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一道好的数学竞赛题,往往蕴藏着丰富的内涵,如果我们能充分挖掘其中的潜能,那将会大大提高其教育价值,下面本人仅以一道竞赛题为例谈一下自己粗浅的看法。命题1:如图1.设AM是△ABC边BC上的中线,任作一条直线分别交AB、AC、AM于P、Q、N,求证:AB/APAM/AV、AC/AQ成等差数列(1978年辽宁省中学数学竞赛复赛题)。围绕着这道竞赛题,可以从以下几个方面展示其教育价值。一、变换视角,增强联想思维能力。视角一,我们先考察命题1结论的整体特征,AB/AP、AM/AN、AC/AQ成等差数列,即AM/AN=1/2(AB/AP+AC/AQ),再联想到几何量成等差数列的有什么定理?容易想到梯形中位线定理。因此,要创造出现梯形的条件,那么,梯形位置应在何处?由特征式等AM/AN=1/2(AB/AP+AC/AQ)得到启示,
A good math contest question often contains rich connotations. If we can fully tap into its potential, it will greatly increase its educational value. I will now only use one contest as an example to talk about my own superficial views. Proposition 1: As shown in Figure 1. Let AM be the midline on BC with △ABC, and make a straight line to AB, AC, and AM respectively for P, Q, and N. Verify that: AB/APAM/AV, AC/AQ are equal Series (question for the second round of the 1978 Liaoning Middle School Mathematics Contest). Around this contest, you can demonstrate its educational value from the following aspects. First, change the perspective and enhance the ability to think. From perspective one, we first examine the overall characteristics of the conclusions of Proposition 1, AB/AP, AM/AN, AC/AQ into equal difference series, that is, AM/AN=1/2(AB/AP+AC/AQ), and then think of What are the theorems of geometrical quantities into arithmetic series? It is easy to think of trapezoidal median line theorem. Therefore, to create a trapezoidal condition, where should the trapezoidal position be? Inspired by the characteristic equation such as AM/AN=1/2(AB/AP+AC/AQ),