重难点14奔驰定理与四心问题五大题型（原卷版）_11.4-2_2026-11-04

      
      
重难点14 奔驰定理与四心问题【五大题型】

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D]𝐴𝐵+ 𝐴𝐶= 4𝑂𝑀+2𝐻𝑀
10]ž23-24 c¥·ĥĦħĨ·ª/¢“ ”µ=:;€Ćq0noćĈ2*+,-./c4¯
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* △𝐴𝐵𝐶%cL2△𝐵𝑀𝐶，△𝐴𝑀𝐶，△𝐴𝑀𝐵0,BŠ®a𝑆𝐴，𝑆𝐵，𝑆𝐶2d𝑆𝐴⋅𝑀𝐴+ 𝑆𝐵⋅𝑀𝐵+ 𝑆𝐶⋅𝑀𝐶=
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A]»𝑆𝐴:𝑆𝐵:𝑆𝐶= 1:1:12g𝑀a △𝐴𝑀𝐶0!"
B]»𝑀a △𝐴𝐵𝐶0%"2g𝐵𝐶⋅𝑀𝐴+𝐴𝐶⋅𝑀𝐵+𝐴𝐵⋅𝑀𝐶= 0
C]»𝑀a △𝐴𝐵𝐶0&"2g 𝑀𝐴+ 𝑀𝐵⋅𝐴𝐵= 𝑀𝐵+ 𝑀𝐶⋅𝐵𝐶= 𝑀𝐴+ 𝑀𝐶⋅𝐴𝐶= 0
D]»𝑀a △𝐴𝐵𝐶0$"23𝑀𝐴+4𝑀𝐵+5𝑀𝐶= 02gcos∠𝐴𝑀𝐵=
6
6
11]ž23-24 c¥·ÞÖēĔ·ª/¢L𝑂N △𝐴𝐵𝐶rN0+,%2ž    ¢
A]»ØL𝑃ef𝑂𝑃= 𝑂𝐴+𝜆
𝐴𝐵
|𝐴𝐵|sin𝐵+
𝐴𝐶
|𝐴𝐶|sin𝐶(𝜆> 0)2gØL𝑃0ÙÚc óÜ △𝐴𝐵𝐶0$"
B]»ØL𝑃ef𝑂𝑃= 𝑂𝐴+𝜆
𝐴𝐵
|𝐴𝐵|cos𝐵+
𝐴𝐶
|𝐴𝐶|cos𝐶(𝜆> 0)2gØL𝑃0ÙÚc óÜ △𝐴𝐵𝐶0!"
C]»2𝑂𝐴+ 𝑂𝐵+3𝑂𝐶= 02𝑆△𝐴𝑂𝐶2𝑆△𝐴𝐵𝐶Š®€ △𝐴𝑂𝐶2 △𝐴𝐵𝐶0,B2g𝑆△𝐴𝑂𝐶:𝑆△𝐴𝐵𝐶= 1:6
D]`[ △𝐴𝐵𝐶?4%@𝐴2𝐵2𝐶05‚Š®*𝑎2𝑏2𝑐2»𝑎⋅𝑂𝐴+𝑏⋅𝑂𝐵+𝑐⋅𝑂𝐶= 02gL𝑂a △𝐴𝐵𝐶
0%"ž%“””"¢
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12]ž23-24 c·Ÿ ·İıIIJ¢`[ O *+,‘c4 L2A2B2C *+,‘?4×˃0L2ØL P
efúû𝑂𝑃= 𝑂𝐴+𝜆(
𝐴𝐵
|𝐴𝐵| +
𝐴𝐶
|𝐴𝐶|)(𝜆∈(0, + ∞))2gL P 0ÙÚc ÛÜ △𝐴𝐵𝐶0      
".
13]ž2024·)Njà·câ`[𝐺a𝛥𝐴𝐵𝐶0!"2ÜL𝐺0•ƒ(‚𝐴𝐵,𝐴𝐶Š®F„6L𝑃,𝑄，»𝐴𝑃=
3
5
𝐴𝐵,g𝛥𝐴𝐵𝐶(𝛥𝐴𝑃𝑄0,Bhia      
.
14]ž2024 c¥·)ÇijĴ·ĵĶ¢“ ”*+,-./c4¯°±²0³´2µa34 5k0_
A(“”(𝑀ercedes-Benz)0𝑙𝑜𝑔𝑜pFq2·Al¸x=a“ ¢.“ ”0%M^¥
^_2`
[𝑂* △𝐴𝐵𝐶%cL2△𝐵𝑂𝐶, △𝐴𝑂𝐶, △𝐴𝑂𝐵0,BŠ®a𝑆𝐴,𝑆𝐵,𝑆𝐶2g𝑆𝐴⋅𝑂𝐴+ 𝑆𝐵⋅𝑂𝐵+ 𝑆𝐶⋅𝑂𝐶= 0.»𝑂
* △𝐴𝐵𝐶¼@%0cL2 𝐴,𝐵,𝐶* △𝐴𝐵𝐶0?4%@2d𝑂Lef𝑂𝐴⋅𝑂𝐵= 𝑂𝐵⋅𝑂𝐶= 𝑂𝐶⋅𝑂𝐴2g¥ÐÑ
ÒĐĭ0*      
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①𝑂* △𝐴𝐵𝐶0&"J②∠𝐵𝑂𝐶+ 𝐴= πJ
③|𝑂𝐴|:|𝑂𝐵|:|𝑂𝐶| = cos𝐴:cos𝐵:cos𝐶J④tan𝐴⋅𝑂𝐴+ tan𝐵⋅𝑂𝐵+ tan𝐶⋅𝑂𝐶= 0
)'8ĸ
15]ž2024 ?·Ÿ ·¡P¢Ĺĺ“ ”289s¥#
(1)L O a △𝐴𝐵𝐶%cL2»𝑆△𝐴𝑂𝐵:𝑆△𝐵𝑂𝐶:𝑆△𝐴𝑂𝐶= 4:3:22𝐴𝑂= 𝜆𝐴𝐵+𝜇𝐴𝐶2Yđ
𝜆C𝜇0ÎJ
(2)» O a △𝐴𝐵𝐶0&"2Ļļ
sin2𝐴𝑂𝐴+ sin2𝐵𝑂𝐵+ sin2𝐶𝑂𝐶= 0.
16]ž23-24 c¥·Þ§ď£·ª/¢N △𝐴𝐵𝐶/2@𝐴,𝐵,𝐶r50‚Š®a𝑎,𝑏,𝑐,𝐻* △𝐴𝐵𝐶%0cL2d
𝐴𝐻=
1
4𝐴𝐵+
1
3𝐴𝐶]
(1)»𝐻* △𝐴𝐵𝐶0$"2Ļļ
7𝑐2−7𝑏2 = 𝑎2J
(2)»𝐻* △𝐴𝐵𝐶0&"2Y∠𝐵𝐴𝐶]
17]ž23-24 ñ‘·‘ÏĽľ·ª/¢N𝛥𝐴𝐵𝐶/2𝐴𝐶= 22𝐵𝐶= 62∠𝐴𝐶𝐵= 60°2L O a𝛥𝐴𝐵𝐶rN+,‘
cL2ef𝑂𝐶= 𝑚𝑂𝐴+𝑛𝑂𝐵ž𝑚,𝑛∈𝑅d𝑚+ 𝑛≠1¢]
ž1¢Ļļ
𝐶𝑂=
𝑚
𝑚+𝑛−1𝐶𝐴+
𝑛
𝑚+𝑛−1𝐶𝐵J
ž2¢»L O a𝛥𝐴𝐵𝐶0!"2Y m'n 0ÎJ
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𝐺𝐴+ 𝐺𝐵+ 𝐺𝐶= 0J
(2)YĻ
𝑂𝐺=
1
3 𝑂𝐴+ 𝑂𝐵+ 𝑂𝐶J
(3)YĻ
𝑂𝐻= 𝑂𝐴+ 𝑂𝐵+ 𝑂𝐶]

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(1)» P * △𝐴𝐵𝐶0%"22𝑏= 3𝑎= 4𝑐2ŕˆ AP „ BC 6L D2Y
𝐴𝑃
𝑃𝐷J
(2)» P *¼@ △𝐴𝐵𝐶0&"2𝐴= 2𝐵2𝑃𝐵= 𝑥𝑃𝐴+𝑦𝑃𝐶2Y𝑥+ 𝑦0ŖÎŗŘ.