# 笛沙格定理

## 证明

(A.B)∩(a.b)，(A.C)∩(a.c)，(B.C)∩(b.c)共线。

${\displaystyle \langle A\times a,B\times b,C\times c\rangle =0}$

${\displaystyle \langle (A\times B)\times (a\times b),(A\times C)\times (a\times c),(B\times C)\times (b\times c)\rangle =0.}$

### 第一个重述

${\displaystyle X\times (Y\times Z)}$

${\displaystyle Y(X\cdot Z)-Z(X\cdot Y),}$

${\displaystyle (X\times Y)\times (Z\times W)=\langle X,Y,W\rangle Z-\langle X,Y,Z\rangle W.}$

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

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

${\displaystyle \langle A\times C,a\times c,b\times c\rangle \langle A\times B,a\times b,B\times C\rangle =\langle A\times C,a\times c,B\times C\rangle \langle A\times B,a\times b,b\times c\rangle .}$

### 第二个重述

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

${\displaystyle \langle C,a,c\rangle \langle b,A,B\rangle =\langle c,A,C\rangle \langle B,a,b\rangle .}$

### 第三个重述

${\displaystyle M_{ij}=u_{i}\cdot v_{j},\qquad \langle u_{1},u_{2},u_{3}\rangle \langle v_{1},v_{2},v_{3}\rangle =|M|.}$

${\displaystyle \left|{\begin{matrix}A\cdot b&a\cdot b&c\cdot b\\A\cdot B&a\cdot B&c\cdot B\\A\cdot C&a\cdot C&c\cdot C\end{matrix}}\right|=\left|{\begin{matrix}a\cdot B&A\cdot B&C\cdot B\\a\cdot b&A\cdot b&C\cdot b\\a\cdot c&A\cdot c&C\cdot c\end{matrix}}\right|}$

${\displaystyle \left|{\begin{matrix}C\cdot b&a\cdot b&c\cdot b\\C\cdot A&a\cdot A&c\cdot A\\C\cdot B&a\cdot B&c\cdot B\end{matrix}}\right|=\left|{\begin{matrix}c\cdot B&A\cdot B&C\cdot B\\c\cdot a&A\cdot a&C\cdot a\\c\cdot b&A\cdot b&C\cdot b\end{matrix}}\right|.}$

### 第四个重述

${\displaystyle (A\cdot b)(a\cdot B)(c\cdot C)+(a\cdot b)(c\cdot B)(A\cdot C)+(c\cdot b)(A\cdot B)(a\cdot C)}$
${\displaystyle -(A\cdot b)(c\cdot B)(a\cdot C)-(a\cdot b)(A\cdot B)(c\cdot C)-(c\cdot b)(a\cdot B)(A\cdot C)}$
${\displaystyle =(a\cdot B)(A\cdot b)(C\cdot c)+(A\cdot B)(C\cdot b)(a\cdot c)+(C\cdot B)(a\cdot b)(A\cdot c)}$
${\displaystyle -(a\cdot B)(C\cdot b)(A\cdot c)-(A\cdot B)(a\cdot b)(C\cdot c)-(C\cdot B)(A\cdot b)(a\cdot c)}$

${\displaystyle (C\cdot b)(a\cdot A)(c\cdot B)+(a\cdot b)(c\cdot A)(C\cdot B)+(c\cdot b)(C\cdot A)(a\cdot B)}$
${\displaystyle -(C\cdot b)(c\cdot A)(a\cdot B)-(a\cdot b)(C\cdot A)(c\cdot B)-(c\cdot b)(a\cdot A)(C\cdot B)}$
${\displaystyle =(c\cdot B)(A\cdot a)(C\cdot b)+(A\cdot B)(C\cdot a)(c\cdot b)+(C\cdot B)(c\cdot a)(A\cdot b)}$
${\displaystyle -(c\cdot B)(C\cdot a)(A\cdot b)-(A\cdot B)(c\cdot a)(C\cdot b)-(C\cdot B)(A\cdot a)(c\cdot b).}$

### 第五个重述

${\displaystyle (A\cdot C)(B\cdot c)(a\cdot b)+(A\cdot B)(C\cdot a)(b\cdot c)}$
${\displaystyle -(A\cdot b)(B\cdot c)(C\cdot a)-(A\cdot C)(B\cdot a)(b\cdot c)}$
${\displaystyle =(A\cdot B)(C\cdot b)(a\cdot c)+(A\cdot c)(B\cdot C)(a\cdot b)}$
${\displaystyle -(A\cdot c)(B\cdot a)(C\cdot b)-(A\cdot b)(B\cdot C)(a\cdot c)}$

${\displaystyle (A\cdot c)(B\cdot C)(a\cdot b)+(A\cdot C)(B\cdot a)(b\cdot c)}$
${\displaystyle -(A\cdot c)(B\cdot a)(C\cdot b)-(A\cdot C)(B\cdot c)(a\cdot b)}$
${\displaystyle =(A\cdot B)(C\cdot a)(b\cdot c)+(A\cdot b)(B\cdot C)(a\cdot c)}$
${\displaystyle -(A\cdot b)(B\cdot c)(C\cdot a)-(A\cdot B)(C\cdot b)(a\cdot c).}$

### 第六个重述

${\displaystyle t_{1}=(A\cdot C)(B\cdot c)(a\cdot b),}$
${\displaystyle t_{2}=(A\cdot B)(C\cdot a)(b\cdot c),}$
${\displaystyle t_{3}=(A\cdot b)(B\cdot c)(C\cdot a),}$
${\displaystyle t_{4}=(A\cdot C)(B\cdot a)(b\cdot c),}$
${\displaystyle t_{5}=(A\cdot B)(C\cdot b)(a\cdot c),}$
${\displaystyle t_{6}=(A\cdot c)(B\cdot C)(a\cdot b),}$
${\displaystyle t_{7}=(A\cdot c)(B\cdot a)(C\cdot b),}$
${\displaystyle t_{8}=(A\cdot b)(B\cdot C)(a\cdot c).}$

${\displaystyle t_{1}+t_{2}-t_{3}-t_{4}=t_{5}+t_{6}-t_{7}-t_{8},}$

${\displaystyle t_{6}+t_{4}-t_{7}-t_{1}=t_{2}+t_{8}-t_{3}-t_{5}.}$

### 第七个重述

${\displaystyle t_{1}+t_{2}-t_{3}-t_{4}-t_{5}-t_{6}+t_{7}+t_{8}=0,}$

${\displaystyle 0=t_{1}+t_{2}-t_{3}-t_{4}-t_{5}-t_{6}+t_{7}+t_{8}.}$