如圖，在四棱錐P－ABCD中，底面ABCD是菱形，∠BAD＝，AB＝2，PA＝1，PA⊥平面ABCD，E是PC的中點，F是AB的中點．

（1）求證：BE∥平面PDF；
（2）求證：平面PDF⊥平面PAB；
（3）求三棱錐P－DEF的體積．
⑴略；⑵略；⑶解析:
（1）取PD的中點為M，連結(jié)ME，MF，因為E是PC的中點，所以ME是△PCD的中位線．所以ME∥CD，ME＝．又因為F是AB的中點，且由于ABCD是菱形，AB∥CD，AB＝CD，所以ME∥FB，且ME＝FB．所以四邊形MEBF是平行四邊形，所以BE∥MF．
連結(jié)BD，因為BE平面PDF，MF平面PDF，所以BE∥平面PDF．
（2）因為PA⊥平面ABCD，DF平面ABCD，所以DF⊥PA．
連結(jié)BD，因為底面ABCD是菱形，∠BAD＝，所以△DAB為正三角形．
因為F是AB的中點，所以DF⊥AB．
因為PA，AB是平面PAB內(nèi)的兩條相交直線，所以DF⊥平面PAB．
因為DF平面PDF，所以平面PDF⊥平面PAB．
（3）因為E是PC的中點，所以點P到平面EFD的距離與點C到平面EFD的距離相等，故＝＝，又＝×2×＝，E到平面DFC的距離h＝＝，所以＝××＝．
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