重难点11解三角形的图形类问题和重要模型九大题型（原卷版）_11.4-2_2026-11-04

      
      
重难点11 解三角形的图形类问题和重要模型【九大题型】

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1
3,𝐴𝐷⊥𝐴𝐶DÃ𝐴𝐷Ì𝐵𝐶Í3𝐷D𝐴𝐷= 3D
sin𝐶= ²   ³
Au
1
3
Bu 3
3
Cu 6
3
Du2 2
3
ÎÏ 1-1²2024·ÐÑÒÓÔÕ·)/³Öh △𝐴𝐵𝐶-;*𝐴,𝐵,𝐶-¯‘2°𝑎,𝑏,𝑐DÃ𝑎=
3,𝐵𝐶‘±,
.𝐴𝐷’ 1D𝑏𝑐×Øف²    ³
Au
7
4
Bu
7
2
Cu 3
Du2 3
ÎÏ 1-2²2024·ÚÒÛÜ·Ý/³S △𝐴𝐵𝐶,D* ADBDC Þ¯-‘2° aDbDcDÂ𝑎cos𝐶= 2𝑐cos
𝐴D
𝑏𝑐
𝑎2-×Øف²    ³
Au 3
Bu
3
2
Cu 3
2
Du3
ÎÏ 1-3²2024·ßàáâ·)/³S △𝐴𝐵𝐶,D𝑎、𝑏、𝑐2° △𝐴𝐵𝐶-;*𝐴、𝐵、𝐶-¯‘D𝑀‘𝐴𝐶
±I3DÄÅ𝑀𝐶= 3𝐴𝑀DÂ𝑎2 + 𝑐2−𝑏2 +𝑎𝑐= 0D𝑐= 2D𝑎= 4D|𝐵𝑀| = ²    ³
Au 7
2
Bu 3
7
Cu
3
7
Du 39
2
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” 2²2024·ãË·/äåæ³S △𝐴𝐵𝐶,D∠𝐴𝐶𝐵-12.d¯‘𝐴𝐵ÌÍ3𝐷D △𝐶𝐴𝐷-%& △𝐶𝐵𝐷
- 2 0DÃ𝐶𝐷= 2,∠𝐴𝐶𝐵= 120°D𝐵𝐶= ²    ³
Au3
Bu4
Cu6
Du8
ÎÏ 2-1²2024·çèéê·Ý/³S△𝐴𝐵𝐶,D3 D S𝐵𝐶‘±D𝐴𝐷12∠𝐵𝐴𝐶D∠𝐵𝐴𝐶= 120°D𝐴𝐵= 2
3D𝐴𝐷= 2 3
3 D𝐴𝐶= ²    ³
      
      
Au2
Bu 3
Cu3
Du2 3
ÎÏ 2-2²2024·ë˒ì·)/³¾ △𝐴𝐵𝐶-;*𝐴,𝐵,𝐶-¯‘2°𝑎,𝑏,𝑐DÖh𝑎= 2,𝑏= 4.
(1)Âcos𝐵+ 2cos𝐴= 𝑐cos𝐶Dr𝐶-Ù]
(2)Â𝐷A‘𝐴𝐵±-I3DÃ𝐶𝐷12∠𝐴𝐶𝐵,cos∠𝐴𝐶𝐵= −
1
9Dr𝐶𝐷-’.
ÎÏ 2-3²2024·íêîï·/äåæ³Öh △𝐴𝐵𝐶;*𝐴,𝐵,𝐶-¯‘2°𝑎,𝑏,𝑐D𝑏(sin𝐵+ sin𝐶) = (𝑎−𝑐)(
sin𝐴+ sin𝐶)u
(1)r A]
(2)A -12.𝐴𝐷Ì𝐵𝐶Í𝐷3D9𝑏+ 𝑐= 64Dr𝐴𝐷-×ØÙu
 3  ()*+,-,./
” 3²2024·ðñ·/äåæ³¾ △𝐴𝐵𝐶-;*∠𝐵𝐴𝐶,∠𝐵,∠𝐶-¯‘2°𝑎,𝑏,𝑐DÖh2𝑏cos𝐵cos2𝐶= 𝑎−2𝑐
cos𝐶cos2𝐵u
(1)r∠𝐵𝐴𝐶u
(2)Â𝑏+ 𝑐= 8DÑ𝐵𝐶±-,.𝐴𝐷=
19
2 Dr △𝐴𝐵𝐶-%&u
ÎÏ 3-1²2024·ë˒ì·)/³¹=DS △𝐴𝐵𝐶,DÖh𝐴𝐵= 3,𝐴𝐶= 6,𝐴ò*D𝐵𝐶,𝐴𝐶‘±-ó
,.𝐴𝑀,𝐵𝑁…ÌÍ3𝑃, △𝐴𝐵𝐶-%&9 3
2 .
      
      
(1)r𝐵𝐶-’[]
(2)r∠𝐴𝑃𝐵-!"Ù.
ÎÏ 3-2²2024·ßààï·)/³S △𝐴𝐵𝐶,,*𝐴,𝐵,𝐶-¯‘A𝑎,𝑏,𝑐,Öh𝑏(1 + cos𝐴) = 𝑐(1−cos2𝐵).
(1)ôõ:𝑏= 𝑐;
(2)Â𝐵𝐶‘±-𝐴𝐷2,𝐴𝐶‘±-,.𝐵𝐸2 7,r △𝐴𝐵𝐶-%&.
ÎÏ 3-3²2024·ö÷øùú·Ý/³S △𝐴𝐵𝐶,D3𝑀,𝑁2°𝐵𝐶,𝐴𝐶-,3D𝐴𝑀d𝐵𝑁ÌÍ3𝐺D
𝐴𝑀= 3,∠𝑀𝐴𝐵= 45°u
(1)Â𝐴𝐶= 5 2Dr,.𝐵𝑁-’]
(2) △𝐴𝐵𝐶Aò*)*+Drû‘+𝐺𝑀𝐶𝑁%&-üÙýþu
 4  ()*+,-0*/
” 4²2024·ßàïÿ·/äåæ³Öhò* △𝐴𝐵𝐶,D*𝐴D𝐵D𝐶Þ¯-‘2°𝑎D𝑏D𝑐D¡,𝑎= 8D
𝑎
𝑐= 1 + sin2𝐴−sin2𝐶
sin2𝐵
DÃ𝑎≠𝑐u
(1)rô
𝐵= 2𝐶]
(2)Öh3𝑀S.Ā𝐴𝐶±DÃ∠𝐴𝐵𝑀= ∠𝐶𝐵𝑀Dr𝐵𝑀-üÙýþu
      
      
ÎÏ 4-1²2024·;āĂ·)/³S △𝐴𝐵𝐶,D;*𝐴,𝐵,𝐶-¯‘2°𝑎,𝑏,𝑐DÃ(𝑎−2𝑏)cos𝐶= 𝑐
( 2cos𝐵−cos𝐴).
(1)r
𝑏
𝑎-Ù]
(2)Â𝐵= 2𝐶Dôõ
 △𝐴𝐵𝐶¦*)*+.
ÎÏ 4-2²2024·ßà㥷/äåæ³Sò* △𝐴𝐵𝐶,.;*𝐴D𝐵D𝐶Þ¯-‘2°A𝑎D𝑏D𝑐DÖh
𝑎−2𝑐cos𝐵= 𝑐.
(1)rô
𝐵= 2𝐶]
(2)rsin𝐵+ 2 3cos2𝐶-üÙýþ.
ÎÏ 4-3²2024·ąĆÊć·Ý/³S △𝐴𝐵𝐶,D*𝐴D𝐵D𝐶-¯‘2°𝑎D𝑏D𝑐DÖh𝑐= 4,𝑏= 3.
(1)Âcos𝐶= −
1
4Dr𝑎-Ù? △𝐴𝐵𝐶-%&]
(2)S²1³-óĈÁDrcos 2𝐶+ π
3 -Ù]
(3)Â𝐴= 2𝐵Dr𝑎-Ù.
      
      
 5  ()*,-*12./
” 5²2024·Êć¸ĉĊ·)/³S △𝐴𝐵𝐶,D;* ADBDC -¯‘2° aDbDcD3 D ‘𝐵𝐶±I
3DÃÄÅ(𝐴𝐷+ 𝐴𝐶) ⋅𝐵𝐶= 0u
(1)ôõ
𝐴𝐷= 𝑏]
(2)Â𝐴𝐷;* A -12.DÃ𝐴𝐷=
1
3𝐴𝐵+
2
3𝐴𝐶Drsin𝐴u
ÎÏ 5-1²2024·ûċČčĎ·)/³ďS①2𝑎−𝑏= 2𝑐cos𝐵D②
3𝑎
𝑐cos𝐵= tan𝐶+ tan𝐵D
③3sin(𝐴+ 𝐵) = 3−2cos2𝐶
2)ĐóĈ,TIĐDđySÁ%-6,D𝐵,𝐶Þ¯-‘2°A𝑎,𝑏,𝑐DÖh
_____u
(1)r*𝐶]
(2)Â𝑏= 4D3𝐷S‘𝐴𝐵±D𝐶𝐷∠𝐴𝐶𝐵-12.Dr‘’𝑎-Ùu
ÎÏ 5-2²2024·ĒêēĔ·/äåæ³Öh △𝐴𝐵𝐶,;* ADBDC -¯‘2° aDbDcDÃÄÅ 3𝑐+ 𝑏
sin𝐴=
3𝑎cos𝐵u
(1)r* A -Øĕ]
(2) D A‘ BC ±I3Dà AD A* A -*12.Dr
𝐵𝐶
𝐴𝐷-×ĕÙu
ÎÏ 5-3²2024·íê·/äåæ³K①
𝑐+2𝑎
𝑏
=
cos(π−𝐶)
cos𝐵
D②
sin𝐴+sin𝐶
sin𝐵+sin𝐶=
𝑏−𝑐
𝑎D③2𝑎sin2𝐵
2 =
3𝑏sin𝐴´)Đ
      
      
óĈ,ĖIĐDđySÁ%-6,.
Öh △𝐴𝐵𝐶-;*𝐴D𝐵D𝐶-¯‘2°𝑎D𝑏D𝑐Ã______.
(1)r*𝐵-Øĕ]
(2)Â𝐴-*12.̑𝐵𝐶Í3𝐷DÃ𝐴𝐷=
6D𝑐= 2Dr‘𝑏.

¹ėTYĐóĈ2°(^DĘęIĐ(^Ě2.
 6  ()*,-/
” 6²2024·ûċ·/äåæ³S △𝐴𝐵𝐶,D;*𝐴D𝐵D𝐶-¯‘2°𝑎D𝑏D𝑐Dà 3𝑐sin𝐵+ 𝑏cos(𝐴+ 𝐵)
= 𝑏.
(1)r*𝐶-Øĕ]
(2)Â𝑎= 8D △𝐴𝐵𝐶-%&4 3Dr𝐴𝐵‘±-.
ÎÏ 6-1²2024·ě¤ĜÜ·/äåæ³ĝ △𝐴𝐵𝐶-;* ADBDC Þ¯-‘2° aDbDcDÃa2𝑏cos 𝐴−π
3
= 𝑎+ 𝑐D
(1)r* B
(2) AC ‘±-ℎ=
3
4 𝑏Drcos𝐴cos𝐶.
ÎÏ 6-2²2024·ÊćĞğĠ·)/³S △𝐴𝐵𝐶,D;*𝐴D𝐵D𝐶Þ¯-‘2°𝑎D𝑏D𝑐D𝐶=
π
3Ã
𝑎+ 𝑏= 7D △𝐴𝐵𝐶-789ġĢ4 3
3 .
      
      
(1)r △𝐴𝐵𝐶-%&]
(2)r △𝐴𝐵𝐶‘𝐴𝐵±-ℎ.
ÎÏ 6-3²2024·ðñ·/äåæ³Öh △𝐴𝐵𝐶-;*𝐴D𝐵D𝐶Þ¯-‘2°𝑎D𝑏D𝑐D𝑎= 1Dsin𝐵+
3
𝑏cos𝐴= 0u
(1)r*𝐴]
(2)ĝ𝐴𝑀A △𝐴𝐵𝐶-Dr𝐴𝑀-×ØÙu
 7  ()*+,-$23/
” 7²23-24 ݱ·ģË·Ĥĥ³S △𝐴𝐵𝐶,D3𝐷.Ā𝐵𝐶-û$23ÃĦL3𝐵,∠𝐵𝐴𝐷d∠𝐵𝐴𝐶ħđ.
(1)r
𝐴𝐶
𝐴𝐷-Ù]
(2)Â∠𝐵𝐴𝐷= 30∘,𝐴𝐵= 4Dr𝐴𝐷-’.
ÎÏ 7-1²2023·ëć·/äåæ³S △𝐴𝐵𝐶,D;* AD𝐵D𝐶-¯‘2°𝑎D𝑏D𝑐DÖh𝑎2(1 + cos𝐴)
= 2𝑏𝑐sin2𝐴u
(1)Ĩĩ △𝐴𝐵𝐶-+Ī]
(2)Öh𝐷𝐵𝐶±I3Dī𝐴=
2π
3 D𝑎= 3 3D𝐴𝐷=
3bD𝐷𝐵𝐶-M$23Ĭ
      
      
ÎÏ 7-2²2024·ëËĭâ·/äåæ³S △𝐴𝐵𝐶,D*𝐴，𝐵，𝐶-¯‘2°𝑎，𝑏，𝑐，Öh𝑎= 𝑏cos𝐶−
3
3 𝑐sin𝐵
(1)r*𝐵
(2)Į𝐵Ž𝐵𝐷⊥𝐵𝐴DÌ.Ā𝐴𝐶Í DDÃ𝐴𝐷= 2𝐷𝐶Dr*𝐶.
ÎÏ 7-3²23-24 )±·ë˒ì·Ĥ,³ĝ aDbDc 2° △𝐴𝐵𝐶-;* ADBDC -¯‘DAD  BC
‘±-,.Dcį1D∠𝐵𝐴𝐶=
2𝜋
3 D2𝑐sin𝐴cos𝐵= 𝑎sin𝐴−𝑏sin𝐵+
1
2𝑏sin𝐶u
(1)r AD -’[]
(2) E  AB ±ĦL B -û$23DG  △𝐴𝐵𝐶-45DÈ8 EG ݃d AC ÌÍ3 FDr AF -’[u
 8  )*+-456
” 8²2024·ÒııÜ·Ý/³¾ △𝐴𝐵𝐶-;*𝐴,𝐵,𝐶-¯‘2°𝑎,𝑏,𝑐DÖh
𝑎+𝑏
𝑐=
sin𝐶−sin𝐵
sin𝐴−sin𝐵u
(1)r*𝐴]
(2)Â𝑎= 6D3𝑀 △𝐴𝐵𝐶-45DÃ𝐴𝑀= 2 3Dr △𝐴𝐵𝐶-%&u
ÎÏ 8-1²2023·ûċ;Ò·I/³ △𝐴𝐵𝐶-;*𝐴:𝐵:𝐶Þ¯-‘2°𝑎:𝑏:𝑐D𝑎= 6D𝑏sin
𝐵+𝐶
2 = 𝑎sin
𝐵u
      
      
(1)r*𝐴-Øĕ]
(2)𝑀 △𝐴𝐵𝐶-45D𝐴𝑀-ƒ.Ì𝐵𝐶Í3𝐷DÃ𝐴𝑀= 2 3Dr △𝐴𝐵𝐶-%&u
ÎÏ 8-2²2023·ÒàIJijĴ·I/³¹=DÖh△ABD -45 CD△ABC );* A:B:C -¯‘2°
 aDbDc.Ãcos2𝐴
2 =
𝑏+𝑐
2𝑐
(1)r∠ACB -Øĕ]
(2)Â∠𝐶𝐴𝐵=
π
6Drsin∠𝐶𝐷𝐴-Øĕ.
ÎÏ 8-3²2023·Ēêĵí·/äåæ³S △𝐴𝐵𝐶,D*𝐴,𝐵,𝐶-¯‘𝑎,𝑏,𝑐,𝑐⋅sin𝐴= 𝑎⋅cos𝐶Dĝ △𝐴𝐵𝐶
-%&𝑆,𝑆=
2
4 𝑏𝑐.
(1)r*𝐵-Øĕ]
(2)Â𝑎= 3DĮ △𝐴𝐵𝐶-453𝐺-¦.𝑙d‘𝑎,𝑐-Ì32°𝐸,𝐹,𝐵𝐶= 𝜆𝐵𝐸D𝐵𝐴= 𝜇𝐵𝐹Dď̜𝜆+ 𝜇-
Ù.
      
      
 9  )*+-789:;<96
” 9²2024·ģËĶķ·Ý/³S △𝐴𝐵𝐶,D*𝐴,𝐵,𝐶-¯‘2°𝑎,𝑏,𝑐DÃ𝑎cos𝐶+
3𝑐sin𝐴= 𝑏+ 𝑐.
(1)r*𝐵-üÙýþ]
(2)Öh △𝐴𝐵𝐶;<9-ġĢ$Í 3
2 Dr △𝐴𝐵𝐶ĸ’-üÙýþ.
ÎÏ 9-1²2023·ÊË·/äåæ³Öh △𝐴𝐵𝐶-75𝑂D3𝑀,𝑁2°S.Ā𝐴𝐵,𝐴𝐶±DÃ𝑂Ł𝑀𝑁-,
3u
(1)Â𝐵𝐶=
3,𝑂𝐴= 1Dr △𝐴𝐵𝐶%&-×ØÙ]
(2)ôõ
𝐴𝑀⋅𝑀𝐵= 𝐴𝑁⋅𝑁𝐶u
ÎÏ 9-2²2024·ÚÒ·/äåæ³¹=DS1%;-ûĐĺ3𝐴D𝐵D𝐶D𝐷ŠĻ-û‘+𝐴𝐵𝐶𝐷,D
𝐴𝐵= 1D𝐵𝐶= 2D𝐶𝐷= 3D𝐴𝐷= 4.
(1)r △𝐴𝐶𝐷%&-üÙýþ]
(2)Âû‘+𝐴𝐵𝐶𝐷ļS789Dr789%&.
      
      
ÎÏ 9-3²2024·ðñ·/äåæ³Öh △𝐴𝐵𝐶,D*𝐴D𝐵D𝐶-¯‘2°A𝑎D𝑏D𝑐D 3𝑏−𝑐sin𝐴=
3𝑎
cos𝐶u
(1)r*𝐴-Øĕ]
(2)Â𝑎= 7D △𝐴𝐵𝐶789-ġĢ𝑅D;<9ġĢ𝑟Dr
𝑅
𝑟-×ĕÙu
I:„
1u²2024·ĽÜľĿŀ·)/³S △𝐴𝐵𝐶,D𝐴𝐵= 2D𝐴𝐶= 3D ∠𝐴=
π
3D △𝐴𝐵𝐶789-ġĢ²ŁŁ³
Au 7
3
Bu 21
3
Cu2 7
3
Du2 21
3
2u²2024·öłŃ·)/³S △𝐴𝐵𝐶,D𝐴𝐵= 2D𝐵𝐶=
7D∠𝐵𝐴𝐶= 120°D𝐷A𝐵𝐶‘I3D𝐴𝐷A∠𝐵𝐴𝐶-
*12.D𝐴𝐷= ²    ³
Au
2
3
Bu1
Cu2
Du 3
3u²2024·ßà·/äåæ³S △𝐴𝐵𝐶,D* ADBDC Þ¯-‘2° aDbDcD𝑐(sin𝐴−sin𝐶) = (𝑎−𝑏)
(sin𝐴+ sin𝐵)D △𝐴𝐵𝐶-%& 3
4 Dĸ’3𝑏D AC ‘±-²    ³
Au 3
3
Bu 3
2
Cu 3
Du2 3
4u²2024·ě¤ěÜ·/äåæ³S △𝐴𝐵𝐶,D*𝐴,𝐵,𝐶Þ¯ń-‘2°𝑎,𝑏,𝑐D3𝑀‘𝐵𝐶-,3DÂ
𝐴𝑀= 𝐴𝐶Dcos2𝐵= cos(𝐴+ 𝐶)Dsin∠𝐵𝐴𝐶= ²    ³
Au 3
3
Bu 6
3
Cu 21
7
Du2 7
7
5u²2024·íà·)/³S △𝐴𝐵𝐶,D;*𝐴,𝐵,𝐶Þ¯-‘2°𝑎,𝑏,𝑐.Öh𝐴=
2π
3 ,𝑏2 + 𝑐2 = 24, △𝐴𝐵𝐶-78
9ġĢ𝑅= 2 3,𝐷A‘𝐴𝐶-,3D𝐵𝐷’²    ³
Au 2 +1
Bu2 3
Cu6 2
Du 21
6u²2024·íêîï·)/³S △𝐴𝐵𝐶,D;*𝐴,𝐵,𝐶Þ¯-‘2°𝑎D𝑏D𝑐DÃ
𝑏sin𝐵
sin𝐴−𝑎=
(𝑐−𝑎)sin𝐶
sin𝐴
Dƒ𝐵𝐶
Ç3𝐷D 𝐵𝐶= 𝐶𝐷DÂ𝐴𝐷= 2 3,𝐴𝐵= 2D𝑎= ²    ³
      
      
Au1
Bu 3
Cu2
Du3
7u²2024·ĒêĒÜ·/äåæ³S △𝐴𝐵𝐶,D*𝐴:𝐵:𝐶-¯‘2°𝑎:𝑏:𝑐DÂ𝑐= 3D𝑏= 2D∠𝐵𝐴𝐶-
12.𝐴𝐷-’4 6
5 D𝐵𝐶‘±-,.𝐴𝐻-’$Ͳ    ³
Au 17
2
Bu4 2
3
Cu 17
4
Du4 3
3
8u²2024·ðñ·/äåæ³ÖhS △𝐴𝐵𝐶,D*𝐴,𝐵,𝐶-¯‘2°𝑎,𝑏,𝑐,2sin𝐴= 𝑎cos𝐶,𝑐= 2uÂ𝐺 △𝐴𝐵𝐶
-45D𝐺𝐴2 +𝐺𝐵2−𝐺𝐶2-×ĕف²    ³
Au12−4 2
9
Bu8+4 2
9
Cu4 2−2
3
Du4+2 2
3
Ý:Y
9u²2024·Ēà·Ý/³Öh △𝐴𝐵𝐶;*𝐴,𝐵,𝐶-¯‘2°𝑎,𝑏,𝑐,𝑂 △𝐴𝐵𝐶-45Dcos𝐴=
1
5,𝐴𝑂= 2D
²    ³
Au𝐴𝑂=
1
4𝐴𝐵+
1
4𝐴𝐶
Bu𝐴𝐵⋅𝐴𝐶≤3
Cu △𝐴𝐵𝐶-%&-×Øف3 6
Du𝑎-×ĕف2 5
10u²2024·ě¤ĜÜ·/äåæ³ △𝐴𝐵𝐶,D;* ADBDC Þ¯-‘2° aDbDcuÖh𝑎= 2D △𝐴𝐵𝐶
-%&𝑆=
3
2 𝐴𝐵⋅𝐴𝐶DŸÁŅ'R-A²    ³
Au𝐴= 30°
Bu △𝐴𝐵𝐶-ĸ’-×Øف 6
CuÂ𝑏𝑐= 4D △𝐴𝐵𝐶R)*+
DuÂ𝐴𝐵‘±-,.’$Í2 3
3 D𝑆=
3
11u²2024·ģËĶķ·/äåæ³S △𝐴𝐵𝐶,D𝐴𝐵= 4D𝐴𝐶= 6D𝐴=
π
3D𝐷‘𝐵𝐶±Iĺ3D²   ³
Au𝐵𝐶= 2 7
Buī𝐴𝐷*𝐴-*12.bD𝐴𝐷= 12 3
5
Cuī𝐷‘𝐵𝐶,3bD𝐴𝐷= 3 2
DuÂ3𝑃 △𝐴𝐵𝐶;ĖI3D𝑃𝐴⋅𝑃𝐵+ 𝑃𝐶-×ĕف−
19
4
):UV
12u²2024·ßààï·/äåæ³Öh △𝐴𝐵𝐶-;*𝐴,𝐵,𝐶Þ¯-‘2°𝑎,𝑏,𝑐,𝐶= 135°Dà △𝐴𝐵𝐶-78
9ġĢ𝑅= 1D △𝐴𝐵𝐶%&-×Øف      
u
      
      
13u²2024·ûċņŇ·)/³¹=DD  △𝐴𝐵𝐶-‘ AC ±I3D|𝐴𝐷| = 2|𝐷𝐶|D∠𝐴𝐵𝐶= 60°D
|𝐴𝐵| + 2|𝐵𝐶| = 4D|𝐵𝐷|-×ĕف      
.
14u²2024·ûċňâ·/äåæ³Sʼn* △𝐴𝐵𝐶,D𝑎D𝑏D𝑐2°A △𝐴𝐵𝐶-;*𝐴D𝐵D𝐶Þ¯-‘D3𝐺
A △𝐴𝐵𝐶-45DÂ𝐴𝐺⊥𝐵𝐺Dcos𝐶-üÙýþA      
.
û:(^
15u²2024·ÚÒŊÜ·/äåæ³Öh △𝐴𝐵𝐶-ĸ’ 20D*𝐴D𝐵D𝐶Þ¯-‘2°𝑎D𝑏D𝑐
(1)Â𝐶=
π
4D𝑐= 7Dr △𝐴𝐵𝐶-%&]
(2) △𝐴𝐵𝐶-;<9ġĢ 3D𝑎= 7Drtan𝐴-Ùu
16u²2024·;āĂŋ?Ōō·Ý/³S △𝐴𝐵𝐶,D¾*𝐴:𝐵:𝐶-¯‘2°𝑎:𝑏:𝑐DÖh 3𝑎=
3𝑐cos𝐵+ 𝑐
sin𝐵u
(1)r*𝐶]
(2)Öh3𝐷S𝐴𝐶‘±DÃ𝐴𝐷= 2𝐷𝐶D𝐵𝐶= 6D𝐵𝐷= 2 7Dr △𝐴𝐵𝐶-%&u
17u²2024·íêŎĠ·)/³ĝ)*+𝐴𝐵𝐶-;*𝐴:𝐵:𝐶-¯‘2°𝑎:𝑏:𝑐Ãsin(𝐵+ 𝐶) = 2 3sin2𝐴
2u
(1)r*𝐴-Øĕ]
(2)Â𝑏= 3D𝐵𝐶‘±-3 21
7 Dr)*+𝐴𝐵𝐶-ĸ’u
      
      
18u²2024·àŏ·/äåæ³Öh △𝐴𝐵𝐶-;* ADBDC -¯‘2° aDbDcDÃ2𝑏sin 𝐴+ π
6 −2𝑎= 𝑐u
(1)r B]
(2)Â∠𝐴𝐵𝐶-12.Ì𝐴𝐶Í3𝐷DÃ𝐵𝐷= 2D𝑎= 3Dr △𝐴𝐵𝐶-%&u
19u²2024·ûċňâ·/äåæ³)*+);*𝐴D𝐵D𝐶-¯‘2°𝑎D𝑏D𝑐DÖh 3𝑏
𝑎=
1−cos𝐵
sin𝐴.
(1)r*𝐵-Øĕ]
(2) △𝐴𝐵𝐶-%&$Í 3D𝐷𝐵𝐶‘-,3Dī,.𝐴𝐷-’×ŐbDr𝐴𝐶‘-’.