帕普斯定理

设

${\displaystyle \alpha =\langle U\times V,X\times W,Y\times Z\rangle }$ 

${\displaystyle \beta =\langle U\times Z,X\times V,Y\times W\rangle }$ 

${\displaystyle \gamma =\langle U\times W,X\times Z,Y\times V\rangle }$ 

我们需要证明如果${\displaystyle \alpha }$  = 0且${\displaystyle \beta }$  = 0，则${\displaystyle \gamma }$  = 0。

第一步

利用恒等式

${\displaystyle \langle A,B,C\rangle =\langle C,A,B\rangle =\langle B,C,A\rangle }$ 

可将${\displaystyle \alpha }$ 、${\displaystyle \beta }$ 及${\displaystyle \gamma }$ 表述为以下形式：

${\displaystyle \alpha =\langle U\times V,X\times W,Y\times Z\rangle }$ 

${\displaystyle \beta =\langle Y\times W,U\times Z,X\times V\rangle }$ 

${\displaystyle \gamma =\langle X\times Z,Y\times V,U\times W\rangle }$ 

第二步

利用恒等式

${\displaystyle \langle A,B,C\rangle =A\cdot (B\times C)}$ 

${\displaystyle A\times (B\times C)=(A\cdot C)B-(A\cdot B)C}$ 

可得

${\displaystyle \alpha =(U\times V)\cdot ((X\times W)\times (Y\times Z))}$ 

${\displaystyle \beta =(Y\times W)\cdot ((U\times Z)\times (X\times V))}$ 

${\displaystyle \gamma =(X\times Z)\cdot ((Y\times V)\times (U\times W))}$ 

以及

${\displaystyle \alpha =(U\times V)\cdot (\langle X,W,Z\rangle Y-\langle X,W,Y\rangle Z)}$ 

${\displaystyle \beta =(Y\times W)\cdot (\langle U,Z,V\rangle X-\langle U,Z,X\rangle V)}$ 

${\displaystyle \gamma =(X\times Z)\cdot (\langle Y,V,W\rangle U-\langle Y,V,U\rangle W)}$ 

第三步

利用数量积的分配律，可得：

${\displaystyle \alpha =\langle X,W,Z\rangle \langle U,V,Y\rangle -\langle X,W,Y\rangle \langle U,V,Z\rangle }$ 

${\displaystyle \beta =\langle U,Z,V\rangle \langle Y,W,X\rangle -\langle U,Z,X\rangle \langle Y,W,V\rangle }$ 

${\displaystyle \gamma =\langle Y,V,W\rangle \langle X,Z,U\rangle -\langle Y,V,U\rangle \langle X,Z,W\rangle }$ 

第四步

利用恒等式

${\displaystyle \langle A,B,C\rangle =\langle C,A,B\rangle =\langle B,C,A\rangle }$ 

${\displaystyle \langle A,B,C\rangle =-\langle A,C,B\rangle =-\langle C,B,A\rangle =-\langle B,A,C\rangle }$ 

可得：

${\displaystyle \alpha =\langle X,W,Z\rangle \langle U,V,Y\rangle -\langle X,W,Y\rangle \langle U,V,Z\rangle }$ 

${\displaystyle \beta =-\langle U,Z,X\rangle \langle Y,W,V\rangle +\langle X,W,Y\rangle \langle U,V,Z\rangle }$ 

${\displaystyle \gamma =\langle U,Z,X\rangle \langle Y,W,V\rangle -\langle X,W,Z\rangle \langle U,V,Y\rangle }$ 

第五步

把这些等式相加，得：

${\displaystyle \alpha +\beta +\gamma =0}$ 

${\displaystyle \gamma =-(\alpha +\beta )}$ 

因此，如果${\displaystyle \alpha }$  = 0且${\displaystyle \beta }$  = 0，则${\displaystyle \gamma }$  = 0。

证毕。

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