分析:由题意先证明ADEF为平行四边形,再由向量加法的平行四边形法则得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203114423.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203130402.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203145415.png)
,同理求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203176403.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203192422.png)
再把三个式子加起来,重新组合利用向量加法的首尾相连法则求解.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231652032236990.png)
证明:连接DE、EF、FD,如图,
∵D、E、F分别是△ABC三边的中点,
∴EF∥AD,DE∥AF,
∴四边形ADEF为平行四边形,
由向量加法的平行四边形法则,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203114423.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203130402.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203145415.png)
①,
同理在平行四边形BEFD中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203317412.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203348403.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203176403.png)
②,
在平行四边形CFDE在中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203395415.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203410427.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203192422.png)
③,将①②③相加,得
(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203145415.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203176403.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203192422.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203114423.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203130402.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203317412.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203348403.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203410427.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203395415.png)
=(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203130402.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203348403.png)
)+(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203114423.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203410427.png)
)+(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203317412.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823165203395415.png)
)
=0
故选D.