実験台 のソース

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$$$
  \begin{array}{l}
         \left| \begin{array}{ccc} a&b&c \\ c&a&b \\ b&c&a \end{array} \right|
       = \left| \begin{array}{ccc} a+b+c&b&c \\ c+a+b&a&b \\ b+c+a&c&a \end{array} \right|
  \\\\ = (a+b+c) \left| \begin{array}{ccc} 1&b&c \\ 1&a&b \\ 1&c&a \end{array} \right|
  \\\\ = (a+b+c) \left| \begin{array}{ccc} 1&b&c \\ 0&a-b&b-c \\ 0&c-b&a-c \end{array} \right|
  \\\\ = (a+b+c) \left| \begin{array}{cc} a-b&b-c \\ c-b&a-c \end{array} \right|
  \\\\ = (a+b+c)\big((a-b)(a-c) - (b-c)(c-b)\big)
  \\\\ = (a+b+c)(a^2 - ab - ac + bc + b^2 - 2bc + c^2)
  \\\\ = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca)
  \end{array}
$$$
 
$$$
  \begin{array}{l}
         \left| \begin{array}{ccc} a&b&c \\ c&a&b \\ b&c&a \end{array} \right|
       = \left| \begin{array}{ccc} a+b+c&b&c \\ c+a+b&a&b \\ b+c+a&c&a \end{array} \right|
       = \left| \begin{array}{ccc} a+b+c&b&c \\ 0&a-b&b-c \\ 0&c-b&a-c \end{array} \right|
  \\\\ = (a+b+c) \left| \begin{array}{cc} a-b&b-c \\ c-b&a-c \end{array} \right|
  \\\\ = (a+b+c)\big((a-b)(a-c) + (b-c)^2 \big)
  \\\\ = (a+b+c)(a^2 - ab - ac + bc - 2bc + b^2 + c^2)
  \\\\ = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca)
  \end{array}
$$$
 
$$$
  \begin{array}{l}
         \left| \begin{array}{ccc} a&b&c \\ c&a&b \\ b&c&a \end{array} \right|
       = \left| \begin{array}{ccc} a+b+c&b&c \\ c+a+b&a&b \\ b+c+a&c&a \end{array} \right|
  \\\\ = (a+b+c) \left| \begin{array}{ccc} 1&b&c \\ 1&a&b \\ 1&c&a \end{array} \right|
       = (a+b+c) \arrs{1 \\ 1 \\ 1}\sx \left( \arrs{b \\ c \\ a} \vx \arrs{c \\ a \\ b} \right)
  \\\\ = (a+b+c) (a^2 + b^2 + c^2 - ab - bc - ca)
  \end{array}
$$$
 
$$$
  \begin{array}{l}
         \left| \begin{array}{ccc} a&b&c \\ c&a&b \\ b&c&a \end{array} \right|
       = (a+b+c) \arrs{1 \\ 1 \\ 1}\sx \left( \arrs{b \\ c \\ a} \vx \arrs{c \\ a \\ b} \right)
  \end{array}
$$$
 
$$$
  \begin{array}{l}
         \left| \begin{array}{ccc} a&b&c \\ c&a&b \\ b&c&a \end{array} \right|
       = \arrs{a \\ b \\ c} \vx \arrs{c \\ a \\ b} \sx \arrs{1 \\ 1 \\ 1} \cdot \,(a+b+c)
  \end{array}
$$$