如圖,△
ABC中,
AC=
BC=
AB,
ABED是邊長為1的正方形,EB⊥底面
ABC,若
G,
F分別是
EC,
BD的中點(diǎn).
(1)求證:
GF∥底面
ABC;
(2)求證:
AC⊥平面
EBC;

(1)先證明GF//AC,再根據(jù)線面平行的判定定理即可證明
(2)先證BE⊥AC,再證AC⊥BC,根據(jù)線面垂直的判定定理即可證明
試題分析:(1)連接
AE,如下圖所示.
∵
ADEB為正方形,∴
AE∩
BD=
F,且
F是
AE的中點(diǎn),
又
G是
EC的中點(diǎn),∴
GF∥AC,
又
AC?平面
ABC,
GF
平面
ABC,
∴
GF∥平面
ABC.
(2)∵
ADEB為正方形,∴
EB⊥
AB,
又∵平面
ABED⊥平面
ABC,平面
ABED∩平面
ABC=
AB,
EB?平面
ABED,
∴
BE⊥平面
ABC,∴
BE⊥
AC.
又∵
AC=
BC=
AB,∴
CA2+
CB2=
AB2,∴
AC⊥
BC.
又∵
BC∩
BE=
B,∴
AC⊥平面
BCE.
點(diǎn)評:要證明線面平行與線面垂直,就要緊扣相應(yīng)的判定定理和性質(zhì)定理,定理中要求的條件缺一不可.
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