(16分)如圖,四棱錐
S-ABCD的底面是正方形,每條側(cè)棱的長都是地面邊長的
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倍,
P為側(cè)棱SD上的點。

(Ⅰ)求證:
AC⊥
SD;
(Ⅱ)若
SD⊥
平面PAC,求二面角
P-AC-D的大小
(Ⅲ)在(Ⅱ)的條件下,側(cè)棱SC上是否存在一點E,使得BE∥平
面PAC。若存在,求SE:EC的值

;若不存在,試說明理由。
解法一:
(
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Ⅰ)連BD,設AC交BD于O,由題意

。在正方形ABCD中,

,所以
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,得
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.
(Ⅱ)設正方形邊長
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,則
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。
又
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,所以
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,
連
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,由(Ⅰ)知
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,所以
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,
且
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,所以
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是二面角

的平面角。
由
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,知
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,所以

,
即二面角

的大小為
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。
(Ⅲ)在棱SC上存在一點E,使
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由(Ⅱ)可得
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,故可在
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上取一點
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,使
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,過
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作
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的平行線與
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的交點即為
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。連BN。在
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中知
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,又由于
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,故平面
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,得
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,由于
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,故
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.
解法二:
(Ⅰ);連
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,設
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交于
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于
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,由題意知
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.以O為坐標原點,
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分別為
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軸、
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軸
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、
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軸正方向,建立坐標系
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如圖。
設
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底面邊長為
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,則高

。
于是
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

故
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從而
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(Ⅱ)由題設知,平面

的一個法向量

,平面

的一個法向量

,設所求二面角為

,則

,所求二面角的大小為


(Ⅲ)在棱

上存在一點

使

.
由(Ⅱ)知

是平面

的一個法向量,
且
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設
則
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而
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即當

時,
而

不在平面
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內(nèi),故
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
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中點
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
;
(2)求證:

;
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
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
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
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