Question

Prove that:
$open vertical bar table row straight a straight b straight c row cell straight a squared end cell cell straight b squared end cell cell straight c squared end cell row bc ca ab end table close vertical bar space equals space left parenthesis ab plus bc plus ca right parenthesis thin space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis$

Solution

Let $increment space equals space open vertical bar table row straight a cell space straight b end cell cell space straight c end cell row cell straight a squared end cell cell space straight b squared end cell cell space straight c squared end cell row bc ca ab end table close vertical bar$
$equals space 1 over abc open vertical bar table row cell straight a squared end cell cell straight b squared end cell cell straight c squared end cell row cell straight a cubed end cell cell straight b cubed end cell cell straight c cubed end cell row abc abc abc end table close vertical bar space equals space abc over abc space open vertical bar table row cell straight a squared end cell cell space space straight b squared end cell cell space space straight c squared end cell row cell straight a cubed end cell cell space straight b cubed end cell cell space space straight c cubed end cell row 1 1 1 end table close vertical bar equals space open vertical bar table row 1 1 1 row cell straight a squared end cell cell straight b squared end cell cell straight c squared end cell row cell straight a cubed end cell cell straight b cubed end cell cell straight c cubed end cell end table close vertical bar equals space open vertical bar table row 1 0 cell space space 0 end cell row cell straight a squared end cell cell space space straight b squared minus straight a squared end cell cell space space space straight c squared minus straight a squared end cell row cell straight a cubed end cell cell straight b cubed minus straight a cubed end cell cell space space straight c cubed minus straight a cubed end cell end table close vertical bar comma space space by space straight C subscript 2 minus straight C subscript 1 comma space space straight C subscript 3 minus straight C subscript 1 equals space open vertical bar table row cell straight b squared minus straight a squared end cell cell space space space space space space straight c squared minus straight a squared end cell row cell straight b cubed minus straight a cubed end cell cell space space space space space straight c cubed minus straight a cubed end cell end table close vertical bar comma space expanding space with space first space row comma equals space open vertical bar table row cell left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight b plus straight a right parenthesis end cell cell space space space space space space space space space space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a right parenthesis end cell row cell left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight b squared plus straight a squared plus ba right parenthesis end cell cell space space space space space space space space space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c squared plus straight a squared plus ca right parenthesis end cell end table close vertical bar equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space open vertical bar table row cell straight b plus straight a end cell cell space space space space space straight c plus straight a end cell row cell straight b squared plus straight a squared plus ba end cell cell space space space space space space space space space space space space space straight c squared plus straight a squared plus ca end cell end table close vertical bar equals space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space open vertical bar table row cell straight b plus straight a end cell cell space space space space space space straight c minus straight b end cell row cell straight b squared plus straight a squared plus ba end cell cell space space space space space space straight c squared minus straight b squared plus ca minus ba end cell end table close vertical bar comma space by space straight C subscript 2 minus straight C subscript 1 equals left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space left parenthesis straight c minus straight b right parenthesis space open vertical bar table row cell straight b plus straight a end cell cell space space space space space space space space space space space space space space 1 end cell row cell straight b squared plus straight a squared plus ba end cell cell space space space space space space space space space space space space straight c plus straight b plus straight a end cell end table close vertical bar equals left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space open square brackets bc plus straight b squared plus ab plus ca plus ab plus straight a squared minus straight b squared minus straight a squared minus ab close square brackets equals left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis ab plus bc plus ca right parenthesis$

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