Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.一、向量有关知识复习（1）向量共线的充要条件:与 共线 （2）向量垂直的充要条件：（3）两向量相等充要条件：且方向相同。（4）平面向量基本定理Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.二、应用向量知识证明平面几何有关定理例1、证明直径所对的圆周角是直角ABCO如图所示，已知O，AB为直径，C 为O上任意一点。求证ACB=90分析：要证ACB=90，只须证向 量 ，即 。即 ，ACB=90 思考：能否用向量坐标形式证明？Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.二、应用向量知识证明平面几何有关定理例2、证明平行四边形四边平方和等于两对角线平方和ABDC已知：平行四边形ABCD。 求证：解：设 ，则分析：因为平行四边形对边平行且相 等，故设 其它线段对应向 量用它们表示。Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.三、应用向量知识证明三线共点、三点共线例3、已知：如图AD、BE、CF是ABC三条高 求证：AD、BE、CF交于一点FABCDEABCDEH分析：思路一：设AD与BE交于H，只要证 CHAB，即高CF与CH重合，即CF 过点H由此可设利用ADBC，BECA，对应向量垂直。Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.三、应用向量知识证明三线共点、三点共线例3、已知：如图AD、BE、CF是ABC三条高 求证：AD、BE、CF交于一点ABCDEH解：设AD与BE交于H，即高CF与CH重合，CF过点H，AD、BE、CF交于一点 。Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.三、应用向量知识证明三线共点、三点共线例4、如图已知ABC两边AB、AC的中点分别为M、N， 在BN延长线上取点P，使NP=BN，在CM延长线上取点Q， 使MQ=CM。求证：P、A、Q三点共线ABC NMQP解：设 则由此可得即 故有 ，且它们有 公共点A，所以P、A、Q三点共线Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.四、应用向量知识证明等式、求值例5、如图ABCD是正方形M是BC的中点，将正方形折起，使点A与M重合，设折痕为EF，若正方形面积为64，求AEM的面积ABCDMNEF 分析：如图建立坐标系，设E(e,0)，M(8,4), N是AM的中点，故N(4,2) =(4,2)-(e,0)=(4-e,2)解得：e=5故AEM的面积为10Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.四、应用向量知识证明等式、求值例5、如图ABCD是正方形M是BC的中点，将正方形折起，使点A与M重合，设折痕为EF，若正方形面积为64，求AEM的面积ABCDMNEF 解：如图建立坐标系，设E(e,0)，由正方形面积为64，可得边长为8由题意可得M(8,4)，N是AM的中点，故N(4,2) =(4,2)-(e,0)=(4-e,2)解得：e=5 即AE=5Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.四、应用向量知识证明等式、求值练习：PQ过OAB的重心G，且OP=mOA,OQ=nOB求证：分析:由题意OP=mOA,OQ=nOB,联想线段的定比分点，利用向量坐标知识进行求解。 OABGPQ由PO=mOA, QO=nOB可知：O分 的比为 ，O分 的比为由此可设 由向量定比分点公式，可求 P、Q的坐标，而G为重心，其坐标也可求出，进而 由向量 ，得到 m n 的关系。-m -n? ?Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.四、应用向量知识证明等式、求值练习：PQ过OAB的重心G，且OP=mOA,OQ=nOB求证：OABGPQ证：如图建立坐标系，设 所以重心G的坐标为 由PO=mOA, QO=nOB可知：即O分 的比为-m，O分 的比为-n 求得由向量 可得：化简得：Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.五、小结、巩固练习：练习1：证明对角线互相垂直平分的四边形是菱形练习2：如图O为ABC所在平面内一点，且满足求证：ABOCABCOEvaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.