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首页 2019-2020年初中数学竞赛专题复习第二篇平面几何第9章三角形试题新人教版

2019-2020年初中数学竞赛专题复习第二篇平面几何第9章三角形试题新人教版.doc

2019-2020年初中数学竞赛专题复习第二篇平面几何第9章三…

沙漠骆驼
2019-05-29 0人阅读 举报 0 0 暂无简介

简介:本文档为《2019-2020年初中数学竞赛专题复习第二篇平面几何第9章三角形试题新人教版doc》,可适用于初中教育领域

年初中数学竞赛专题复习第二篇平面几何第章三角形试题新人教版..★已知等腰直角三角形是斜边.的角平分线交于过作与垂直且交延长线于求证:.解析如图延长、设交于.则得.又平分故平分为中点所以...★在中已知、、分别为、、的中点、为形外两点使若求的长.解析如图连结、则故.又故所以又所以于是...★在梯形的底边上有一点若、、的周长相等求.解析作平行四边形则若与不重合则在(或延长线)上但由三角形不等式易知在上时的周长EMBEDEquationDSMT的周长在延长线上时的周长周长均与题设矛盾故与重合同理...★★内、分别在边、上并且、分别是、的角平分线.求证:.解析延长到使连结.易知所以.因所以.于是...★★设等腰直角三角形中是腰的中点在斜边上并且.求证:.解析如图作的平分线在上.由于故故.又于是于是...★★设、都是等腰直角三角形、是各自的斜边是的中点求证:也是等腰直角三角形.解析如图作、、、分别垂直于直线垂足为、、、.由故有.同理所以.又得且故.又由故结论成立...★★已知、在上(靠近)求证:的充要条件是.解析如图作且则又故且.若则因得则.反之若由得.又故又于是...★★两三角形全等且关于一直线对称求证:可以将其中一个划分成块每一块通过平移、旋转后拼成另一个三角形.解析如图设与关于对称分别找到各自的内心、分别向三边作垂线、、与、、于是个四边形……均为轴对称的筝形且四边形四边形所以两者可通过平移、旋转后重合同理另外两对筝形也可通过平移、旋转后重合...★★★已知:两个等底等高的锐角三角形可以将每个三角形分别分成四个三角形分别涂上红色、蓝色、黄色和绿色使得同色三角形全等.解析如图设至距离等于至距离取各自的中位线、则.由、均为锐角三角形可在、上各取一点、使图中标相同数字的角相等于是.评注还有一种旋转而不是对称的构造法...★已知与中与是否一定全等?解析如图让与重合与重合、在同侧若与重合则否则由条件知四边形为梯形和圆内接四边形于是它是一个等腰梯形于是.综上可知与全等.评注本题也可以运用三角形面积公式、余弦定理结合韦达定理来证明...★★如图所示已知、均为正三角形、、分别为、和的中点求证:为正三角形.解析如图设、中点分别为、连结、、、.则四边形为平行四边形设则又故于是为正三角形.评注注意有时在另一侧此时不影响最终结论...★★★中.是中点、分别在、上(可落在端点)满足求的最小值(用、、表示).解析如图延长至使连结、、、由于是、的中点故又垂直平分故.取中点(图中未画出)则于是的最小值为取到等号仅当即四边形为矩形时...★★★已知为内一点由作、的垂线垂足分别是、.设为中点求证:.解析如图所示取中点中点连、、、.显然四边形是平行四边形所以..又由所以同理.由所以从而所以...★★在中已知、分别是边、上的点且求的度数.解析如图延长到使连、.因为所以..于是.又因为所以.在和中所以故.于是...★★在中、为锐角、、分别为边、、上的点满足且.求证:.解析若则在上取一点使.连结并延长交于连结.在与中故.于是有所以.又易知因此.但另一方面由知所以.从而.矛盾故假设不成立.若同法可证此假设不成立.综上所述于是由知从而...★★如图为边长是的等边三角形为顶角是的等腰三角形以为顶点作一个角角的两边分别交、于、连结形成一个.求的周长.解析延长到使连结.易知在与中有从而.所以.于是在与中有.从而故.所以...★★★为等腰直角三角形点、分别为边和的中点点在射线上且点在射线上且求证:.解析取中点连.在与中故.于是有.同样易知于是有.在与中由知所以.于是有.从而在与中有故.于是有.总之即...★★★已知延长至使连结与交于为的外心则、、、共圆.解析如图连好辅助线由于故设则又故于是于是因此、、、共圆...★★★已知和且和分别是、的中点问两个三角形是否必定全等?解析如图作出外心(及相应的、图中未画出).若在上则此时与未必全等.若不与重合则.当、、共线则所以从而.当、、不共线则于是(或)于是由三角形全等可得(或)(或)故有(或).评注此题亦可用中线长公式证明...★★如果两个三角形满足“”它们不一定全等此时称它们是相近的现在有一三角形作与之“相近”……一般有与相近问是否存在一个使与相做且不全等?解析这是不可能的.因为由正弦定理与有等大的外接圆(它们有一对内角相等或互补)从而推出与x有等大的外接圆它们不可能只相似不全等...★★★是否存在两个全等的三角形与可划分为两个三角形与可划分成两个三角形与使与却不全等?解析这样的两个三角形是存在的如图(a)、(b)设不等边三角形其中不妨设是各自的最长边则、为各自的最短边.在、上分别找、使则由于故所以又因为因此而显然不与全等.(若还可避免相似.)..★★★已知中是内心的垂直平分线分别交、于、、在上求证:.解析如图连结、、、.易诮与为全等之正三角形.两端延长至与使则于是同理因此.而、将三等分、将三等分于是由平行线分线段成比例知().评注读者可以考虑:如果是否有...★★★已知锐角三角形的垂心和外心分别为和分别与、交于、证明:的周长为.解析如图连结、、、.由可知在一侧在一侧.因故而于是.又故为正三角形.又故又故.于是.又做.§.特殊三角形..★在直角三角形中是斜边是中点是上一点求.解析如图连结.设因则.故...★已知中为在平分线上的射影为中点求.解析延长交于.由.知.又故...★等腰三角形中为直线上一点则(在上)(在外).解析如图设在上且较靠近.作于则为中点于是.当在外时的结论同理可证.评注这是斯图沃特定理在等腰三角形的特殊情形具有十分广泛的用途(例如题..)亦可用相交弦定理证明...★★已知锐角三角形中、是高为垂心是的中点求证:.解析如图连结则.于是EMBEDEquationDSMT.由于故...★已知斜边为的直角三角形中在上的投影为.若以、、为三边可以构成一个直角三角形求的所有可能值.解析显然由、、构成的直角三角形中不是斜边且.若则为斜边.设则由的面积知又故.易知则由前式知得故.同理若可得.所以的可能值为或...★★已知中为高在上以下哪些条件能判定:():()().解析设则.先看条件():.若则否则不妨设则.得于是矛盾.故.再看见条件():.则于是故.最后条件():.于是.若则仍有矛盾故.所以三个条件都能判定...★已知是等腰直角三角形的斜边上任意一点求.解析如图作于.不妨设.在上则于是.又.故.评注请读者考虑若对上任一点有为定值是否可认为为等腰直角三角形...★★在中是内一点过点向的三边、、分别垂线、、垂足分别为、、且求的长.解析如图由于于是此即.而故.所以...★★已知中是的中垂线求.解析如图不妨设则.作的平分线由于故.因此从而所以.设则因此(舍).于是...★★正三角形内有一点关于、的对称点分别为、作平行四边形求证:.解析如图设与交于连结则垂直平分为正三角形于是四边形为等腰梯形的中垂线即的中垂线.于是...★★与相切于点与相交于、若求.解析如图由题意可得作于则又故.再作于设则.于是...★已知大小相等的等边与等边有三组边分别平行一个指向上方一个指向下方相交部分是一个六边形则这个六边形的主对角线共点.解析如图设两个三角形的边的交点依次为、、、、、.设、的高为则正的高(与的距离)正的高于是、互相平分同理、互相平分于是、、的中点为同一点结论成立...★★★★求证:过正三角形的中心任作一条直线则、、三点至的距离平方和为常数.解析如图不妨设与、相交且与延长线交于(平行容易计算).由中位线及重心性质知.故.连结、作易知故.对于等腰三角形有.因此(定值)这里用到了.于是、、三点至的距离平方和为结论得证.§.三角形中的巧合点..★已知:是内一点、、延长后分别交对边于、、若则是的垂心解析如图由条件知故同理故.又故这样可得故为之垂心...★★求证:到三角形三顶点的距离平方和最小的点是三角形的重心.解析设中、、是中线是重心是任一点.由斯图沃特定理并考虑到结论成立.得.①又由中线长公式有.代入式①得.结论成立...★★★已知是锐角的垂心是中点过作的垂线交、于、求证:是中点.解析设两条高为、.又不妨设在上.由于故于是同理又故...★★★的边、、上分别有点、、且求证:的重心与的重心是同一点.解析在上取一点使则所以四边形为平行四边形设与交于又设的中点为连结、、与交于于是由得于是于是所以为与之重心...★★★已知是重心求证:是正三角形.解析设三条中线分别为、、.连为中位线.于是由条件知、、、共圆故于是.由于代入得.在外作等腰使连结.由圆心角与圆周角的关系故、、三点共线故于是又故为正三角形...★★★已知是上一点、、都是正三角形、在同侧在另一侧求证:以这三个正三角形的中心为顶点的三角形是正三角形且它的中心在上.又问此题如何推广?解析如图设、、分别为、和的中心则由题..知为正三角形.过、、分别作的垂线、、则又故.又设中点为(图中未画出)于则且.设与交于则所以为的中点.评注此题不难推广只需此时、、为各自对应的重心则必有之重心位于上...★★★内有一点连结、、并延长分别与对边相交把分成六个小三角形若这六个小三角形中有三个面积相等则点是否必为之重心?解析如图设、、交于.由对称性可分四种情况讨论.().于是由梅氏定理(或添平行线)得为中心.().此时故、分别为、中点为重心.().此时有由塞瓦定理于是回到情形().()见题...综上所知答案是肯定的...★★★设有一个三角形三角之比为作两较大角的平分线分别交对边于、.求证:这个三角形的重心在上.解析如图(a)设为最小角作中线交于于是只要证明.分别作、在直线上则故问题变成或.不妨设在上找一点使又作在上则各角大小如图(b)所示.于是故...★★★不等边锐角中、分别是其垂心和重心求证:若.解析设的一条中线与高分别为、则欲证结论等价于.熟知.于是结论变为.设则由中线长及余弦定理知欲证式左端右端整理得于是剩下的任务是证明这个等价条件.同理有另两式于是条件变为由正弦及余弦定理知上式即EMBEDEquationDSMT或化简即得...★★已知凸四边形中是否一定为之外心?解析当固定.由题设、固定于是、外接圆固定它们的交点、固定又若为外心时确为的外接圆和的外接圆之异于的交点因此结论成立...★★★已知锐角的外接圆与内切圆的半径分别为、是外心至三边距离之和为试用、表示.解析易知.设三边分别为、、由于等则于是.①又等可得故式①的右端.于是...★★★★:已知、分别在、上、交于求证:、、、的外心四点共圆.解析如图设、的外心分别为、为的外心于是垂直平分.垂直平分.设则由垂径定理知于是.易知过中点(由塞瓦定理或面积比)作在上则又故.又设的外心分别为、(图中未画出)于是、分别在直线与上且于是于是、、、四点共圆...★★★已知:中是中点为重心为外心求证:.解析如图延长交于则.连结并延长分别交、于、则为重心易见.又对应边垂直所以.解析为外心故而由中线公式于是于是...★★★设和分别是的内心和外心求证:的充分必要条件是.解析延长与外接圆交于点连结、、则.由内心性质知结合托勒密定理得所以所以故的充要条件是.评注本题的关键是先把转换为然后再用托勒密定理.托勒密定理是:圆内接四边形的对角线的乘积等于对边乘积的和...★★★设是的外接圆是三角形重心延长、、分别交于、、则.解析设、、的中点分别为、、则由中线长公式及相交弦定理有(此处三边分别设为、、).同理有.三式相加即得结论...★★在内平分求证:是内心.解析如图作在上在上则.又故于是.而故所以为内心...★★已知:中是内心与垂直于求的值.解析设三边长分别为、、则.易知若设则.于是...★★设中最长在其上分别找两点、使又设为内心求(用、、及其组合表示).解析如图连结、、、.易知同理为的外心因此...★★★★的边上有一点与的内心与、四点共圆求证:.解析如图设与的内心分别为与.连结、、、、两端延长分别交、于、则由条件知同理也是此值于是.又设与交于则由角平分线性质知故由梅氏定理(直线截及直线截)得(此处、分别为、延长后与、之交点)又由角平分线性质知于是结论成立...★★★已知中、分别为其外心与内心在上求证:.解析如图不妨设在内且在“之上”(在形外、之下类似处理)连结、则故、、、共圆于是.这里为、直线之交点.由于故于是...★★设为的重心已知且求的面积.解析由题意可画出图(a)令为中点垂足为点因为重心可知.由勾股定理可知令.由①与②可得化简后可得即代入③得再代入①式可得解方程可得故的面积=的面积.解析由题意可画出图(b)令为中点在的延长线上取点使得因此之面积为之面积的一半.此时因与互相平分可知四边形为平行四边形也因此可知即的三边长为、、故可知为直角三角形故的面积为所以的面积的面积...★★★已知为异于的任一点求证:.解析如图在外作正三角形由于故四边形的内角均小于是凸四边形.对于中任一异于的点将、均以点为中心顺时针旋转至和则与均为正三角形.由全等知这是因为是一条折线而、、、四点共线且仅对于满足四点共线.评注当内角均小于时满足条件的点称为的费马点(当有内角比如时到、、距离之和最小的点正是点).unknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunknownunk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