Question

四棱柱ABCD-A1B1C1D1的底面是菱形,且角C1CB=角C1CD=角BCD=60度.证明CC1垂直BD

Answer 1

令AC∩BD＝O.

∵四棱柱ABCD－AB1C1D1的底面ABCD是菱形,∴BC＝DC.

∵CC1＝CC1、∠C1CB＝C1CD、BC＝DC,∴△C1CB≌△C1CD,∴C1B＝C1D,∴BD⊥C1O.

∵ABCD是菱形,∴BD⊥CO.

∵BD⊥C1O、BD⊥CO、C1O∩CO＝O,∴BD⊥平面C1CO,∴CC1⊥BD.