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向量在三角形中的应用经常出现在各级各类考试中,有些试题往往与三角形的面积(或面积之比)有关,这类试题因向量“形”与“数”的双重特征与三角形边角关系的巧妙结合,往往使学生感到难以把握.其实,解决这类问题的关键是通过向量的运算及位置关系,特别是利用向量的平行(共线)关系,以揭示点的共线关系,或利用向量的平行关系,揭示相关三角形边之间的平行关系,从而使问题巧妙化解.现举几例说明如下:
The application of vectors in triangles often appears in various types of examinations. Some questions are often related to the area (area ratio) of triangles. These types of questions are due to the dual nature of the vector “shape” and “number”. The clever combination of triangular corner relationships often makes students feel difficult to grasp. In fact, the key to solving this type of problem is to use vector operations and positional relationships, especially to use the parallel (collinear) relationships of vectors to reveal the total Line relationships, or the use of parallel relationships between vectors, reveal the parallel relationship between the edges of the relevant triangles, so that the problem can be subtlely solved. Here are a few examples to illustrate: