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'
