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在平面几何里有一类证明线段成比例题,数学证明通常要作辅助线,这比较难掌握,可利用物理中的杠杆平衡原理求证,方法新颖,而且较简单. 例1 如图1,在△ABC(AB>AC)的边AB上取一点D,在边AC、上取一点E,使AD=AE,直线DE和BC的延长线交于点P,求证:BP:CP=BD:CE. 证明设点A、B、C分别放有质量为m1、m2、m3的物体,由杠杆平衡原理得:m1·AD=m2·BD, m1·AE=m3·EC,m2·BP=m3·CP.因为AD=AE,
In plane geometry, there is a class of proofs for proportional problems in line segments. Mathematical proofs are usually used as auxiliary lines. This is difficult to grasp. The principle of lever balance in physics can be used for verification. The method is novel and relatively simple. Example 1 As shown in Figure 1, Take a point D on the edge AB of ABC(AB>AC), and take a point E on the edge AC, so that AD=AE, the extension line of the straight lines DE and BC intersect at point P, verifying: BP:CP=BD:CE. Prove that set points A, B, and C have masses with masses m1, m2, and m3, respectively. The principle of lever balance is: m1·AD=m2·BD, m1·AE=m3·EC, m2·BP=m3·CP. Because AD=AE,