(1)证明:∵BC⊥CC1,BC⊥AC,AC∩CC1=C,∴BC⊥平面ACC1A1, C1D⊂平面ACC1A1,∴BC⊥C1D, A1C1=A1D=AD=AC,∴∠A1DC1=∠ADC=, ∴∠C1DC=,即C1D⊥DC, 又BD∩CD=C,∴C1D⊥平面BDC, (2)三棱锥C-BC1D即三棱锥C1-BCD,由(1)知BC⊥CD, CD=a,BC=a ∴△BCD的面积S=×BC×CD=a2, 由(1)知,C1D是三棱锥C1-BCD底面BDC上的高, ∴体积V=Sh=×S×C1D=×a2×a=a3. |