题目内容
①点P在△ABC所在的平面内,且
;②点P为△ABC内的一点,且使得
取得最小值;③点P是△ABC所在平面内一点,且
,上述三个点P中,是△ABC的重心的有( )
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232044354041149.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435420696.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435498656.png)
A.0个 | B.1个 | C.2个 | D.3个 |
D
①根据向量加法的平行四平行法则,由
得,
在
的中线上,由
得,
在
边的中线上,所以
是两条中线的交点,
是△ABC的重心。②设
,
,
,
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435420696.png)
=
当
时,
取最小值,此时
恰是重心。
③设
是线段
的中点,由向量的加法法则的
,代入
得
,这说明
共线,且
把中线CD分为2:1两段,
是重心。△ABC的重心的有3个
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435700902.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435716289.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435763395.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435794884.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435716289.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435903402.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435716289.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435716289.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435966639.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435981666.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204436106671.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204436122615.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435420696.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232044361531746.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232044361844073.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232044362001002.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435420696.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435716289.png)
③设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204436246315.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204436278396.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204436293689.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204436324732.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204436340604.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204436371482.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435716289.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823204435716289.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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