如图,在正方体ABCD-A1B1C1D1中,点N在BD上,点M在B1C上,且CM=DN,求证:MN∥平面AA1B1B.
问题描述:
如图,在正方体ABCD-A1B1C1D1中,点N在BD上,点M在B1C上,且CM=DN,求证:MN∥平面AA1B1B.
答
解:如图,作MP∥BB1,交BC于点P,连接NP.
∵MP∥BB1,∴
=CM MB1
.CP PB
∵BD=B1C,DN=CM,∴B1M=BN.
∵
=CM MB1
,∴DN NB
=CP PB
.DN NB
∴NP∥CD∥AB.∴面MNP∥面AA1B1B.
∴MN∥面AA1B1B.