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第13届“希望杯”高一培训题第84题为: 有两幅同一地区的,比例尺不同的长方形地图.将其中较小的一幅随意地平放在另一幅地图内.证明这两幅地图中一定有一个相同的地点是在某一个重合的点上(即:上下平贴的两张图中必有一个点是两幅地图中的同一个位置). 参考解答上只给了用相似三角形的方法寻找这个重合点,很多同学看完解答都问同一个问题:为什么?这道很难的题的命题背景很深,它牵涉到数学分析中的闭矩形域套定理.设大地图为矩形ABCD,小地图为矩形A1B1C1D1相似比为
The 84th title of the 13th “Hope Cup” high-level training question is: There are two rectangular maps of the same area with different scales. Place the smaller one of them flatly on the other map. Prove these two images. There must be one and the same place in the map at a certain coincident point (ie, there must be one point in two maps that are flatly attached to the top and bottom is the same position in the two maps). The reference answer only gives similar triangles. The method looks for this coincidence point. Many students have asked the same question after reading the answer: Why? The difficult proposition has a very deep proposition background. It involves the closed rectangular domain set theorem in mathematical analysis. Let the large map be rectangular ABCD The similarity ratio of the small map to the rectangular A1B1C1D1 is