The application of the plane vector iso-sum line theorem in triangles
平面向量等和线定理在三角形中的应用

摘要

平面向量是解决数学问题的重要工具，其中三点共线定理是平面向量模块的重要知识点。平面向量等和线定理作为平面向量三点共线定理的拓展，在解决向量线性表示中的系数和相关问题时具有简洁性。本文将从平面向量三点共线定理出发，探究等和线的概念和相关性质，并以等和线定理在三角形中的应用为例，从数形结合的角度展现其在解决平面向量线性表示的系数和、最值、取值范围题型中的高效性和直观性。最后总结解决系数和相关问题的一般逻辑和注意点，旨在帮助学生优化上述一类题目的解题思路，提高学生的解题效率，提升直观想象等方面的数学学科核心素养。

Abstract

Plane vectors are important tools for solving mathematical problems, with the collinearity theorem of three points being a key concept in the plane vector module. The plane vector iso-sum line theorem, as an extension of the collinearity theorem of three points in plane vectors, is concise in solving problems related to coefficients and other issues in vector linear representation. This article will start from the collinearity theorem of three points in plane vectors, explore the concept and properties of iso-sum lines, and use the iso-sum line theorem in triangles as an example to demonstrate its efficiency and intuitiveness in solving coefficient sum, maximum value, and value range problems in plane vector linear representation from the perspective of numerical and geometric combination. Finally, it will summarize the general logic and key points for solving coefficient-related problems, aiming to help students optimize their problem-solving strategies for such types of questions, improve their problem-solving efficiency, and enhance their core mathematical skills in aspects such as intuitive imagination.