Question

正三棱柱A1B1C1-ABC中,点D是BC中点,BC=根号2BB1,设B1D交BC1=F,求证：BC1与平面AB1D垂直

Answer 1

设BC1,B1D交于E,tan∠C1BB1＝√2＝tan∠BDB1.∴∠C1BB1＝∠BDB1.

∴∠C1BB1+∠BB1D＝∠BDB1+∠BB1D＝90°.∴∠B1EB＝180°-90°＝90°.

∴BC1⊥B1D.

又AD⊥AA1‖CC1.∴AD⊥CC1.而AD⊥BC（三合一）,∴AD⊥平面BCC1B1.

∴AD⊥BC1.

∵BC1⊥B1D.BC1⊥AD.∴BC1⊥平面AB1D