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在国内外众多的数学竞赛中,几何占有相当的比例。仔细分析这些试题,发现它们有一个共同的特点:于传统内容中渗透近代数学思想和方法。几何变换就是典型的例子。不少竞赛题,乍看起来,难度较大,但应用几何变换的思想进行处理,可以化繁为简,迅速找到解题思路。常见的初等几何变换有反射,平移,旋转,中心对称,位似等等。本文从不同国家和国际奥林匹克竞赛题中,选出几例,用变换思想进行处理,供读者参考。 1.反射变换例1 已知A为平面上两半径不等的⊙0_1,和⊙o_2的一个交点,外公切线P_1P_2的切点为P_1、P_2,另一条外公切线的切点为Q_1,Q_2,M_1、M_2分别为P_1Q_1、P_2Q_2的中
In many mathematical competitions at home and abroad, geometry holds a considerable proportion. A careful analysis of these questions reveals that they have a common feature: infiltrating modern mathematical thinking and methods in traditional content. Geometric transformation is a typical example. A lot of competition questions, at first glance, are more difficult, but applying the idea of geometric transformation can simplify the process and quickly find solutions to problems. Common elementary geometric transformations include reflections, translations, rotations, center symmetries, and similarities. This article selects several cases from different national and international Olympic competition questions, and uses transformation thinking to handle it for readers’ reference. 1. Example 1 of reflection transformation It is known that A is an intersection of ⊙0_1 and ⊙o_2 with two unequal radii in the plane. The tangent point of the tangent P_1P_2 is P_1, P_2, and the tangent point of the other tangent is Q_1, Q_2. M_1 and M_2 are the middle of P_1Q_1 and P_2Q_2, respectively.