Question

如圖,已知α∩β=EF,AC、BD是異面直線,且AC⊥β, BD⊥α,AB是AC、BD的公垂線,求證:AB∥EF.

Answer 1

證明:過點A引平面α的垂線AH,垂足為H,則AC和AH確定一平面ACH.

∵BD⊥α,

∴BD∥AH.

又∵AB⊥BD,

∴AB⊥AH,

AB⊥AC.

∴AB⊥平面ACH.

又∵BD⊥α,AC⊥β,EF=α∩β,

∴BD⊥EF,AC⊥EF.

又∵BD∥AH,

∴AH⊥EF.

∴EF⊥平面ACH.

∴AB∥EF.