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初中几何第二册P.92有这样一道习题:“两圆相交于点A和B,经过交点B的任一直线和两圆分别相交于点C和D,求证AC:AD等于定值”。初三学生刚接触这类定值问题时往往感到束手无策。即使是按教材提示完成了,却也不知所以然。究其原因还是由于学生对在研究问题的过程中变量与常量之问的相依性缺乏认识。如果我们抓住矛盾的对立统一法则,揭示变动元素在“变”的过程中有“不变”的规律,把握住从运动的特殊状态去窥测一般,即“以动求静,以静窥动”的方法去思考,那么对解除学生求解这类定值问题的难点是有所启发的。现在让我们利用这种思考方法来探求这道习题的定值及其证明。
P.92 in junior high school geometry book P.92 has such an exercise: “The two circles intersect at points A and B. Any line and two circles passing through point B intersect at points C and D, respectively, verifying that AC:AD equals the fixed value”. Junior high school students often feel helpless when they come into contact with this kind of fixed value problem. Even if it was completed according to teaching materials, it did not know why. The reason for this is because students lack understanding of the interdependence of variables and constants in the process of studying problems. If we grasp the contradictory principle of the unity of opposites, we will reveal that the elements of change have a “unchanged” pattern in the process of “changing” and grasp the general state from the special state of movement. The way to think about it is to instruct students about the difficulty of solving such fixed-value problems. Now let us use this thinking method to explore the value of this exercise and its proof.