✶

✶


❑❍❆■ ❚❍⑩❈ ❚Ù ❉■➏◆ ❱❯➷◆●
❈❍❖ ❇⑨■ ❚❖⑩◆ ❇❻❚ ✣➃◆●
❚❍Ù❈ ❱⑨ ❈Ü❈ ❚❘➚
❑✐➲✉ ✣➻♥❤ ▼✐♥❤1
◆❣➔② ✶ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✶
❚â♠ t➢t ♥ë✐ ❞✉♥❣
❚ù ❞✐➺♥ ❧➔ ✤è✐ t÷ñ♥❣ ❝ì ❜↔♥ ❝õ❛ ❤➻♥❤ ❤å❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥✳ ❚r➯♥
t↕♣ ❝❤➼ ❚❍❚❚ sè ✸✻✼✱ t❤→♥❣ ✶ ♥➠♠ ✷✵✵✽✱ ✤➣ ❣✐î✐ t❤✐➺✉ ❜➔✐ ✈✐➳t ✧❙ü
♣❤➙♥ ❧♦↕✐ tù ❞✐➺♥ ✈➔ ù♥❣ ❞ö♥❣✧✳ ❚r♦♥❣ ❜➔✐ ❜→♦ ♥➔② ❝❤ó♥❣ tæ✐ ①✐♥ ✤÷ñ❝
tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ❦❤❛✐ t❤→❝ tù ❞✐➺♥ ✈✉æ♥❣ ❝❤♦ ❜➔✐ t♦→♥ ❜➜t
✤➥♥❣ t❤ù❝ ✈➔ ❝ü❝ trà✳ ✣➸ t✐➺♥ ❝❤♦ ✈✐➺❝ t❤❡♦ ❞ã✐ ❝õ❛ ❜↕♥ ✤å❝ ❝❤ó♥❣ tæ✐
♥❤➢❝ ❧↕✐ ♠ët sè ❦➳t q✉↔ q✉❛♥ trå♥❣✳
❈❤♦ OABC ❧➔ tù ❞✐➺♥ ✈✉æ♥❣ t↕✐ O✱ OA = a, OB = b, OC = c, S =
SABC , S1 = SOAB , S2 = SOBC , S3 = SOAC ✳ ●å✐ OH ❧➔ ✤÷í♥❣ ❝❛♦
R, r, R1 ❧➛♥ ❧÷ñt ❧➔ ❜→♥ ❦➼♥❤ ♠➦t ❝➛✉ ♥❣♦↕✐ t✐➳♣✱ ♥ë✐ t✐➳♣ tù ❞✐➺♥ ✈➔
❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABC; α, β, γ ❧➔ sè ✤♦ ❝→❝ ❣â❝
♥❤à ❞✐➺♥ ❝↕♥❤ AB, BC, CA✳ ❑❤✐ ✤â✿
✶✳ ❚❛♠ ❣✐→❝ ABC ❝â ❜❛ ❣â❝ ♥❤å♥❀
✷✳ OH1 = a1 + b1 + c1 ❀
✸✳ cos2 α + cos2 β + cos2 γ = 1❀
✹✳ S12 = SHAB .S, S22 = SHBC .S, S32 = SHCA.S ❀
✺✳ S12 + S22 + S32 = S 2 ✭✣à♥❤ ❧þ P②t❤❛❣♦r❡✮❀
√
✻✳ VOABC = 61 abc; Stp = 12 (ab + bc + ca + a2 b2 + b2 c2 + c2 a2 )✳
❚❛ ❝❤✐❛ t❤➔♥❤ ❝→❝ ♥❤â♠ ❜→✐ t♦→♥ s❛✉✿ ✭❇↕♥ ✤å❝ tü ✈➩ ❤➻♥❤ ❝❤♦ ❝→❝
❜➔✐ t♦→♥ tr♦♥❣ ❜➔✐ ✈✐➳t ♥➔②✮✳

⇒
OB
=
⇒
tan
γ
=
=
OB 2
a2 c 2
a2 + c 2
OB 2
a2 c 2
√
b a2 + c 2
ac
⇒ tanγ =
⇒ cotγ = √
.
ac
b a2 + c 2

❚÷ì♥❣ tü t❤➻

ab
bc
cotα = √
; cotβ = √
.
c a2 + b 2

1
( b + b2b+c2 )
2 a2 +b2

+

(b +c )(c +a )
2
2
1
( c + c2c+a2 )
2 b2 +c2

a2
(a2 +b2 )(c2 +a2 )
2

+ 12 ( c2a+a2 +

a2
)
a2 +b2

= 32 .

❇➔✐ t➟♣ ✷✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝

M = tan2 α + tan2 β + tan2 γ + cot2 α + cot2 β + cot2 γ.
▲í✐ ❣✐↔✐


=
.
+
+
+
+
+
)+(
+
+
)
≥
6+
b2 a2 b2 c2 a2 c2
a2 b 2 + a2 c 2 b 2 c 2 + b 2 a2 c 2 a2 + c 2 b 2
2
2

❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a = b = c✳ ❱➟② ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ M ❧➔ 152
❦❤✐ OABC ❧➔ tù ❞✐➺♥ ✈✉æ♥❣ ❝➙♥ t↕✐ O✳
❈→❝❤ ✷✳ ✣➦t cos2α = x; cos2β = y; cos2γ = z ⇒ x + y + z = 1
⇒ tan2 α

1−x
y+z
1 − cos2 α
=
=
.
2

z y
x x
y+z z+x x+y

M=

15
3
≥ 6+ = .
2
2

❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y = z ⇔ α = β = γ ⇔ OABC ❧➔ tù
❞✐➺♥ ✈✉æ♥❣ ❝➙♥ t↕✐ O.
❇➔✐ t➟♣ ✸✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

cosα
cosβ

2

cosβ
+
cosγ

▲í✐ ❣✐↔✐

2

+


x
−1
y

2

+

x
y

2

+

y
z

2

+

z
x

2

+


z
x

2

+2

x z y
+ +
z y x

≥9


❇➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ✤ó♥❣ t❤❡♦ ❆▼✲●▼ ❝❤♦ ❜❛ sè ❞÷ì♥❣✳ ⑩♣ ❞ö♥❣ (∗) ❝❤♦
x = cos2 α; y = cos2 β; z = cos2 γ t❛ ✤÷ñ❝
cosα
cosβ

2

+

cosβ
cosγ

2

+


▲í✐ ❣✐↔✐

❈→❝❤ ✶✳ ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼✲●▼✱ ✤à♥❤ ❧þ s✐♥ ✈➔ ❦➳t q✉↔ sin2A +
sin2 B + sin2 C ≤ 94 ✱ t❛ ❝â
1
1
1
9
1
=
+
+
≥
OH 2
a2 b 2 c 2
a2 + b2 + c2
a2 + b2 + c2
⇒ OH 2 ≤
9
2
(a
+
b2 ) + (b2 + c2 ) + (c2 + a2 )
2
⇒ 2OH ≤
9

▼➔ t❛ ❧↕✐ ❝â✿
(a2 + b2 ) + (b2 + c2 ) + (c2 + a2 )
AB 2 + BC 2 + CA2

+ SOBC
+ SOCA
2

1
1
AB.BC.CB
1
OA2 .OB 2 + OB 2 .OC 2 + OC 2 .OA2
=
4R1
4
4
4
2
2
2
2
2
2
(OA + OB )(OB + OC )(OC + OA )
⇒
= OA2 .OB 2 + OB 2 .OC 2 + OC 2 .OA2
4R12
⇒

✣➦t OA2 = x; OB 2 = y; OC 2 = z t❤➻
R12 =

(x + y)(y + z)(z + x)

xy + yz + zx
4(xy + yz + zx)
⇔ (x + y)(y + z)(z + x) ≥ 8xyz.

❇➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ✤ó♥❣ t❤❡♦ ❆▼✲●▼✳ ❱➟② t❛ ❝â OH ≤ √R2 ✳ ❉➜✉ ❜➡♥❣ ①↔②
r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ OA = OB = OC.
1

❇➔✐ t➟♣ ✺✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

VOABC ≤

AB.BC.CA
√
12 2

✳
▲í✐ ❣✐↔✐

❚❛ ❝â

OA =

CA2 + AB 2 − BC 2
; OB =
2

✻

AB 2 + BC 2 − CA2

(CA2 + AB 2 − BC 2 )(AB 2 + BC 2 − CA2 ) ≤ CA4

VOABC ≤

1
6

AB.BC.CA
AB 2 BC 2 CA2
√
=
8
12 2

❇➔✐ t➟♣ ✻✳ ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝

√
S1
S2
S3
3 3
√
+
+
≤
S + 2S1 S + 2S2 S + 2S3
3+2 3

▲í✐ ❣✐↔✐


+
+
≥
S + 2s1 S + 2s2 S + 2s3
3S + 2(S1 + S2 + S3
9
9
√
≥
=
2
2
2
(3 + 2 3)S
3S + 2 3(S1 + S2 + S3

✼


❙✉② r❛✿
√
√
9
6 3
3 3
√
√ ⇒P ≤
√
2P ≤ 3 − S
=

cos A =
=
=

AB 2 + AC 2 − BC 2
2AB.AC
x + y + y + z − (y + z)
2 (x + y)(x + z)
x
(x + y)(x + z)

❚÷ì♥❣ tü
y

cos B =

(y + z)(y + x)

; cos C =

z
(z + x)(z + y)

❉♦ ✤â
x
(x + y)(x + z)

+

y

3
2


▲í✐ ❣✐↔✐

❚❤❡♦ ❜➔✐ t♦→♥ ✼✱ ❣✐↔ sû xy + yz + zx = 1 t❤➻
√

x
+
1 + x2

y
1 + y2

+√

z
3
≤
2
1 + z2

✣➦t x = a1 ✱ y = 1b ✱ z = 1c ✱ t❛ ❝â ❜➔✐ t♦→♥ ✽✳
❇➔✐ t➟♣ ✾✳ ❈❤♦ ①✱②✱③ ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

1
1
9


◆➳✉ ❣✐↔ t❤✐➳t xy + yz + zx = 1 t❤➻ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ q✉❡♥ t❤✉ë❝ s❛✉✿
x2

1
1
9
1
+ 2
+ 2
≤
+1 y +1 z +1
4

❇➔✐ t➟♣ ✶✵✳ ❈❤♦ a, b, c > 0 : ab + bc + ca = (a + b)(b + c)(c + a)✳ ❈❤ù♥❣

♠✐♥❤ r➡♥❣

√
a
3 3
b
c
√
≤
+√
+√
(∗)
4
c+a


a
(a + b)(a + c)

√
3 3
≤
4
(a + b)(b + c)(c + a)

√
c ab + bc + ca
c+a

c

+

b

ab + bc + ca
+
(b + a)(b + c)

(c + a)(c + b)

ab + bc + ca
+
(c + b)(c + a)



❚÷ì♥❣ tü ❝ô♥❣ ❝â
cos B =

cos C =

b
(b + a)(b + c)
c
(c + a)(c + b)

; sin B =

ab + bc + ca
(b + a)(b + c)

; sin C =

ab + bc + ca
(c + a)(c + b)

❑❤✐ ✤â ✭✯✮ trð t❤➔♥❤

√
3 3
sin A cos C + sin C cos B + sin B cos A ≤
(∗ ∗ ∗)
4
√



❙✉② r❛

❚❤➳ t❤➻ (sin A − sin C)(cos C − cos B) ≤ 0 ✈➔ 0 < C < π2

√
3 3
1
P ≤ sin A cos B + sin B cos A + sin C cos C = sin C + sin 2C ≤
2
4

❚❛ t❤➜② r➡♥❣ ✭✯✯✯✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤ù♥❣ tä ✭✯✯✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
❱➟② ❜➜t ✤➥♥❣ t❤ù❝ ✭✯✮ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳

✹ ❇➔✐ t➟♣ ❝õ♥❣ ❝è
❚r➯♥ ✤➙② ❧➔ ♥❤ú♥❣ ❦❤❛✐ t❤→❝ ❜❛♥ ✤➛✉ ❝õ❛ ❝❤ó♥❣ tæ✐ ✈➲ tù ❞✐➺♥ ✈✉æ♥❣ ❝❤♦ ❝→❝
❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ✈➔ ❝ü❝ trà✳ ❍② ✈å♥❣ r➡♥❣ ❝→❝ ❜↕♥ s➩ ❦❤❛✐ t❤→❝ t❤➯♠
✤÷ñ❝ ♥❤✐➲✉ ❜➔✐ t♦→♥ ♠î✐ ✈➔ ♥❤✐➲✉ ✤✐➲✉ t❤ó ✈à ✈➲ tù ❞✐➺♥ ✤➦❝ ❜✐➺t ♥➔②✳ ❈✉è✐
❝ò♥❣ ①✐♥ ♠í✐ ❝→❝ ❜↕♥ ❧➔♠ ♠ët sè ❜➔✐ t♦→♥ s❛✉✿
❇➔✐ ✶✿ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ (2 + tan2 α)(2 + tan2 β)(2 + tan2 γ) ≥ 64
❇➔✐ ✷✿ ❚➻♠ ❣✐→ trà ❧î♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝✿
M = (cot α. cot β + cot β. cot γ + cot γ. cot α)2 + 6(cos α. cos β. cos γ)2

❇➔✐ ✸✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝
P =

cos α + cos β cos β + cos γ cos γ + cos α
+
+

CM
+
+
.
2
2
AO
BO
CO2

❇➔✐ ✻✳ ❳→❝ ✤à♥❤ ✈à tr➼ ✤✐➸♠ M ✤➸ ❜✐➸✉ t❤ù❝ s❛✉ ❧➔ ♥❤ä ♥❤➜t
√

3M O + M A + M B + M C.

❇➔✐ ✼✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
S12
S22
S32
3
+
+
≤ .
2
2
2
2
2
2
S + S1

+
x

(y + z)(y + x)
+
y

✶✷

(z + x)(z + y)
≥ 6.
z