Question

正三棱锥ABC-A1B1C1,D为CC1上的中点,AB=AA1,证明BD垂直于AB1

Answer 1

ABC－A1B1C1是正三棱柱,且AB＝AA1,∴ABBA1是正方形,∴AB1⊥A1B.

∵AC＝B1C1,CD＝C1D,∠ACD＝∠B1C1D＝90°,∴△ACD≌△B1C1D,∴AD＝B1D.

令AB1与A1B的交点为E,显然有：AE＝B1E,而AD＝B1D,∴AB1⊥DE.

由AB1⊥A1B,AB1⊥DE,A1B∩DE＝E,∴AB1⊥面A1BD,∴BD⊥AB1.