Question

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．

Answer 1

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