# 帕普斯定理

（1）U与V的交点，X与W的交点，Y与Z的交点共线，且
（2）U与Z的交点，X与V的交点，Y与W的交点共线，

${\displaystyle \langle U\times V,X\times W,Y\times Z\rangle =0}$

${\displaystyle \langle U\times Z,X\times V,Y\times W\rangle =0}$

${\displaystyle \langle U\times W,X\times Z,Y\times V\rangle =0.}$

## 证明

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

### 第一步

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

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

### 第二步

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

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

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

### 第三步

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

### 第四步

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

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

### 第五步

${\displaystyle \alpha +\beta +\gamma =0}$
${\displaystyle \gamma =-(\alpha +\beta )}$