证明(1):因为四边形AA1C1C是菱形,所以有AA1=A1C1=C1C=CA=1. 从而知△AA1B是等边三角形. 设D是AA1的中点、连接BD,C1D, 则BD⊥AA1,由 = . 知C1到AA1的距离为 .∠AA1C1=60°, 所以△AA1C1是等边三角形, 且C1D⊥AA1,所以AA1⊥平面BC1D. 又BC1 平面BC1D,故AA1⊥BC1. (2)由(1)知BD⊥AA1,又侧面ABB1A1⊥侧面AA1C1C, 所以BD⊥平面AA1C1C, 即B到平面AA1C1C的距离为BD. 又 =![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021232505-37656.png) ![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021232506-28541.png) ,BD= . 所以 = =![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021232507-33102.png) BD= × × = . 故三棱锥A1﹣ABC的体积为 . |