解:(I)证明:连结BD,在菱形ABCD中,∠BAD=60
0,∴△ABD为正三角形,
∵E为AB的中点,∴ED⊥AB (1分)
在直六面体ABCD-A
1B
1C
1D
1中:
平面ABB
1A
1⊥平面ABCD且交于AB,∵ED
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151455911135.gif)
面ABCD∴ED⊥面ABB
1A
1(3分)
∴平面A
1ED⊥平面ABB
1A
1(4分)
(II)解:由(I)知:ED⊥面ABB
1A
1∵A
1E
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151455911135.gif)
面ABB
1A
1∴A
1E⊥ED
又在直平行六面体ABCD-A
1B
1C
1D
1中:AA
1⊥面ABCD,
由三垂线定理的逆定理知:AE⊥ED,∴∠A
1EA=60
0(5分)
取BB
1的中点F,连EF.AB
1,则EF
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151455942141.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151455973329.gif)
,在直平行六面体ABCD-A
1B
1C
1D
1中:AB
1![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151455942141.gif)
DC
1∴EF
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151455942141.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151456020341.gif)
∴E.F.C
1、D四点共面(6分)
∵ED⊥面ABB
1A
1且EF
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151455911135.gif)
面ABB
1A
1∴EF⊥ED∴∠A
1EF为二面角A
1-ED-C
1的平面角(7分)
在Rt△A
1AE中:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151456051658.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151456067687.gif)
在Rt△EBF中:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151456098709.gif)
,
在Rt△A
1B
1F中:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151456114783.gif)
∴在Rt△A
1EF中:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231514561291321.gif)
,∴二面角A
1-ED-C
1的余弦值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151455864281.gif)
(9分)
(III)过F作FG⊥A
1E交A
1E于G点∵平面A
1ED⊥面ABB
1A
1且平面A
1ED∩面ABB
1A
1=A
1E∴FG⊥平面A
1ED,
即:FG是点F到平面A
1ED的距离(11分)
在Rt△EGF中:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151456160608.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151456176656.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231514561921145.gif)
(13分)
∵EF
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151455942141.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151456363333.gif)
且E.D∈面A
1ED∴点C
1到平面A
1ED的距离为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151455895439.gif)
(14分)