1)已知,如图甲,MN是平行四边形ABCD外的一条直线,AA’、BB’、CC’、DD’都垂直于MN,A’、B’、C’、D’为垂足.求证AA’+CC'=BB'+DD'. (2)若直线MN向上移动,使
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![1)已知,如图甲,MN是平行四边形ABCD外的一条直线,AA’、BB’、CC’、DD’都垂直于MN,A’、B’、C’、D’为垂足.求证AA’+CC'=BB'+DD'. (2)若直线MN向上移动,使](/uploads/image/z/2449547-35-7.jpg?t=1%29%E5%B7%B2%E7%9F%A5%2C%E5%A6%82%E5%9B%BE%E7%94%B2%2CMN%E6%98%AF%E5%B9%B3%E8%A1%8C%E5%9B%9B%E8%BE%B9%E5%BD%A2ABCD%E5%A4%96%E7%9A%84%E4%B8%80%E6%9D%A1%E7%9B%B4%E7%BA%BF%2CAA%E2%80%99%E3%80%81BB%E2%80%99%E3%80%81CC%E2%80%99%E3%80%81DD%E2%80%99%E9%83%BD%E5%9E%82%E7%9B%B4%E4%BA%8EMN%2CA%E2%80%99%E3%80%81B%E2%80%99%E3%80%81C%E2%80%99%E3%80%81D%E2%80%99%E4%B8%BA%E5%9E%82%E8%B6%B3.%E6%B1%82%E8%AF%81AA%E2%80%99%2BCC%27%3DBB%27%2BDD%27.+++++++++++++++++++++++++++++++++++++++++++++++++++%EF%BC%882%EF%BC%89%E8%8B%A5%E7%9B%B4%E7%BA%BFMN%E5%90%91%E4%B8%8A%E7%A7%BB%E5%8A%A8%2C%E4%BD%BF)
1)已知,如图甲,MN是平行四边形ABCD外的一条直线,AA’、BB’、CC’、DD’都垂直于MN,A’、B’、C’、D’为垂足.求证AA’+CC'=BB'+DD'. (2)若直线MN向上移动,使
1)已知,如图甲,MN是平行四边形ABCD外的一条直线,AA’、BB’、CC’、DD’都垂直于MN,A’、B’、C’、D’为垂足.求证AA’+CC'=BB'+DD'.
(2)若直线MN向上移动,使点C在直线一侧,A、B、D三点在直线另一侧(如图乙),则垂线段AA'、BB'、CC'、DD'之间存在什么关系?先对结论进行猜想,然后加以证明.
(3)MN继续向上移动,当过B点时;在B、D之间时;过D点时;在A、D之间时;过A点时.AA'、BB'、CC'、DD'各存在什么关系,得出结论并加以证明.
1)已知,如图甲,MN是平行四边形ABCD外的一条直线,AA’、BB’、CC’、DD’都垂直于MN,A’、B’、C’、D’为垂足.求证AA’+CC'=BB'+DD'. (2)若直线MN向上移动,使
如图,过点B'做B'B''∥AB交AA'于点B'',过点C做CC''∥CD交DD'于点C''
∴四边形ABB'B''和四边形CC'C''D是平行四边形,且B'B''∥C'C''
∴B'B''=AB=CD=C'C'',BB'=AB'',CC'=DC'',∠B''B'A'=∠C''C'D'
又∵∠B''A'B'=∠C''D'C'=90º
∴△B''B'A'≌△C''C'D'
∴A'B''=D'C''
∴A'B''+AB''+CC'=D'C''+B'B+C''D
∴AA'+CC'=BB'+DD'
(2)CC'+BB'+DD'=AA'
如图,过点B'做B'B''∥AB交AA'于点B'',过点C做CC''∥CD交D'D的延长线于点C''
∴四边形ABB'B''和四边形CC'C''D是平行四边形,且B'B''∥C'C''
∴B'B''=AB=CD=C'C'',BB'=AB'',CC'=DC'',∠B''B'A'=∠C''C'D'
又∵∠B''A'B'=∠C''D'C'=90º
∴△B''B'A'≌△C''C'D'
∴B''A'=C''D'
∵B''A'=AA'-AB''=AA'-BB',C''D'=C''D+DD'=CC'+DD'
∴AA'-BB'=C''D+DD'=CC'+DD'
∴AA'=BB'+C''D+DD'
(3)AA'=CC'+DD'
如图,过点C做CD''∥CD交D'D的延长线于点D''
∴四边形CC'D''D'是平行四边形,且C'D''∥CD∥AB
∴AB=CD=C'D'',∠ABA'=∠D''C'D',且CC'=DD''
又∵∠AA'B=∠D''D'C'=90º
∴△ABA'≌△D''C'D'
∴AA'=D'D''
∴AA'=DD''+DD'
∴AA'=CC'+DD'
在B、D之间时,AA'+CC'=BB'+DD'
如图,同理可证,△B'B''A''≌△C'D''D'
∴A'B''=D'D''
∴AA'-BB'=CC'-DD'
∴AA'+CC'=BB'+DD'
在A、D之间时与在B,D之间相同
过A点时,CC'=BB'+DD'
如图,过D'点做D'C''∥CD交CC'于点C''
则△ABB'≌△D'C''C'
∴CC''=BB'
∴CC'-CC''=CC'-DD'=BB'
∴CC'=BB'+DD'