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1994199419941994————2009200920092009年全国联赛中的平面几何题1.1.1.1.如图,,,,设的外接圆的半径为,,,,内心为,,,,,,,,,,,,的外角平分线交圆于....ABCABCABCABC∆∆∆∆OOOORRRRIIII°°°°====∠∠∠∠60606060BBBBCCCCAAAA∠∠∠∠∠∠∠∠AAAA∠∠∠∠OOOOEEEE证明:(1):(1):(1):(1)(2)(2)(2)(2)(1994(1994(1994(1994年))));;;;AEAEAEAEIOIOIOIO====....))))33331111((((2222RRRRICICICICIAIAIAIAOIOIOIOIRRRR++++++++++++IEOCBA2.2.2.2.如图,,,,菱形的内切圆与各边分别相切于在与上分别作圆的切线交ABCDABCDABCDABCDOOOO,,,,,,,,,,,,,,,,HHHHGGGGFFFFEEEE⌒⌒⌒⌒EFEFEFEF⌒⌒⌒⌒GHGHGHGHOOOOABABABAB于,,,,交于,,,,交于,,,,交于....证明::::.(1995.(1995.(1995.(1995年))))MMMMBCBCBCBCNNNNCDCDCDCDPPPPDADADADAQQQQNPNPNPNPMQMQMQMQ////////OQPNMGFHEDCBA3.3.3.3.如图示,,,,圆与圆和的三边所在的直线都相切,,,,为切点,,,,直线和相交于1111OOOO2222OOOOABCABCABCABC∆∆∆∆HHHHGGGGFFFFEEEE,,,,,,,,,,,,EGEGEGEGFHFHFHFHPPPP点....证明::::.(1996.(1996.(1996.(1996年))))BCBCBCBCPAPAPAPA⊥⊥⊥⊥HGO2O1FECBAP4.4.4.4.如图,,,,已知两个半径不相等圆与圆相交于两点,,,,且圆,,,,圆分别与圆内切于两点....1111OOOO2222OOOONNNNMMMM,,,,1111OOOO2222OOOOOOOOTTTTSSSS,,,,求证::::三点共线.(1997.(1997.(1997.(1997年))))TTTTNNNNSSSSMNMNMNMNOMOMOMOM,,,,,,,,⇔⇔⇔⇔⊥⊥⊥⊥MNTSO2O1O5.5.5.5.如图,,,,分别为的外心和内心,,,,是边上的高,,,,在线段上....求证::::的外接圆半IIIIOOOO,,,,ABCABCABCABC∆∆∆∆ADADADADBCBCBCBCIIIIODODODODABCABCABCABC∆∆∆∆径等于边上的旁切圆半径....注::::的边上的旁切圆是与边的延长线以及边都相切BCBCBCBCABCABCABCABC∆∆∆∆BCBCBCBCACACACACABABABAB,,,,BCBCBCBC的圆.(1998.(1998.(1998.(1998年))))CBADIO6.6.6.6.如图,,,,在凸四边形中,,,,平分,,,,是线段上的一点....,,,,....ABCDABCDABCDABCDACACACACBADBADBADBAD∠∠∠∠EEEECDCDCDCDFFFFACACACACBEBEBEBE====∩GGGGBCBCBCBCDFDFDFDF====∩求证::::.(1999.(1999.(1999.(1999年))))GACGACGACGACEACEACEACEAC∠∠∠∠====∠∠∠∠GFEDCBA7.7.7.7.如图,,,,锐角的边上有两点,,,,满足,,,,作((((是ABCABCABCABC∆∆∆∆BCBCBCBCFFFFEEEE,,,,CAFCAFCAFCAFBAEBAEBAEBAE∠∠∠∠====∠∠∠∠ACACACACFNFNFNFNABABABABFMFMFMFM⊥⊥⊥⊥⊥⊥⊥⊥,,,,NNNNMMMM,,,,垂足),),),),延长交的外接圆于点....证明::::四边形的面积与的面积相等.(2000.(2000.(2000.(2000年))))AEAEAEAEABCABCABCABC∆∆∆∆DDDDAMDNAMDNAMDNAMDNABCABCABCABC∆∆∆∆DMNFECBA8.8.8.8.如图,,,,中,,,,为外心,,,,三条高交于点,,,,直线和交于点,,,,直线和交ABCABCABCABC∆∆∆∆OOOOCFCFCFCFBEBEBEBEADADADAD,,,,,,,,HHHHEDEDEDEDABABABABMMMMFDFDFDFDACACACAC于点,,,,求证:(1):(1):(1):(1)(2)(2)(2)(2).(2001.(2001.(2001.(2001年))))NNNN;;;;,,,,DEDEDEDEOCOCOCOCDFDFDFDFOBOBOBOB⊥⊥⊥⊥⊥⊥⊥⊥MNMNMNMNOHOHOHOH⊥⊥⊥⊥NMHOFEDCBA9.9.9.9.如图,,,,在中,,,,,,,,,,,,点是的外心,,,,高交于点,,,,点分别在ABCABCABCABC∆∆∆∆°°°°====∠∠∠∠60606060AAAAACACACACABABABABOOOOABCABCABCABC∆∆∆∆CFCFCFCFBEBEBEBE,,,,HHHHNNNNMMMM,,,,线段上,,,,且满足....求的值.(2002.(2002.(2002.(2002年))))HFHFHFHFBHBHBHBH,,,,CNCNCNCNBMBMBMBM====OHOHOHOHNHNHNHNHMHMHMHMH++++NMOEFHCBA10.10.10.10.过圆外一点作圆的两条切线和一条割线,,,,切点为,,,,所作割线交圆于两点,,,,在之间....在PPPPBBBBAAAA,,,,DDDDCCCC,,,,CCCCDDDDPPPP,,,,弦上取一点使....求证::::.(2003.(2003.(2003.(2003年))))CDCDCDCD,,,,QQQQPBCPBCPBCPBCDAQDAQDAQDAQ∠∠∠∠====∠∠∠∠PACPACPACPACDBQDBQDBQDBQ∠∠∠∠====∠∠∠∠OQDCPBA11.11.11.11.如图,,,,在锐角中,,,,上的高与上的高相交于点,,,,以为直径的圆分别与两边ABCABCABCABC∆∆∆∆ABABABABCECECECEACACACACBDBDBDBDHHHHDEDEDEDE相交于两点,,,,求的长.(2004.(2004.(2004.(2004年))))ACACACACABABABAB,,,,GGGGFFFF,,,,....7777,,,,20202020,,,,25252525,,,,================BEBEBEBEBDBDBDBDBCBCBCBCKKKKAHAHAHAHFGFGFGFG∩AKAKAKAKKGFHDEABC12.12.12.12.如图,,,,在中,,,,过作外接圆的切线,,,,又以点为圆心,,,,为半径作圆分别交ABCABCABCABC∆∆∆∆,,,,ACACACACABABABABAAAAABCABCABCABC∆∆∆∆llllAAAAACACACAC线段于点交直线于点....证明::::直线分别通过的内心与一个旁心.(2005.(2005.(2005.(2005年))))ABABABAB,,,,DDDDllllFFFFEEEE,,,,DFDFDFDFDEDEDEDE,,,,ABCABCABCABC∆∆∆∆FEDCBA13.13.13.13.如图,,,,以为焦点的椭圆与的边交于....在的延长线上任取点,,,,以11110000,,,,BBBBBBBB11110000BBBBABABABAB∆∆∆∆iiiiABABABAB))))1111,,,,0000((((====iiiiCCCCiiii0000ABABABAB0000PPPP0000BBBB为圆心,,,,为半径作圆弧交的延长线于点;;;;以为圆心,,,,为半径作圆弧交00000000PPPPBBBB00000000QQQQPPPP00001111BBBBCCCC0000QQQQ1111CCCC00001111QQQQCCCC00001111QQQQPPPPAAAABBBB1111的延长线于点;;;;以为圆心,,,,为半径作圆弧交的延长线于点;;;;以为圆心,,,,为半1111PPPP1111BBBB11111111PPPPBBBB11111111QQQQPPPP00001111CCCCBBBB1111QQQQ0000CCCC11110000QQQQCCCC径作圆弧交的延长线于点....试证明:(1):(1):(1):(1)与点重合,,,,且圆弧与圆弧相内切于点;;;;22221111PPPPQQQQ0000ABABABAB2222PPPP2222PPPP0000PPPP00000000QQQQPPPP11110000QQQQPPPP0000PPPP(2)(2)(2)(2)四点共圆.(2006.(2006.(2006.(2006年))))1111000011110000,,,,,,,,,,,,QQQQQQQQPPPPPPPPP2Q1P1Q0P0B0AC1B1C014.14.14.14.如图,,,,在锐角中,,,,,,,,是边上的高,,,,是线段上一点,,,,过作,,,,垂足ABCABCABCABC∆∆∆∆ACACACACABABABABADADADADBCBCBCBCPPPPADADADADPPPPACACACACPEPEPEPE⊥⊥⊥⊥为,,,,作,,,,垂足为,,,,分别是的外心....求证::::四点共圆的充要条EEEEABABABABPFPFPFPF⊥⊥⊥⊥FFFF22221111,,,,OOOOOOOOCDECDECDECDEBDFBDFBDFBDF∆∆∆∆∆∆∆∆,,,,FFFFEEEEOOOOOOOO,,,,,,,,,,,,22221111件为是的垂心.(2007.(2007.(2007.(2007年))))PPPPABCABCABCABC∆∆∆∆O2O2EFPDCBA15.15.15.15.如图,,,,凸四边形,,,,,,,,,,,,其中是平面上的ABCDABCDABCDABCD°°°°∠∠∠∠++++∠∠∠∠180180180180DDDDBBBBABABABABPCPCPCPCCACACACAPDPDPDPDBCBCBCBCPAPAPAPAPPPPffff⋅⋅⋅⋅++++⋅⋅⋅⋅++++⋅⋅⋅⋅====))))((((PPPP一动点....(1)求证::::当达到最小值时,,,,四点共圆;;;;))))((((PPPPffffCCCCBBBBAAAAPPPP,,,,,,,,,,,,(2)设是外接圆的弧上一点,满足:,且圆EEEEABCABCABCABC∆∆∆∆OOOOABABABAB,,,,11113333,,,,22223333−−−−========ECECECECBCBCBCBCABABABABAEAEAEAEECAECAECAECAECBECBECBECB∠∠∠∠====∠∠∠∠22221111的两条切线为,,求的最小值.(2008.(2008.(2008.(2008年))))OOOODCDCDCDCDADADADA,,,,2222====ACACACAC))))((((PPPPffffOPEDCBA16.16.16.16.如图,,,,已知分别为锐角三角形(((())))的外接圆上弧,,,,弧的中点....过点作NNNNMMMM,,,,ABCABCABCABC∆∆∆∆BBBBAAAA∠∠∠∠∠∠∠∠ΓΓΓΓBCBCBCBCACACACACCCCC交圆于点,,,,为的内心,,,,连接并延长交圆于....MNMNMNMNPCPCPCPC/