚❙ ❚r➛♥ ❱✐➺t ❈÷í♥❣✱ t❤➛② ✤➣ trü❝ t ữợ t t t tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ✈ø❛ q✉❛✳ ❳✐♥ ❝❤➙♥ t ỡ Pỏ t qỵ t ổ ợ rữớ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ✤➸ t→❝ ❣✐↔ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s tợ ữớ t ỗ ♥❣❤✐➺♣ ❧✉æ♥ ❦❤✉②➳♥ ❦❤➼❝❤ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ ✈➔ ✈✐➳t ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✷✶ ❚→❝ ❣✐↔ ❇ò✐ ❱➠♥ ❍ò♥❣ ✐v ❉❛♥❤ s→❝❤ ❤➻♥❤ ✈➩ ✶✳✶ ▼ët t❛♠ ❣✐→❝ ✈ỵ✐ ❝→❝ t tr ỵ sin ✳ ✳ ✶✳✷ ▼ët t❛♠ ❣✐→❝ ✈ỵ✐ ❝→❝ ❣â❝ α✱ ữủt ố ợ a✱ b✱ c✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❚ù ❣✐→❝ ABCD ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❚ù ❣✐→❝ ABCD ❝â AB, CD ❝➢t ♥❤❛✉ t↕✐ P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ✣✐➸♠ Q ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦ QDA = BDC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ✣✐➸♠ S ữủ s t ỗ ✳ ✳ ✳ ✳ ✳ ✶✳✼ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ BAC = BDC ✳ ✳ ✳ ✳ ✶✳✽ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ AP.P B = DP.P C ✳ ✳ ✷✳✶ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ CAD + DCA = CBA ✷✳✷ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ DAB − ABC = BAC − ABD ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝→❝ ✤÷í♥❣ ♣❤➙♥ ❣✐→❝ t↕✐ ❊ ✈➔ P s♦♥❣ s♦♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A C ⑤⑤ BD ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ EI = EI ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻ P❤➛♥ ♠ð rë♥❣ ❝õ❛ M Q ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ CD ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ SP.P T = U P.P V ✳ ✳ ✷✳✽ ❍➻♥❤ ❝❤✐➳✉ tr➯♥ ❝→❝ ✤÷í♥❣ ❝❤➨♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ CEFπ+ CF E = A ✳ ✳ ✷✳✶✵ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ψ = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✶ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ W XY Z ❧➔ ❤➻♥❤ t❤♦✐ ✳ ✷✳✶✷ ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ ABJF ✈➔ CBJF ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✸ ABCD ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ K ≡ L ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✹ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ EM F + EN F = π ✳ ′ ′ ′ a ′ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ✷✵ ✷✶ ✷✸ ✷✹ ✷✺ ✷✻ ✈ ✷✳✶✺ ●â❝ ❣✐ú❛ ❤❛✐ ❝↕♥❤ ✤è✐ ❞✐➺♥ ✈➔ ✤÷í♥❣ ❝❤➨♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✻ ❚ù ❣✐→❝ ABCD ❦❤æ♥❣ ♥ë✐ t✐➳♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✼ ✣✐➸♠ C ❞✐ ❝❤✉②➸♥ tr➯♥ ✤÷í♥❣ ❝❤➨♦ AC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✽ ❈→❝ ♣❤➛♥ ✤÷í♥❣ ❝❤➨♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✾ EF GH ❧➔ t✐➳♣ ①ó❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ABCD ♥ë✐ t✐➳♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✵ P❤➙♥ ❣✐→❝ tr♦♥❣ ❣â❝ A ✈➔ tr✉♥❣ trü❝ BC ❝➢t ♥❤❛✉ t↕✐ D ✳ ✳ ✳ ✷✳✷✶ P ✱ Q✱ O✱ K ❝ò♥❣ t❤✉ë❝ ♠ët ✤÷í♥❣ trá♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✶ ✸✸ ✸✹ ✹✵ ✹✶ ✹✷ ▼ư❝ ❧ư❝ vi ▲í✐ ❝❛♠ ✤♦❛♥ ❉❛♥❤ ♠ư❝ ❤➻♥❤ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈✐ ✶ ✸ ❈❤÷ì♥❣ ✷✳ ▼ët sè ❦➳t q✉↔ ♠ỵ✐ ✈➲ tù ❣✐→❝ ♥ë✐ t✐➳♣ ✶✶ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✸ ✹✹ ✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳ ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳ ✷✳✷✳ ✷✳✸✳ ✷✳✹✳ ✷✳✺✳ ✷✳✻✳ ▼ët sè ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ❣â❝ ❤♦➦❝ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣ ▼ët sè ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ỗ ✳ ✳ ✳ ✳ ▼ët sè ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ♠ð rë♥❣ ❝õ❛ ❝↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ët sè ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ sè ❧÷đ♥❣ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ët sè ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤÷í♥❣ ❝❤➨♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ët sè ❦➳t q✉↔ ❦❤→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ i✈ ✸ ✸ ✹ ✹ ✹ ✹ ✶✶ ✶✺ ✷✵ ✷✼ ✸✸ ✹✵ ▼ð ✤➛✉ ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ ❧➔ ♠ët ♥ë✐ ❞✉♥❣ q✉❛♥ trå♥❣ ❝õ❛ ❚♦→♥ ❤å❝✱ t❤÷í♥❣ ✤÷đ❝ ù♥❣ ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ❤➻♥❤ ❤å❝ ♣❤➥♥❣✳ ▼ët tù ❣✐→❝ ✤÷đ❝ ❣å✐ ❧➔ ♥ë✐ t✐➳♣ ♠ët ✤÷í♥❣ trá♥ ✭t❤÷í♥❣ ❣å✐ ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣✮ ♥➳✉ ❜è♥ ✤➾♥❤ ❝õ❛ ♥â ✤➲✉ ♥➡♠ tr➯♥ ✤÷í♥❣ trá♥✳ ✣÷í♥❣ trá♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ tù ❣✐→❝✳ ❚➙♠ ❝õ❛ ✤÷í♥❣ trá♥ ❝→❝❤ ✤➲✉ ❝→❝ ✤➾♥❤✱ ✤÷đ❝ ❣å✐ ❧➔ t➙♠ ♥❣♦↕✐ t✐➳♣✳ ❚r♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ t♦→♥ ❤å❝✱ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✤➸ ♠ët tù ❣✐→❝ trð t❤➔♥❤ tù ❣✐→❝ ♥ë✐ t✐➳♣ t❤÷í♥❣ ✤÷đ❝ ✤÷❛ r ữợ ự tr ❜➔✐ t➟♣✳ ❚ù❝ ❧➔✱ tr♦♥❣ ♠ët ❜➔✐ t➟♣ ❤♦➦❝ ♠ët ✤à♥❤ ❧➼✱ ✈ỵ✐ ♠ët tù ❣✐→❝ ♥ë✐ t✐➳♣✱ t❛ s➩ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t✱ ❝→❝ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ✤➾♥❤✱ ❝↕♥❤✱ ❦❤♦↔♥❣ ❝→❝❤ ❝õ❛ tù ❣✐→❝ ✤â✳ ❚❤æ♥❣ t❤÷í♥❣✱ ✤à♥❤ ❧➼ ✤↔♦ ❝õ❛ ♥❤ú♥❣ ✤à♥❤ ❧➼ ✤â ❝ơ♥❣ ú ự ởt tự ỗ t t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤÷đ❝ ♥➯✉ tr♦♥❣ ✤à♥❤ ❧➼✱ tr♦♥❣ ❜➔✐ t♦→♥ t❤➻ tù ❣✐→❝ ➜② s➩ trð t❤➔♥❤ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ ❝â ♥❤✐➲✉ t➼♥❤ ❝❤➜t ♥ê✐ t✐➳♥❣✱ ❝→❝ t➼♥❤ ❝❤➜t ♥➔② ❧➔ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✤➸ ♠ët tù ❣✐→❝ trð t❤➔♥❤ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ❈â t❤➸ t❤➜② r➡♥❣✱ ❝→❝ ✤➦❝ ✤✐➸♠ ❤❛② ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✤â ❧↕✐ ❝❤➼♥❤ ❧➔ ✤✐➲✉ ❦✐➺♥ tự õ tỗ t ợ ố ♥❣❤✐➯♥ ❝ù✉ ❦➽ ❤ì♥ ✈➲ ♥❤ú♥❣ ✤➦❝ ✤✐➸♠✱ t➼♥❤ ❝❤➜t ❝õ❛ tù ❣✐→❝ ♥ë✐ t✐➳♣✱ ❝ơ♥❣ ♥❤÷ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ởt tự ỗ tr t tù ❣✐→❝ ♥ë✐ t✐➳♣✱ tæ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐ ✳ ◆❣♦➔✐ ✈✐➺❝ ♥❤➢❝ ❧↕✐ ♠ët sè ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t ✈➲ tù ❣✐→❝ ♥ë✐ t✐➳♣✱ ❧✉➟♥ ✈➠♥ ✤➣ tê♥❣ ❤ñ♣ ữủ ởt số t q ợ tự t✐➳♣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tæ✐ tr➻♥❤ ♠ët sè ❦➳t q✉↔ ✈➲ ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➲ t➔✐✱ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ❈→❝ ♥ë✐ tr ữủ tờ ủ tứ ỗ t ✏▼ët sè ❦➳t q✉↔ ♠ỵ✐ ❝õ❛ tù ❣✐→❝ ♥ë✐ t✐➳♣ ữỡ ởt số t q ợ tự ❣✐→❝ ♥ë✐ t✐➳♣✳ ❈❤÷ì♥❣ ♥➔② ❧➔ ♥ë✐ ❞✉♥❣ trå♥❣ t➙♠ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tỉ✐ tê♥❣ ❤đ♣ ♠ët sè ❦➳t q✉↔ ♠ỵ✐ ✈➲ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❧✐➯♥ q✉❛♥ ✤➳♥ ❣â❝ ❤♦➦❝ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s♦♥❣✱ ❧✐➯♥ q✉❛♥ ỗ q rở ❧✐➯♥ q✉❛♥ ✤➳♥ ✤÷í♥❣ ❝❤➨♦✱ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ sè ❧÷đ♥❣ ❣✐→❝✱✳✳✳ ❈→❝ ♥ë✐ ❞✉♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✹❪✱ ❬✺❪✳ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➲ t➔✐ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ tù ❣✐→❝ ♥ë✐ t✐➳♣✱ ❝→❝ ♥ë✐ ❞✉♥❣ ♥➔② ✤÷đ❝ tê♥❣ ❤đ♣ tø ❝→❝ t➔✐ ❧✐➺✉✳ ✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❳➨t t❛♠ ❣✐→❝ ABC ✈ỵ✐ ✤ë ❞➔✐ ❝→❝ ❝↕♥❤ a = BC ✱ b = AC ✱ c = AB ✈➔ ❝→❝ ❣â❝ ❝õ❛ t❛♠ ❣✐→❝ ✤÷đ❝ ❦➼ ❤✐➯✉ A B C ỵ s r ữủ ỵ sin ổ tự sin ởt ♣❤÷ì♥❣ tr➻♥❤ ❜✐➸✉ ❞✐➵♥ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❝❤✐➲✉ ❞➔✐ ❝→❝ ❝↕♥❤ ❝õ❛ ♠ët t❛♠ ❣✐→❝ ❜➜t ❦➻ ✈ỵ✐ sin õ tữỡ ự ỵ sin ữủ t ữợ b c a = = sin A sin B sin C ụ õ t t ữợ ♥❣❤à❝❤ ✤↔♦ sin C sin A sin B = = a b c ❍➻♥❤ ✶✳✶✿ ▼ët t❛♠ ❣✐→❝ ✈ỵ✐ ❝→❝ t tr ỵ sin ỵ s r ữủ ỵ cosin sỹ ❧✐➯♥ q✉❛♥ ❣✐ú❛ ❝❤✐➲✉ ❞➔✐ ❝õ❛ ❝→❝ ❝↕♥❤ ❝õ❛ ♠ët t ợ cosin õ tữỡ ự c2 = a2 + b2 − 2ab cos γ ❈æ♥❣ t❤ù❝ tr ụ õ t ữủ t ữợ a2 + b2 c2 cos C = 2ab ỵ cosin ❞ò♥❣ ✤➸ t➼♥❤ ❝↕♥❤ t❤ù ❜❛ ❦❤✐ ❜✐➳t ❤❛✐ ❝↕♥❤ ❝á♥ ❧↕✐ ✈➔ ❣â❝ ❣✐ú❛ ❤❛✐ ❝↕♥❤ ✤â✱ ❤♦➦❝ t➼♥❤ ❝→❝ ❣â❝ ❦❤✐ ❜✐➳t ❝❤✐➲✉ ❞➔✐ ❜❛ ❝↕♥❤ ❝õ❛ ♠ët t ỵ cosin ữủ tữỡ tỹ ❝❤♦ ❤❛✐ ❝↕♥❤ ❝á♥ ❧↕✐✿ a2 = b2 + c2 − 2bc cos α b2 = a2 + c2 − 2ac cos β ❍➻♥❤ ✶✳✷✿ ▼ët t❛♠ ❣✐→❝ ✈ỵ✐ ❝→❝ õ ữủt ố ợ ❝→❝ ❝↕♥❤ a✱ b✱ c✳ ✶✳✷✳ ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❬✶❪ ▼ët tù ❣✐→❝ ❝â ❜è♥ ✤➾♥❤ ♥➡♠ tr➯♥ ♠ët ✤÷í♥❣ trá♥ ✤÷đ❝ ❣å✐ ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥✳ ✶✳✷✳✷ ❚➼♥❤ ❝❤➜t ❚➼♥❤ ❝❤➜t ✶✳✶✳ ❬✶❪ ▼ët tù ❣✐→❝ ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✸✵ ✈➔ ❤❛✐ ✈➳ ❦❤→❝ ✶✳ ❱➟② ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ▼➺♥❤ r tự ỗ õ t✐➳♣ a✱ b✱ c ✈➔ d✱ ❝æ♥❣ t❤ù❝ ❝❤✐❛ ✤æ✐ ❝õ❛ cosin ✤÷đ❝ ❝❤♦ ❜ð✐ A B cos C cos D cos cos ❑❤✐ ✈➔ ❝❤➾ tù ❣✐→❝ ❈❤ù♥❣ ♠✐♥❤✳ ABCD = = = = (s − b)(s − c) , ad + bc (s − c)(s − d) , ab + cd (s − d)(s − a) , ad + bc (s − a)(s − b ab + cd ♥ë✐ t✐➳♣✱ tr♦♥❣ ✤â s ❧➔ ❜→♥ ❦➼♥❤✳ ✿ ✣➛✉ t✐➯♥ t❛ ❝❤ù♥❣ ♠✐♥❤ ❝æ♥❣ t❤ù❝ t❤ù ❜❛ tr♦♥❣ tù ❣✐→❝ ♥ë✐ ✣✐➲✉ ❦✐➺♥ ❝➛♥ t✐➳♣✳ ỵ cosin tr t BCD ABD t❛ ✤÷đ❝ b2 + c2 − 2bc cos C = a2 + d2 − 2ad cos A ✣✐➲✉ ✤â ❞➝♥ ✤➳♥ b2 + c2 − a2 − d2 = 2(ad + bc) cos C ✈➻ cos A = cos(π − C) = − cos C tr♦♥❣ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❧÷đ♥❣ ❣✐→❝ ❝❤✐❛ ✤ỉ✐✱ t❛ ♥❤➟♥ ✤÷đ❝ cos2 C b2 + c2 − a2 − d2 1+ 2(ad + bc) (b + c) − (a − d)2 = 4(ad + bc) (b + c + a − d)(b + c − a + d) = 4(ad + bc) (s − d)(s − a) = ad + bc = ✸✶ ❙✉② r❛✱ t❛ ❝â cos ✳ C = (s − d)(s − a) ad + bc ❍➻♥❤ ✷✳✶✻✿ ❚ù ❣✐→❝ ABCD ❦❤æ♥❣ ♥ë✐ t✐➳♣ ✿ ◆➳✉ ABCD ❦❤æ♥❣ ♥ë✐ t✐➳♣✱ ❣✐↔ sû ✤➾♥❤ A ♥➡♠ ❜➯♥ ♥❣♦➔✐ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ BCD✳ ●å✐ A ❧➔ ✤✐➸♠ ♠➔ ✤÷í♥❣ ❝❤➨♦ AC ❝➢t ✤÷í♥❣ trá♥✳ ❑❤✐ ✤â B AD ≡ A > A ≡ BAD ✭❤➻♥❤ ✷✳✶✻✮✱ ❦➨♦ t❤❡♦ cos A = − cos C ✈➻ cosin ✤❛♥❣ ❣✐↔♠ tr➯♥ ✤♦↕♥ [0, π]✳ ❉♦ ✤â cos A = cos C ✈➻ A BCD ♥ë✐ t✐➳♣✱ ♥➯♥ cos A > − cos C õ ỵ cosin ′ ′ ′ ′ ′ ′ b2 + c2 − 2bc cos C = a2 + d2 − 2ad cos A < a2 + d2 + 2ad cos C; ❉♦ ✈➟② b2 + c2 − a2 − d2 cos C > 2(ad + bc) ▲➔♠ t÷ì♥❣ tü tr♦♥❣ ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ trü❝ t✐➳♣✱ t❛ ♥❤➟♥ ✤÷đ❝ cos C > (s − a)(s − d) ad + bc ✣è✐ ✈ỵ✐ A✱ ♥â ❝❤♦ r➡♥❣ cos C > − cos A✱ ♥➯♥ a2 + d2 − 2ad cos A < b2 + c2 + 2bc cos A ❉♦ ✤â ✈➔ t❛ ✤÷đ❝ ✸✷ a2 + d2 − b2 − c2 < 2(ad + bc) cos A cos (s − b)(s − c) ad + bc A > ✣è✐ ✈ỵ✐ ❤❛✐ ❣â❝ ❝á♥ ❧↕✐✱ ❝â B > B ≡ CBA ✈➔ D > D ≡ CDA ✭❤➻♥❤ ✷✳✶✻✮✳ ❑❤✐ ✤â ′ ❚ø ✤â ❱➻ t❤➳ ′ ′ B + D > B ′ + D′ = π ❚❤❡♦ ✤â ❚❛ ✤÷đ❝ ′ cos D < cos(π − B) = − cos B a2 + b2 − 2ab cos B > c2 + d2 + 2cd cos B a2 + b2 − c2 − d2 > 2(ad + cd) cos B ⇒ cos B < a2 + b2 − c2 − d2 2(ab + cd B < (s − c)(s − d) ab + cd cos ❚÷ì♥❣ tü✱ t❛ ❝â cos D < (s − a)(s − b) ab + cd ❑❤✐ ✤➾♥❤ A ♥➡♠ ❜➯♥ tr♦♥❣ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ BCD✱ t❤➻ t➜t ❝↔ ❜➜t ✤➥♥❣ t❤ù❝ ✤↔♦ ♥❣÷đ❝✱ ❤♦➔♥ t❤➔♥❤ ❝❤ù♥❣ ♠✐♥❤✳ ✸✸ ✷✳✺✳ ▼ët sè ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤÷í♥❣ ❝❤➨♦ ự ỗ ABCD t ✈➔ ❝❤➾ ❦❤✐ AB sin CAD + AD sin CAB = AC sin BAD ❈❤ù♥❣ ♠✐♥❤✳ ✿ ❚r♦♥❣ tù ❣✐→❝ t ABCD ợ ữớ trỏ t ❦✐➺♥ ❝➛♥ ❦➼♥❤ R✱ ❝â CD = 2R sin CAD ữ ỵ BC = 2R sin CAB BD = 2R sin BAD Pt♦❧❡♠② ✱ t❛ ❝â AB.CD + BC.AD = AC.BD ⇔ AB.2R sin CAD + 2R sin CAB.AD = AC.2R sin BAD ⇔ AB sin CAD + AD sin CAB = AC sin BAD ✿ ◆❣÷đ❝ ❧↕✐✱ ①➨t ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABD✳ ❑❤✐ t❤❛② ✤ê✐ ✤✐➸♠ C ❞å❝ tr➯♥ ✤÷í♥❣ ❝❤➨♦ AC ✱ ✈➳ tr tự tr ỵ ổ t ✈➳ ♣❤↔✐ ❝õ❛ ✤➥♥❣ t❤ù❝ t➠♥❣ ❦❤✐ C ❞✐ ❝❤✉②➸♥ ❜➯♥ ♥❣♦➔✐ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ ✈➔ ❣✐↔♠ ❦❤✐ C ❞✐ ❝❤✉②➸♥ tr♦♥❣ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ ▼➦t ❦❤→❝✱ t❛ ❝â ❣â❝ ABD ❦❤æ♥❣ ✤ê✐ ✭❤➻♥❤ ✷✳✶✼✮✳ ❱➻ ✈➟② ✤➸ ✤➥♥❣ t❤ù❝ ❝è ✤à♥❤✱ ✤✐➸♠ C ♣❤↔✐ ♥➡♠ tr➯♥ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABD✱ ♥➯♥ tù ❣✐→❝ ABCD ♥ë✐ t✐➳♣✳ ✣✐➲✉ ❦✐➺♥ ✤õ ❍➻♥❤ ✷✳✶✼✿ ✣✐➸♠ C ❞✐ ❝❤✉②➸♥ tr➯♥ ✤÷í♥❣ ❝❤➨♦ AC ✸✹ ▼➺♥❤ ✤➲ ✷✳✷✵✳ ❚r♦♥❣ tự ỗ õ t a b✱ c ✈➔ d✳ ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❣â❝ ♥❤å♥ θ ❣✐ú❛ ❤❛✐ ✤÷í♥❣ ❝❤➨♦ t❤ä❛ ♠➣♥ |a2 − b2 + c2 − d2 | cos θ = 2(ac + bd) ●å✐ e✱ f ✱ g✱ h ữớ ỵ cosin tr♦♥❣ ❜è♥ t❛♠ ❣✐→❝ t↕♦ ❜ð✐ ❝→❝ ✤÷í♥❣ ❝❤➨♦ ✭❤➻♥❤ ✷✳✶✽✮✱ t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ a2 = e2 + f − 2ef cos(π − θ), b2 = f + g − 2f g cos θ, c2 = g + h2 − 2gh cos(π − θ), d2 = h2 + e2 − 2he cos θ, ❚r♦♥❣ ✤â p ✈➔ q ❧➔ ✤ë ❞➔✐ ✤÷í♥❣ ❝❤➨♦✳ ❑❤✐ ✤â✱ ❝❤÷❛ ①→❝ ✤à♥❤ ✤÷đ❝ ❣â❝ ❣✐ú❛ ❤❛✐ ✤÷í♥❣ ❝❤➨♦ ❧➔ õ tr tự ỗ õ t❤ä❛ ♠➣♥ |a2 − b2 + c2 − d2 | = 2pq cos θ ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ pq = ac + bd ❚❤❛② ✈➔♦ ❜✐➸✉ t❤ù❝ ✈➔ ❣✐↔✐ cosin t❛ ♥❤➟♥ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝✳ |a2 − b2 + c2 − d2 | cos θ = 2(ac + bd) ❍➻♥❤ ✷✳✶✽✿ ❈→❝ ♣❤➛♥ ✤÷í♥❣ ❝❤➨♦ ✭✷✳✾✮ r tự ỗ ABCD õ ❝→❝ ❝↕♥❤ ❧✐➯♥ t✐➳♣ ❧➔ a✱ b✱ c ✈➔ d✳ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✤ë ❞➔✐ ❝õ❛ ✤÷í♥❣ ❝❤➨♦ AC ✈➔ BD ❧➛♥ ❧÷đt ❧➔ ❈❤ù♥❣ ♠✐♥❤✳ p= (ac + bd)(ad + bc) ab + cd q= (ab + cd)(ac + bd) ad + bc ỵ cosin tr t ABD ✈➔ BCD t❛ ❝â q = a2 + d2 − 2ad cos A, q = b2 + c2 − 2bc cos C ◆❤➙♥ ❜✐➸✉ t❤ù❝ ✤➛✉ t✐➯♥ ✈ỵ✐ bc ✈➔ ❜✐➸✉ t❤ù❝ t❤ù ❤❛✐ ✈ỵ✐ ad✱ t❛ ❝â bcq = a2 bc + bcd2 − 2abcd cos A, adq = adb2 + adc2 − 2abcd cos C ❈ë♥❣ ❤❛✐ ✈➳✱ t❛ ❝â (bc + ad)q = ac(ab + cd) + bd(cd + ab) − 2abcd(cos A + cos C) ❑❤✐ ✤â q2 = (ab + cd)(ac + bd) − 2abcd(cos A + cos C) (ad + bc) ✭✷✳✶✵✮ ❚❤❡♦ ♠➺♥❤ ✤➲ ✷✳✶✻✱ tù ❣✐→❝ ABCD ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ cos A + cos C = ✭✷✳✶✶✮ ❚÷ì♥❣ tü✱ t❛ s✉② r❛ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ✤ë ❞➔✐ ✤÷í♥❣ ❝❤➨♦ AC tr♦♥❣ tù ỗ ABCD õ t õ (ac + bd)(ad + bc) − 2abcd(cos B + cos D) p = ✭✷✳✶✷✮ (ab + cd) ❚❤❡♦ ♠➺♥❤ ✤➲ ✷✳✶✻✱ tù ❣✐→❝ ABCD ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ⇔ q2 = (ac + bd)(ad + bc) ab + cd cos B + cos D = ✸✻ p2 = ⇔ (ac + bd)(ad + bc) ab + cd ✭✷✳✶✸✮ ▼➺♥❤ r tự ỗ õ t✐➳♣ a✱ b✱ c ✈➔ d✱ t❤÷ì♥❣ ❝õ❛ ❤❛✐ ✤÷í♥❣ ❝❤➨♦ p ✈➔ q t❤ä❛ ♠➣♥ p ad + bc = q ab + cd ❑❤✐ ✈➔ ❝❤➾ ❦❤✐ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ❈❤ù♥❣ ♠✐♥❤✳ ✿ ◆➳✉ tù ❣✐→❝ ABCD ♥ë✐ t✐➳♣ t❤➻ ❝→❝ t❛♠ ❣✐→❝ ABD✱ BCA✱ CDB ✈➔ DAC ❝â ❝ị♥❣ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣✱ ❜→♥ ❦➼♥❤ R✳ ❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ abc ❝❤♦ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝ ❝â ❝→❝ ❝↕♥❤ a✱ b✱ c ✈➔ ❜→♥ ❦➼♥❤ R✱ t❛ ✤÷đ❝ 4R ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❉♦ ✤â✱ t❛ ❝â adq bcq abp cdp + = + 4R 4R 4R 4R q(ad + bc) = p(ab + cd) ❈❤✐❛ ❤❛✐ ✈➳ ✭✷✳✶✸✮ ❝❤♦ ✭✷✳✶✶✮ t❛ ✤÷đ❝✿ p2 (ac + bd)(ad + bc)(ad + bc) = q (ab + cd)(ab + cd)(ac + bd) (ad + bc)2 p2 = ⇔ q2 (ab + cd)2 p ad + bc ⇒ = q ab + cd ✿ ◆➳✉ tù ❣✐→❝ ❦❤æ♥❣ ♥ë✐ t✐➳♣✱ ❣✐↔ sû A + C > π✳ ❑❤✐ ✤â B + D < π ✈➔ ❜➡♥❣ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ✷✳✶✻ t❛ ❝â ✣✐➲✉ ❦✐➺♥ ✤õ cos A + cos C < ✈➔ ❚ø ✭✷✳✶✷✮ ✈➔ ✭✷✳✶✵✮ t❛ ✤÷đ❝ cos B + cos D > p2 < (ac + bd)(ad + bc) ab + cd ✸✼ ✈➔ q2 > ❈❤✐❛ ❤❛✐ ✈➳✱ t❛ ❝â (ab + cd)(ac + bd) ad + bc p2 (ac + bd)(ad + bc) ad + bc (ad + bc)2 < = q2 ab + cd (ab + cd)(ac + bd) (ab + cd)2 ❉♦ ✤â✱ t❛ ❝â p ad + bc < q ab + cd ✭✷✳✶✹✮ p ad + bc > q ab + cd ✭✷✳✶✺✮ ◆➳✉ A + C < π ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✤ê✐ ❝❤✐➲✉✱ ✈➻ ✈➟② t❛ ♥❤➟♥ ✤÷đ❝ ❉♦ ✤â ABCD ♥ë✐ t✐➳♣✳ ▼➺♥❤ ự ỗ tự t ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✤ë ❞➔✐ ❝õ❛ ❤❛✐ ✤÷í♥❣ ❝❤➨♦ p ✈➔ ♠➣♥ q ✈➔ ❝→❝ ❝↕♥❤ ❧➛♥ ❧÷đt ❧➔ a b c d tự ỗ t❤ä❛ |p − q| |a − c| |b − d| = p+q a+c b+d ❱➻ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ỵ trỹ t ự ữủ ố t s➩ ❝❤➾ ✤÷❛ r❛ ❝❤ù♥❣ ♠✐♥❤ ♥❣÷đ❝✳ ◆➳✉ tù ❣✐→❝ ❦❤æ♥❣ ♥ë✐ t✐➳♣✱ t❤➻ ❈❤ù♥❣ ♠✐♥❤✳ p ad + bc < q ab + cd ❤♦➦❝ p ad + bc > q ab + cd A+C > π A+C < π P❤ư t❤✉ë❝ ✈➔♦ ♥➳✉ ❤♦➦❝ ❧➛♥ ❧÷đt ❝❤ù♥❣ ♠✐♥❤ t❤❡♦ ♠➺♥❤ ✤➲ ✭✷✳✷✷✮✳ ❇➜t ✤➥♥❣ t❤ù❝ ✤➛✉ t✐➯♥ t÷ì♥❣ ✤÷ì♥❣ ❉♦ ✤â✱ t❛ ❝â q ab + cd > p ad + bc q ab + cd 1− |p − q| |ad + bc − (ab + cd)| |(a − c)(d − b)| p ad + bc = < = = q ab + cd p+q ad + bc + ab + cd (a + c)(d + b) 1+ 1+ p ad + bc 1− ✸✽ ✣✐➲✉ ♥➔② ❝❤ù♥❣ ♠✐♥❤✱ ♥➳✉ A + C > π✱ t❤➻ |a − c|.|d − b| |p − q| < p+q (a + c)(d + b) ❚÷ì♥❣ tü✱ t❛ ❝❤ù♥❣ ♠✐♥❤ tr÷í♥❣ ❤ñ♣ t❤ù ❤❛✐ A + C < π ❝â ❦➳t q✉↔ ❧➔ |p − q| |a − c|.|d − b| > p+q (a + c)(d + b) ❱➟② tù ❣✐→❝ ABCD ♥ë✐ t✐➳♣✳ ▼➺♥❤ ✤➲ ✷✳✷✹✳ ◆➳✉ ✤÷í♥❣ ❝❤➨♦ ❝õ❛ tự ỗ ABCD t t P t ❣✐→❝ ABP ✱ BCP ✱ CDP ✈➔ S4 ✱ ABCD ✈➔ q ✈➔ DAP ❝â ❞✐➺♥ t➼❝❤ ❧➛♥ ❧÷đt ❧➔ ✈➔ S1 ✱ S2 ✱ S3 ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ t❤÷ì♥❣ ❝õ❛ ❤❛✐ ✤÷í♥❣ ❝❤➨♦ t❤ä❛ ♠➣♥ p √ √ p S1 S4 + S2 S3 √ =√ q S1 S2 + S3 S4 ●✐↔ sû θ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❣â❝ ❣✐ú❛ ❝→❝ ✤÷í♥❣ ❝❤➨♦✳ ❇è♥ ❤➻♥❤ t❛♠ ❣✐→❝ ❝♦♥ ❝â ❞✐➺♥ t➼❝❤ ✭❤➻♥❤ ✷✳✶✽✮ ❈❤ù♥❣ ♠✐♥❤✳ ✈➻ sin(π − θ) = sin θ✳ ❑❤✐ ✤â ❉♦ ✤â✱ t❛ ❝â S1 = ef sin θ, S2 = S3 = gh sin θ, S4 = S1 S4 = e f h sin θ, √ S1 S2 = f eg sin θ, f g sin θ he sin θ S2 S3 = g f h sin θ √ S3 S4 = h eg sin θ √ √ √ S1 S4 + S2 S3 f h(e + g) √ √ =√ = eg(f + h) S1 S2 + S3 S4 fh p eg q ỵ t ❝â f h = eg ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ▼➺♥❤ ✤➲ ✷✳✷✺✳ ❚ù ❣✐→❝ ỗ ABCD tự t ❦❤✐ t❤÷ì♥❣ ❝õ❛ ❤❛✐ ✤÷í♥❣ ❝❤➨♦ p ✈➔ q t❤ä❛ ♠➣♥ B D p cos cos = A C q cos cos 2 ✸✾ ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ t õ r tự ỗ ABCD sû ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ tø ♠➺♥❤ ✤➲ ✷✳✶✽ D B cos 2 = A C cos cos 2 cos (s − c)(s − d) ab + cd (s − b)(s − c) ad + bc (s − a)(s − b) ad + bc p ab + cd = = ab + cd q (s − a)(s − d) ad + bc (s − c)(s − d) ab + cd (s − b)(s − c) ad + bc (s − a)(s − b) ad + bc p ab + cd = < ab + cd q (s − a)(s − d) ad + bc tr♦♥❣ ✤â t❛ sû ❞ư♥❣ ♠➺♥❤ ✤➲ ✷✳✷✷ tr♦♥❣ ✤➥♥❣ t❤ù❝ ❝✉è✐ ❝ị♥❣✳ ✿ ◆➳✉ tù ❣✐→❝ ❦❤æ♥❣ ♥ë✐ t✐➳♣✱ ❣✐↔ sû ✤➾♥❤ A ♥➡♠ ♥❣♦➔✐ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ BCD✳ ❑❤✐ ✤â✱ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ rót r❛ tø ♠➺♥❤ ✤➲ ✷✳✶✽✱ t❛ ❝â ✣✐➲✉ ❦✐➺♥ ✤õ B D cos 2 = A C cos cos 2 cos ❇➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ❝ò♥❣ ❧➔ ❞♦ ✭✷✳✶✺✮✱ ✈➻ A + C < π✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤➾♥❤ A ♥➡♠ ❜➯♥ tr♦♥❣ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ BCD✱ t➜t ❝↔ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✤ê✐ ❝❤✐➲✉✳ ❉♦ ✤â D B cos 2 > ad + bc > p A C ab + cd q cos cos 2 cos tr♦♥❣ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ❝ò♥❣ ❧➔ ❞♦ ✭✷✳✶✹✮ ❙✉② r❛✱ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ✹✵ ✷✳✻✳ ▼ët sè ❦➳t q✉↔ ❦❤→❝ ▼➺♥❤ ✤➲ ✷✳✷✻✳ ❚r♦♥❣ tù ỗ ABCD ữớ t t P ✱ ❣å✐ E✱ F ✱ G ✈➔ H ❧➔ ❝❤➙♥ ❝→❝ ✤÷í♥❣ ✈✉ỉ♥❣ ❣â❝ ❤↕ tø ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ABCD P✳ ❑❤✐ ✤â EF GH ❧➔ tù ❧➔ ♠ët tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ●å✐ E✱ F ✱ G ✈➔ H ❧➛♥ ❧÷đt ♥➡♠ tr➯♥ ❝→❝ ❝↕♥❤ AB✱ BC ✱ CD ✈➔ DA✳ ❈→❝ tù ❣✐→❝ AEP H ✈➔ BF HE ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ✈➻ ❝❤ó♥❣ ❝â ♠ët ❝➦♣ ❣â❝ ✈✉ỉ♥❣ ✤è✐ ❞✐➺♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❍➻♥❤ ✷✳✶✾✿ EF GH ❧➔ t✐➳♣ ①ó❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ABCD ♥ë✐ t✐➳♣ ✿ ❑❤✐ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣ ✭❤➻♥❤ ✷✳✶✾✮✱ t❛ ❝â ✣✐➲✉ ❦✐➺♥ ❝➛♥ HEP = HAP = DAC = CBD = F BP = F EP ♥➯♥ EP ❧➔ t✐❛ ♣❤➙♥ ❣✐→❝ ❝õ❛ ❣â❝ HEF ✳ ❚÷ì♥❣ tü F P ✱ GP ✈➔ HP ❧➛♥ ❧÷đt ❧➔ ❝→❝ t✐❛ ♣❤➙♥ ❣✐→❝ ❝õ❛ ❝→❝ ❣â❝ EF G✱ F GH ✈➔ GHE ✳ ✣✐➲✉ ♥➔② ❧➔♠ ❝❤♦ EF GH ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✈➻ ♠ët tù ❣✐→❝ ♥ë✐ t✐➳♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝→❝ ✤÷í♥❣ õ ỗ q ✿ ◆➳✉ EF GH ❧➔ tù ❣✐→❝ ♥❣♦↕✐ t✐➳♣ ✭❤➻♥❤ ✷✳✶✾✮ ✣✐➲✉ ❦✐➺♥ ✤õ ✹✶ ❑❤✐ ✤â✱ t❛ ❝â DAC = HAP = HEP = F EP = F BP = CBD ❱➟② ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ❇➔✐ t♦→♥ ✷✳✶✳ ❬✶❪ ABC ✈ỵ✐ AB < AC ✳ P❤➙♥ ❣✐→❝ tr♦♥❣ ❣â❝ A ♥❤❛✉ t↕✐ D ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ABDC ❧➔ tù ❣✐→❝ ❈❤♦ t❛♠ ❣✐→❝ ✈➔ tr✉♥❣ trü❝ ✤♦↕♥ BC ❝➢t ♥ë✐ t✐➳♣✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â D A ố ợ ữớ t BC ●å✐ E✱ F ❧➛♥ ❧÷đt ❧➔ ❤➻♥❤ ❝❤✐➳✉ ✈✉ỉ♥❣ ❣â❝ ❝õ❛ D tr➯♥ AB✱ AC ❳➨t ∆ADE ✈➔ ∆ADF ❝â AED = AF D = 900 AD ❝❤✉♥❣ ✭❚➼♥❤ ❝❤➜t t✐❛ ♣❤➙♥ ❣✐→❝ ❝õ❛ ♠ët ❣â❝✮ ✭❝❤ ✲ ❝❣✈✮ DE = DF ∆ADE = ∆ADF ◆➯♥ ❙✉② r❛ AE = AF ❍➻♥❤ ✷✳✷✵✿ P❤➙♥ ❣✐→❝ tr♦♥❣ ❣â❝ A ✈➔ tr✉♥❣ trü❝ BC ❝➢t ♥❤❛✉ t↕✐ D ✹✷ ❉♦ ✤â ∆DBE = ∆DCF ⇒ BE = CF ◆➳✉ E✱ F ❝ò♥❣ ♥➡♠ tr♦♥❣ ❤♦➦❝ ❝ò♥❣ ♥➡♠ ♥❣♦➔✐ ✤♦↕♥ AB✱ AC t AB = AC ổ ỵ õ B ❣✐ú❛ A ✈➔ E❀ F ♥➡♠ tr♦♥❣ ✤♦↕♥ AC ✭❉♦ AB < AC ✮✳ ❑❤✐ ✤â ACD = EBD ❱➟② tù ❣✐→❝ ABDC ♥ë✐ t✐➳♣✳ ❇➔✐ t♦→♥ ✷✳✷✳ ❬✶❪ ✭❚r➼❝❤ ■▼❖ s❤♦rt❧✐st ✷✵✶✹ ✲ ❇➔✐ ❤➻♥❤ ✤➲ ♥❣❤à ●✸✮✳ ❈❤♦ t❛♠ ❣✐→❝ ABC tr♦♥❣ ❣â❝ B BM t↕✐ P❀ ❝➢t ợ (O) BM t ữớ trỏ t O✳ P❤➙♥ ❣✐→❝ M ✳ P❤➙♥ ❣✐→❝ ❣â❝ AOB ❝➢t ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ ❣â❝ BOC ❝➢t ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ BM t↕✐ Q✳ ●å✐ K t↕✐ ♣❤➙♥ ❣✐→❝ ❧➔ tr✉♥❣ ✤✐➸♠ AB > BC ✱ ❈❤ù♥❣ ♠✐♥❤ ✹ ✤✐➸♠ P ✱ Q✱ O ✱ K ❝ị♥❣ t❤✉ë❝ ♠ët ✤÷í♥❣ trá♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ E ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ OP ✈➔ AB❀ D ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ OQ ✈➔ BC ❈→❝ tù ❣✐→❝ OEBK ✈➔ OKDB ♥ë✐ t✐➳♣✱ ♥➯♥ xOE = ABK ✈➔ KOD = KBD ▼➔ ABK = KBD ♥➯♥ xOE = KOD ❙✉② r❛✱ OK ❧➔ ♣❤➙♥ ❣✐→❝ ♥❣♦➔✐ ❝õ❛ P OQ ▼➦t ❦❤→❝✱ K ❧➔ t➙♠ ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ BM ♥➯♥ KP = KQ ❚❤❡♦ ❜➔✐ t♦→♥ ✷✳✶✱ ❜è♥ ✤✐➸♠ O✱ P ✱ Q✱ K ❝ò♥❣ t❤✉ë❝ ♠ët ✤÷í♥❣ trá♥✳ ❍➻♥❤ ✷✳✷✶✿ P ✱ Q✱ O✱ K ❝ị♥❣ t❤✉ë❝ ♠ët ✤÷í♥❣ trá♥ ✹✸ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✧▼ët sè ❦➳t q✉↔ ♠ỵ✐ ❝õ❛ tù ❣✐→❝ ♥ë✐ t✐➳♣✧ ✤➣ tr➻♥❤ ❜➔② ♥❤ú♥❣ ✈➜♥ ✤➲ s❛✉✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➲ t➔✐✳ ✷✳ ❚r➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t ✈➲ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ✸✳ ❚ê♥❣ ❤ñ♣ ♠ët sè ❦➳t q ợ tự t ỗ ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ❣â❝ ❤♦➦❝ ✤÷í♥❣ t❤➥♥❣ s♦♥❣ s t q q ỗ t q ❧✐➯♥ q✉❛♥ ✤➳♥ ♠ð rë♥❣ ❝õ❛ ❝↕♥❤✱ ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤÷í♥❣ ❝❤➨♦✱ ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ❧÷đ♥❣ ❣✐→❝✱ ✳✳✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❚➠♥❣ ❱ơ ✭✷✵✶✾✮✱ ❈❤✉②➯♥ ✤➲ ❤➻♥❤ ❤å❝ ❧ỵ♣ ✾✱ ◆❳❇ ●✐→♦ ❞ư❝✳ ❬✷❪ ◆❣✉②➵♥ ❚➠♥❣ ❱ơ✱ ▲➯ P❤ó❝ ▲ú✱ ◆❣✉②➵♥ ❈ỉ♥❣ ❚❤➔♥❤ ✭✷✵✷✵✮✱ ▲í✐ ❣✐↔✐ ✈➔ ❜➻♥❤ ❧✉➟♥ ❱▼❖ ✷✵✷✵✳ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❈✳ ❆❧s✐♥❛ ❛♥❞ ❘✳ ❇✳ ◆❡❧s❡♥ ✭✷✵✵✼✮✱ ❧❛t❡r❛❧✱ ❖♥ t❤❡ ❞✐❛❣♦♥❛❧s ♦❢ ❛ ❝②❝❧✐❝ q✉❛❞r✐✲ ❋♦r✉♠ ●❡♦♠✳ ❱♦❧✳ ✼ ✭✷✵✵✼✮✱ ♣♣✳ ✶✹✼✲✶✹✾✳ ❬✹❪ ▼❛rt✐♥ ❏♦s❡❢ss♦♥ ✭✷✵✶✾✮✱ ❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧s✱ ■♥✲ t❡r♥❛t✐♦♥❛❧ ❥♦✉r♥❛❧ ♦❢ ❣❡♦♠❡tr②✱ ❱♦❧✳ ✽ ✭✷✵✶✾✮✱ ◆♦✳ ✶✱ ♣♣✳ ✺✲✷✶✳ ❬✺❪ ▼❛rt✐♥ ❏♦s❡❢ss♦♥ ✭✷✵✶✾✮✱ ❛❧s✱ ▼♦r❡ ❝❤❛r❛❝t❡r✐③❛t✐♥♦♥s ♦❢ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r✲ ■♥t❡r♥❛t✐♦♥❛❧ ❥♦✉r♥❛❧ ♦❢ ❣❡♦♠❡tr②✱ ❱♦❧✳ ✽ ✭✷✵✶✾✮✱ ◆♦✳ ✷✱ ♣♣✳ ✶✹✲✸✷✳ ❬✻❪ ▼❛♥❧✐♦ ❛♥❞ ▲❛❣r❛♥❣✐❛ ✭✉s❡r♥❛♠❡s✮✱ q✉❛❞ ✵✷✱ ❆rt ♦❢ Pr♦❜❧❡♠ ❙♦❧✈✐♥❣✱ ✷✵✵✹✱ ❤tt♣✿✴✴✇✇✇✳❛rt♦❢♣r♦❜❧❡♠s♦❧✈✐♥❣✳❝♦♠✴❝♦♠♠✉♥✐t②✴❝✻❤✶✹✵✺✽ ... q✉❛♥ ✤➳♥ ✤➲ t➔✐✱ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ tr ữủ tờ ủ tứ ỗ t➔✐ ❧✐➺✉✳ ✏▼ët sè ❦➳t q✉↔ ♠ỵ✐ ❝õ❛ tù ❣✐→❝ t ữỡ ởt số t q ợ ✈➲ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ❈❤÷ì♥❣ ♥➔② ❧➔ ♥ë✐ ❞✉♥❣ trå♥❣ t➙♠ ❝õ❛... ✳ ◆❣♦➔✐ ✈✐➺❝ ♥❤➢❝ ❧↕✐ ♠ët sè ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t ✈➲ tù ❣✐→❝ ♥ë✐ t✐➳♣✱ ❧✉➟♥ ✈➠♥ ✤➣ tờ ủ ữủ ởt số t q ợ tự ❣✐→❝ ♥ë✐ t✐➳♣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝... B = DP.P C t❤➻ t❛♠ ❣✐→❝ CAP t BDP ỗ ✤â✱ CAB = CDB ◆➯♥ ABCD ❧➔ tù ❣✐→❝ ♥ë✐ t t ỵ ữỡ ởt số t q ợ tự t r ữỡ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ♠ỵ✐ ✈➲ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ❈→❝ ❦➳t q✉↔ ♥➔②