(2010•永州)探究问题:
(1)阅读理【解析】
①如图(A),在已知△ABC所在平面上存在一点P,使它到三角形顶点的距离之和最小,则称点P为△ABC的费马点,此时PA+PB+PC的值为△ABC的费马距离;
②如图(B),若四边形ABCD的四个顶点在同一圆上,则有AB•CD+BC•DA=AC•BD.此为托勒密定理;
![](https://img.xintiku.com/upload/37/33/3733f7359ff7671f33219eb572d37994.png)
(2)知识迁移:
①请你利用托勒密定理,解决如下问题:
如图(C),已知点P为等边△ABC外接圆的
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAAWCAYAAAArdgcFAAAAAXNSR0IArs4c6QAAAARnQU1BAACx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)
上任意一点.求证:PB+PC=PA;
②根据(2)①的结论,我们有如下探寻△ABC(其中∠A、∠B、∠C均小于120°)的费马点和费马距离的方法:
第一步:如图(D),在△ABC的外部以BC为边长作等边△BCD及其外接圆;
第二步:在
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAAWCAYAAAArdgcFAAAAAXNSR0IArs4c6QAAAARnQU1BAACx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)
上任取一点P′,连接P′A、P′B、P′C、P′D.易知P′A+P′B+P′C=P′A+(P′B+P′C)=P′A+______;
第三步:请你根据(1)①中定义,在图(D)中找出△ABC的费马点P,并请指出线段______的长度即为△ABC的费马距离.
![](https://img.xintiku.com/upload/a3/c9/a3c92b6f7367026d2ede57cdcead8f1a.png)
(3)知识应用:
2010年4月,我国西南地区出现了罕见的持续干旱现象,许多村庄出现了人、畜饮水困难,为解决老百姓的饮水问题,解放军某部来到云南某地打井取水.
已知三村庄A、B、C构成了如图(E)所示的△ABC(其中∠A、∠B、∠C均小于120°),现选取一点P打水井,使从水井P到三村庄A、B、C所铺设的输水管总长度最小,求输水管总长度的最小值.