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2022年三角形四心的向量性质及应用20120516 .pdf

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1、-1-三角形“四心”的向量性质及其应用三角形“四心”的概念介绍(1)重心三条中线的交点：重心将中线长度分成2：1；(2)外心三边中垂线的交点(外接圆的圆心)：外心到三角形各顶点的距离相等；(3)垂心三条高线的交点：高线与对应边垂直；(4)内心三条内角平分线的交点(内切圆的圆心)：角平分线上的任意点到角两边的距离相等工具：O为ABC内一点，则有：0OCSOBSOASOABOCAOBC证明：延长AO交BC于D，如图必有：|OAODSSSOABOCAOBC，|BCBDSSSOABOCAOAB，|BCCDSSSOABOCAOCA；-（*）由DOA,共线，得：0|ODODOAOA进而得：0|ODOAOA

2、OD -由CDB,共线，得：OCBCBDOBBCCDOD|-由得：OAOAOD|0|OCBCBDOBBCCD代入（*）结论得OASSSOABOCAOBCOBSSSOABOCAOCA0OCSSSOABOCAOAB消去分母得：0OCSOBSOASOABOCAOBC证毕.另证：作ACOGABOH/,/，如图：AGOH为平行四边形；由OCSOBSOASOABOCAOBC)()(ACOASABOASOASOABOCAOBCACSABSOASOABOCAABC)(ACSSABSSOASABCOABABCOCAABC)(ACACAHABABAGOASABC)(AHAGOASABC0)(AOOASABCABC

3、ODABCODHFEG-2-反方向思考：设O在ABC的内部，若有正实数321,满足：0321OCOBOA，必有：AOBCOABOCSSS:321证明：作：OAOA1，OBOB2，OCOC3则OAOB0OC，则O为CBA的重心，则：OBAOACOCBSSS.设为S又SSSSSSSSSAOBOBACOAOACBOCOCB2!1332从而得：AOBCOABOCSSSSSS:211332321验证式思考：先证引理：若ba,不共线，对p，有0pa且0pb，必有.0p证明：若.0p必有pa且pb，得ba/，与题设矛盾，故必有.0p再证：设BOC，COA，则2AOB；由)(OCSOBSOASOAOABOCA

4、OBCOCOASOBOASOASOABOCAOBC2cos)2sin(21)2cos(sin21sin212OCOAOBOAOBOAOAOCOAOCOBcos)sin()cos(sinsin212OCOBOA)(sinsin212OCOBOA0)sin(sin212OCOBOA；有对称性知：0)(OCSOBSOASOBOABOCAOBC，又OA，OB不共线，故：必有0OCSOBSOASOABOCAOBC成立一、三角形的重心的向量表示及应用知识：G 是ABC的重心)(31ACABAG0GCGBGA)(31OCOBOAOG (O为该平面上任意一点)略证：1:1:1:GABGCAGBCSSS，得：0

5、GCGBGA变式：已知 DEF，分别为ABC的边 BCACAB，的中点则0CFBEAD二、三角形的外心的向量表示及应用知识：O是ABC的外心222|OCOBOAOCOBOAABCOABCABCO文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编

6、码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q

7、8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编

8、码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q

9、8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编

10、码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q

11、8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8-3-02sin2sin2sinOCCOBBOAA略证：CBASSSOABOCAOBC2sin:2sin:2sin:，得：02sin2sin2sinOCCOBBOAA

12、常用结论：O是ABC的外心.2|;2|22ACAOACABAOAB三、三角形的垂心的向量表示及应用知识：H是ABC的垂心HAHCHCHBHBHA222222|ABHCCAHBBCHA0tantantanHCCHBBHAA略证：CBASSSHABHCAHBCtan:tan:tan:，得：0tantantanHCCHBBHAA扩展：若O是ABC的外心，点H满足：OCOBOAOH，则H是ABC的垂心证明：如图：BE为直径，H为垂心，O为外心，D为BC中点；有：为平行四边形AHCEEACHABEAABCHECAHBCECBCAH/进而得到：,/ECAH且ECAH，即：ECAH；又易知：OCOBODEC

13、2；故：OAOHOCOBAH，即：OCOBOAOH又：OGOCOBOA3（G为重心），故：OGOH3；故：得到欧拉线：ABC的外心O，重心G，垂心H三点共线(欧拉线)，且GHOG21证毕四、三角形的内心的向量表示及应用知识：I是ABC的内心0|0|0|CBCBCACACIBCBCBABABIACACABABAI0|0|0|CACABCBCCIBABACBCBBIACACBABAAI0ICcIBbIAacbaOCcOBbOAaOI0sinsinsinICCIBBIAA注：式子中|,|,|ABcCAbBCa，O为任一点ABDOHCE文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H

14、8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H

15、5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H

16、8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H

17、5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H

18、8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H

19、5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H

20、8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8-4-略证：CBAcbaSSSIABICAIBCsin:sin:sin:，得之五欧拉线：ABC的外心O，重心G，垂心H三点共线(欧拉线)，且GHOG21（前已证）测试题一选择题1O是ABC所在平面上一定点，动点P满足)(ACABOAOP，,0，则点P的轨迹一定通过ABC的()A外心B内心C重心D垂心2(03 全国理 4)O是ABC所在平面上一定点，动点P满足)(ACACABABOAOP，,0，则点P的轨迹一定通过ABC的(

21、)A外心B内心C重心D垂心3O是ABC所在平面上一定点，动点P满足)coscos(CACACBABABOAOP，R，则点P的轨迹一定通过ABC的()A外心B内心C重心D垂心4O是ABC所在平面上一定点，动点P满足)sinsin(CACACBABABOAOP，,0，则点P的轨迹一定通过ABC的()A外心B内心C重心D垂心5O是ABC所在平面上一定点，动点P满足2coscosOBOCABACOPABBACC，R，则点P的轨迹一定通过ABC的()A外心B内心C重心D垂心6O是ABC所在平面上一定点，动点P满足)21()1()1(31OCOBOAOP，*R，则点P的轨迹一定通过ABC的()A内心B垂心

22、C重心DAB 边的中点7已知O是ABC的重心，动点P满足)22121(31OCOBOAOP,则点P一定为ABC的()AAB 边中线的中点BAB 边中线的三等分点(非重心)C重心DAB 边的中点8在ABC中，动点P满足：CPABCBCA222，则P点轨迹一定通过ABC的()外心内心C重心D垂心9已知ABC三个顶点CBA、及平面内一点P，满足0PCPBPA，若实数满足：APACAB，则的值为()A2 B23C3 D6 10设点P是ABC内一点，用ABCS表示ABC的面积，令ABCPBCSS1，ABCPCASS2，ABCPABSS3文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8

23、V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5

24、V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8

25、V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5

26、V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8

27、V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5

28、V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8

29、V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8-5-B C A M N G 定义),()(321Pf，若)61,31,21()(),31,31,31()(QfGf则()A点Q在ABG内B点Q在BCG内C点Q在CAG内D以上皆不对11若ABC的外接圆的圆心为O，半径为1，0OCOBOA，则OBOA()A21B 0 C1 D2112O是平面上一定点，CBA、是平面上不共线的三个点，若222OBBCOA222ABOCCA，则O是ABC的()A外心B内心C重心D垂心13(06

30、陕西)已知非零向量AB与AC满足0|BCACACABAB且21|ACACABAB,则 ABC 为()A三边均不相等的三角形B直角三角形C等腰非等边三角形D等边三角形14已知ABC三个顶点CBA、，若CABCCBABACABAB2，则ABC为()A等腰三角形B等腰直角三角形C直角三角形D既非等腰又非直角三角形二填空题15ABC的外接圆的圆心为O，两条边上的高的交点为H，)(OCOBOAmOH，则实数m=1 16ABC中，7,3,1BCACAB，O为重心，则ACAO2717点O在ABC内部且满足032OCOBOA，则:ABCSAOCS3 18点O在ABC内部且满足ACABAO5152，则:ABCS

31、AOBS4 19已知ABC中，6,5 BCACAB，I为ABC的内心，且BCABAI，则1615.20已知ABC中，1,1,2ACABACAB，O为ABC的外心，且BCyABxAO，则yx2721已知O为锐角ABC的外心，30A，若AOmBCACCBAB2sincossincos，则m2122在ABC中，1,3,ADBDBCABAD，则ADAC3三解答题23 如图，已知点G是ABC的重心，过G作直线与ACAB,两边分别交于NM,两点，且AMxAB，ANyAC，求证：113xy解：由NGM,三点共线，得：ANtAMtAG)1(ACtyABxt)1(-又G是ABC的重心文档编码:CY2Y4H6H5

32、V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8

33、V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5

34、V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8

35、V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5

36、V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8

37、V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5

38、V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8-6-得：ACABAG3131 -由得：3131)1(tyxt，消去t得：113xy.24设O在ABC的内部，若有正实数321,满足：0321OCOBOA，求证：AOBCOABOCSSS:321证明：作：OAOA1，OBOB2，OCOC3则OAOB0OC，则O为CBA的重心，则：OBAOACOCBSSS.设为S又SSSSSSSSSAOBOBACOAOACBOCOCB2!1332从而得：

39、AOBCOABOCSSSSSS:21133232125已知向量1OP，2OP，3OP 满足条件1OP+2OP+3OP=0，|1OP|=|2OP|=|3OP|=1，求证：321PPP为正三角形证明：由1OP+2OP+3OP=01OP+2OP=3OP平方得：1212112121OPOPOPOP从而得：3211)(|2121222121OPOPOPOPPPPP同理可得：3|1332PPPP，即321PPP为正三角形26在ABC中，60,5,2AACAB，求从顶点BA,出发的两条中线BEAD,的夹角的余弦值解：设bABaAC,，则,560cos25,4,2522baba且baBEbaAD21),(21

40、；则,3)8525(41)2(41)21()(2122bbaababaBEAD2394102521221|)(|21|22bbaabaAD221162025214421|)2(|21|22bbaabaBE故：.919149142212393|,cosBEADBEADBEADABCOABCABEDC文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR

41、9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H

42、6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR

43、9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H

44、6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR

45、9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H

46、6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8-7-27已知H是ABC的垂心，且|BCAH，试求A的度数解：

47、设ABC的外接圆半径为R，点O是ABC的外心。H是ABC的垂心OCOBOAOHOCOBOAOHAH,AOCOB2,)2cos1(2)(|2222AROCOBAHAHOBOCBC，)2cos1(2)(|2222AROBOCBCBC|BCAH，AA2cos12cos102cos A而A为ABC的内角，36020A从而902A或2702AA的度数为45或135.28已知),(),0,1(),0,0(cbCBO，试写出OBC的重心G，外心F，和垂心H的坐标，并证明HFG,三点共线(2002 全国)解：易知)3,31(cbG；设),(0ybH，由BCOH，且),1(),(0cbBCybOH，得0)1()

48、,1(),(00cybbcbybBCOH，得cbby)1(0，即)1(,(cbbbH；设),21(yF，则由cbcbycybyFCFO2)()21(41|22222,即)2,21(22cbcbF.而且：)233,212(),633,612(2222ccbbbFHccbbbFG易知：FGFH3（又有公共端点F），故HGF,三点共线.29已知G、M分别为不等边ABC的重心与外心，)0,1(A、B)0,1(且GMAB，(1)求点C的轨迹方程；(2)若直线l过点(0，1)，并与C的轨迹曲线交于P、Q两点，且满足0OQOP(O为坐标原点)，求直线l的方程解：（1）设),(yxC，则)3,3(yxG，再设

49、),0(0yM，由/GMAB，易得：.30yy故外心)3,0(yM；由|MBMC，代入坐标得：91)3()0(222yyyxABCHO文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H

50、8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H5V1 HS5C7Q8Q6D3 ZR9H8V8C9M8文档编码:CY2Y4H6H

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