Question

在三棱柱ABC-A1B1C1中,AB=AC=2AA1,∠BAA1=∠CAA1=60°,D,E分别是AB,A1C中点,求证

Answer 1

设AA1=1,AB=AC=2,

在三角形A1AB中,〈A1AB=60°,

根据余弦定理,A1B=√3,

AA1^2+A1B^2=4,AB^2=4,

根据勾股逆定理,△A1AB是RT△,

〈AA1B=〈B1BA1=90°,则BB1⊥A1B,

同理,〈AA1C=90°,AA1⊥A1C,BB1//AA1,则BB1⊥A1C,

A1C∩A1B=A1,

∴BB1⊥平面A1BC.