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如何添加辅助线历来是证明几何题中的一个难关。在证明诸线段平方或积的和差关系时这个问题显得更加突出。若从所证关系式结构上的特征去分析,运用数形结合的方法进行大胆猜想,常能收到奇效,现举几例供参考。例1 在等腰三角形ABC的底边BC上任取一点P,则有AB~2-AP~2=BP·CP。分析欲证等式左端是两线段的平方差,右端是两线段的乘积。由此我们猜想BP·CP也是也是某两条线段的平方差,即
How to add an auxiliary line has historically been a difficult problem in proving geometric problems. This problem becomes even more pronounced when the sum or square of the line segments is proved. If we analyze the characteristics of the structure of the relationship between the certificate, using a combination of number and shape to make bold conjecture, can often receive bizarre effects, to give a few examples for reference. Example 1 Any point P on the bottom edge BC of the isosceles triangle ABC has AB~2-AP~2=BP•CP. The left side of the equation for analysis is the product of the square difference between the two segments, and the right end is the product of the two segments. From this we guess that BP·CP is also the square difference of some two line segments, ie