题目

The view over a valley of a tiny village with thatched (草盖的) roof cottages around a church, a drive through a narrow village street lined with thatched cottages painted pink or white, the sight over the rolling hills of a pretty collection of thatched farm buildings―these are still common sights in parts of England. Most people will agree that the thatched roof is an essential part of the attraction of the English countryside.         Thatching is in fact the oldest of all the building crafts（工艺）practised in the British Isles. Although thatch has always been used 如图，抛物线与x轴交于A（-1，0），B （3，0）两点，与y轴交点C（0，-3）（1）求抛物线的解析式以及顶点D的坐标；（2）若M是线段BD的中点，连接CM，猜想线段CM与线段BD之间有怎样的数量关系，并证明你的猜想；（3）在坐标轴上是否存在点P，使得以P、A、C为顶点的三角形与△BCD相似？若存在，请直接写出点P的坐标；若不存在，请说明理由．