问题
解答题
已知E、F、G、H分别是空间四边形ABCD的边AB,BC,CD,DA的中点.
(1)证明E,F,G,H四点共面;
(2)证明BD∥平面EFGH.
答案
如图,连结EG,BG.
(1)∵BG是△BCD的中线,可得
=BG
(1 2
+BC
)BD
∴
=EG
+EB
=BG
+EB
(1 2
+BC
)BD
∵
=BF 1 2
,BC
=EH 1 2 BD
∴
=EG
+EB
+BF
=EH
+EF
,EH
根据向量共面的充要条件,得
可得E,F,G,H四点共面.
(2)∵
=EH
+EA
,AH
=EH
+EG GH
∴
=BD
+BA
=2AD
+2EA
=2AH
=2(EH
+EG
)=2GH
+2EG
,GH
结合
,EG
不共线,可得GH
与BD
,EG
共面.GH
又∵BD?面EFGH,∴BD∥面EFGH.