lclblc
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共回答了17个问题采纳率:94.1% 举报
证明略
方法一 如图所示,作PM∥AB交BE于M,作QN∥AB交BC于N,连接MN.
∵正方形ABCD和正方形ABEF有公共边AB,∴AE=BD.
又∵AP=DQ,∴PE=QB,
又∵PM∥AB∥QN,
![](https://img.yulucn.com/upload/5/f5/5f58400c55c548ca183880562e8e9197_thumb.jpg)
∴
![](https://img.yulucn.com/upload/b/4b/b4b6b1c1522ce5b26d43d03cee240320_thumb.jpg)
,
![](https://img.yulucn.com/upload/7/72/7721b126dd45ee79f8edd8949dd03f93_thumb.jpg)
,
![](https://img.yulucn.com/upload/d/16/d16094268bb4b1886aa68b86c42ba5ea_thumb.jpg)
,∴PM QN,
∴四边形PMNQ为平行四边形,∴PQ∥MN.
又MN
![](https://img.yulucn.com/upload/0/e3/0e3a812be06dfbf627d440cfe54063f5_thumb.jpg)
平面BCE,PQ
![](https://img.yulucn.com/upload/f/34/f34f50086d18e50744a7943b98adfb34_thumb.jpg)
平面BCE,
∴PQ∥平面BCE.
方法二 如图所示,连接AQ,并延长交BC于K,连接EK,
∵AE=BD,AP=DQ,
∴PE=BQ,
∴
![](https://img.yulucn.com/upload/2/00/200a565319e8ff492948c663fc12a17a_thumb.jpg)
=
![](https://img.yulucn.com/upload/7/df/7dfa841312c58ece1f29d36d6b410a1b_thumb.jpg)
①
又∵AD∥BK,∴
![](https://img.yulucn.com/upload/7/df/7dfa841312c58ece1f29d36d6b410a1b_thumb.jpg)
=
![](https://img.yulucn.com/upload/9/26/9264af0f4bcd684014f65e6da8fb4e9d_thumb.jpg)
②
由①②得
![](https://img.yulucn.com/upload/2/00/200a565319e8ff492948c663fc12a17a_thumb.jpg)
=
![](https://img.yulucn.com/upload/9/26/9264af0f4bcd684014f65e6da8fb4e9d_thumb.jpg)
,∴PQ∥EK.
又PQ
![](https://img.yulucn.com/upload/f/34/f34f50086d18e50744a7943b98adfb34_thumb.jpg)
平面BCE,EK
![](https://img.yulucn.com/upload/0/e3/0e3a812be06dfbf627d440cfe54063f5_thumb.jpg)
平面BCE,
∴PQ∥平面BCE.
方法三 如图所示,在平面ABEF内,过点P作PM∥BE,交AB于点M,
连接QM.
∵PM∥BE,PM
![](https://img.yulucn.com/upload/f/34/f34f50086d18e50744a7943b98adfb34_thumb.jpg)
平面BCE,
即PM∥平面BCE,
∴
![](https://img.yulucn.com/upload/2/00/200a565319e8ff492948c663fc12a17a_thumb.jpg)
=
![](https://img.yulucn.com/upload/b/6d/b6d38c0a153fd6a84dc3b7866a2141a5_thumb.jpg)
①
又∵AP=DQ,∴PE=BQ,
∴
![](https://img.yulucn.com/upload/2/00/200a565319e8ff492948c663fc12a17a_thumb.jpg)
=
![](https://img.yulucn.com/upload/7/df/7dfa841312c58ece1f29d36d6b410a1b_thumb.jpg)
②
由①②得
![](https://img.yulucn.com/upload/b/6d/b6d38c0a153fd6a84dc3b7866a2141a5_thumb.jpg)
=
![](https://img.yulucn.com/upload/7/df/7dfa841312c58ece1f29d36d6b410a1b_thumb.jpg)
,∴MQ∥AD,
∴MQ∥BC,又∵MQ
![](https://img.yulucn.com/upload/f/34/f34f50086d18e50744a7943b98adfb34_thumb.jpg)
平面BCE,∴MQ∥平面BCE.
又∵PM∩MQ=M,∴平面PMQ∥平面BCE,
PQ
![](https://img.yulucn.com/upload/0/e3/0e3a812be06dfbf627d440cfe54063f5_thumb.jpg)
平面PMQ,∴PQ∥平面BCE.
1年前
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