hdu 4773 几何反演 线->圆

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给定不相交的两个圆以及圆外一点，找一个经过给定点的圆与其他两个圆相切

首先来看反演变换，首先是给定一个圆圆心为O，半径为R

1、圆外一点P与圆内一点P‘会一一对应的反演OP*OP'=R*R

2、经过O的圆，反演后成为不经过O的一条直线

3、不经过O的圆，反演后成为另一个圆，且圆心并不对应

4、不经过O的直线反演后成为一个经过O的圆

5、过O的直线反演后不变

那么这道题就把两个圆反演之后求两圆的公切线，然后反演回去，就会成为一个过O的圆，且与另两圆相切（由反演的过程可看出）

求反演后的圆的圆心，可以将过O的直线所在的那条直径两端点反演，然后再求圆心.

对于一个不过反演中心的圆，怎样求它的反形圆？

很容易知道我们只需要求出反形圆的圆心和半径就可以了。

对于上图我们设圆C1的半径为，C2的半径为，反演半径为

那么根据反演的定义有：

那么，消去得到：

这样我们就得到了反形圆的半径，那么还要求反形圆的圆心。

由于C1和O两点的坐标已知，而且我们知道O,C1，C2位于同一直线上，那么很明显对于C2的坐标，我们可以这样计算：

设O的坐标为，C1的坐标为，C2的坐标为

那么有：

至于由上面解处可以很容易得到，这样我们就完成了圆的反演变换。

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