如图，在正三棱柱（底面为正三角形的直棱柱）ABC-A1B1C1中，F是A1C1的中点．（1）求证：BC1∥平面AFB1； 如图，在正三棱柱（底面为正三角形的直棱柱）ABC-A 1 B...
\u5982\u56fe\u6240\u793a,\u5728\u6b63\u4e09\u68f1\u67f1(\u5e95\u9762\u662f\u6b63\u4e09\u89d2\u5f62\u7684\u76f4\u68f1\u67f1)ABC-A1B1C1\u4e2d,\u5df2\u77e5D\u662fBC\u7684\u4e2d\u70b9,\u6c42\u8bc1A1D\u5782\u76f4BC\u94fe\u63a5A1B\uff0cA1C\u3002\u77e5\u9053A1B=A1C\u3002\u5373\u4e09\u89d2\u5f62A1BC\u4e3a\u7b49\u8170\u4e09\u89d2\u5f62\uff0c\u800cA1D\u4e3a\u5176\u5e95\u8fb9\u4e0a\u7684\u4e2d\u7ebf\u3002\u6545\u77e5\uff1aA1D\u5782\u76f4\u4e8eBC\u3002
\u8bc1\u660e\uff1a\uff081\uff09\u8fde\u63a5A 1 B\u4e0eAB 1 \u4ea4\u4e8e\u70b9E\uff0c\u8fde\u63a5EF\uff0e\u5728\u6b63 \u4e09\u68f1\u67f1ABC-A 1 B 1 C 1 \u4e2d\uff0c\u53ef\u5f97\u56db\u8fb9\u5f62ABB 1 A 1 \u662f\u77e9\u5f62\uff0c\u2234A 1 E=EB\uff0e\u53c8A 1 F=FC 1 \uff0c\u2234EF \u2225 BC 1 \uff0e\u2235EF?\u5e73\u9762AB 1 F\uff0cBC 1 ?\u5e73\u9762AB 1 F\uff0c\u2234BC 1 \u2225 \u5e73\u9762AFB 1 \uff1b\uff082\uff09\u7531\u6b63\u4e09\u68f1\u67f1ABC-A 1 B 1 C 1 \u4e2d\uff0c\u53ef\u5f97AA 1 \u22a5\u5e95\u9762A 1 B 1 C 1 \uff0c\u2234AA 1 \u22a5B 1 F\uff0e\u7531F\u662f\u6b63\u25b3A 1 B 1 C 1 \u7684A 1 C 1 \u7684\u4e2d\u70b9\uff0c\u2234B 1 F\u22a5A 1 C 1 \uff0e\u53c8A 1 A\u2229A 1 C 1 =A 1 \uff0c\u2234B 1 F\u22a5\u5e73\u9762ACC 1 A 1 \uff0c\u2234\u5e73\u9762AFB 1 \u22a5\u5e73\u9762ACC 1 A 1 \uff0e
证明：（1）连接A1B与AB1交于点E，连接EF．在正三棱柱ABC-A1B1C1中，可得四边形ABB1A1是矩形，∴A1E=EB．又A1F=FC1，∴EF∥BC1．
∵EF?平面AB1F，BC1?平面AB1F，
∴BC1∥平面AFB1；
（2）由正三棱柱ABC-A1B1C1中，可得AA1⊥底面A1B1C1，∴AA1⊥B1F．
由F是正△A1B1C1的A1C1的中点，∴B1F⊥A1C1．
又A1A∩A1C1=A1，∴B1F⊥平面ACC1A1，
∴平面AFB1⊥平面ACC1A1．
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