题目

如图所示，将矩形沿折叠，使点恰好落在上处，以为边作正方形，延长至，使，再以、为边作矩形．(1).试比较、的大小，并说明理由．(2).令，请问是否为定值？若是，请求出的值；若不是，请说明理由．为定值．(3).在（2）的条件下，若为上一点且，抛物线经过、两点，请求出此抛物线的解析式．(4).在（3）的条件下，若抛物线与线段交于点，试问在直线上是否存在点，使得以、、为顶点的三角形与相似？若存在，请求直线与轴的交点的坐标；若不存在，请说明理由．            答案：解：（1），理由如下：由折叠知：   在中，为斜边  故····························（2）···································································································· 3分（3），，        为等边三角形，················································································ 4分作于．        的坐标为·································································· 5分抛物线过点，，    所求抛物线解析式为········································································ 6分（4）由（3）：当时，·························································· 7分方法1：若与相似，而．则分情况如下时为或····························· 8分时    为或（0，1）······································ 9分故直线与轴交点的坐标为或或或（0，1）···············10分方法2：与相似时，由（3）得则或，过点作垂直轴于则或当时，当 ，， …………………10分 解析:略 

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