《高一数学必修向量在平面几何解题中的应用》由会员分享，可在线阅读，更多相关《高一数学必修向量在平面几何解题中的应用（13页珍藏版）》请在金锄头文库上搜索。
1、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
2、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=
3、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
4、、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
5、 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 Profi
6、le 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 Asp
7、ose.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 w
8、ith 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
9、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
10、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 Clien
11、t 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.E
12、valuation 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=n
13、OB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
14、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.
