题目内容
(本小题满分14分)
如图,在四棱锥
中,
,
,底面
是菱形,且
,
为
的中点.
(Ⅰ)证明:
平面
;
(Ⅱ)侧棱
上是否存在点
,使得
平面
?并证明你的结论.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231601598751583.gif)
如图,在四棱锥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159656428.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159672424.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159687392.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159703389.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159734457.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159750197.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159765226.gif)
(Ⅰ)证明:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159781244.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159797354.gif)
(Ⅱ)侧棱
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159812227.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159843194.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159843241.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159797354.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231601598751583.gif)
略
(Ⅰ)
是菱形,
,
,
为正三角形, ………………2分
又
为
的中点,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200015439.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231602000312151.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200109553.gif)
,
则有
,
,
,
………………4分
又
,
底面
,
由
,
,
,
平面
…………7分
(Ⅱ)
为侧棱
的中点时,
平面
. ………………8分
证法一:设
为
的中点,连
,则
是
的中位线,
且
,又
且
,
且
,
四边形
为平行四边形, ……………11分
,
平面
,
平面
,
平面
. ………………14分
证法二:设
为
的中点,连
,则
是
的中位线,
,
平面
,
平面
,
平面
. ………………10分
同理,由
,得
平面
.
又
,
平面
平面
, ………………12分
又
平面
,
平面
. ……………14分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159906325.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159921491.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159953536.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159968416.gif)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159984204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160159999242.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200015439.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231602000312151.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200109553.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200218444.gif)
则有
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200233547.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200265556.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200280326.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200296319.gif)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200327398.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200343258.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200374303.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200389334.gif)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200405439.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200436326.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200452491.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200467264.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200499373.gif)
(Ⅱ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200514200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200530239.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200545258.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200499373.gif)
证法一:设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200592211.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200608232.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200623518.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200639347.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200670394.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200686457.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200717528.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200733263.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200795508.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200857448.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200873434.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200889128.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200920414.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201013450.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201029388.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200499373.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201060268.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200499373.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201107265.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200499373.gif)
证法二:设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201138327.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201154235.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201169522.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201185358.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200670394.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201247466.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201279266.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200499373.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201325405.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200499373.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201372398.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200499373.gif)
同理,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201403453.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201419388.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200499373.gif)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201528573.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200889128.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201575422.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200499373.gif)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201622268.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201637402.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160201107265.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823160200499373.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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