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.

: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.