Question

Prove that:
$open vertical bar table row 1 cell space straight a end cell cell space space bc end cell row 1 cell space straight b end cell cell space space ca end cell row 1 cell space straight c end cell cell space ab end cell end table close vertical bar space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis$

Solution

Let space increment space equals space space open vertical bar table row 1 cell space space straight a end cell cell space space space bc end cell row 1 cell space space straight b end cell cell space space space ca end cell row 1 cell space space straight c end cell cell space space space ab end cell end table close vertical bar space equals space 1 over abc open vertical bar table row straight a cell space space straight a squared end cell cell space space abc end cell row straight b cell space space straight b squared end cell cell space space abc end cell row straight c cell space space straight c squared end cell cell space space abc end cell end table close vertical bar

by multiplying

straight R subscript 1 comma space straight R subscript 2 comma space straight R subscript 3 space with space straight a comma space straight b comma space straight c space respectively

equals space abc over abc open vertical bar table row straight a cell space space space straight a squared end cell cell space space 1 end cell row straight b cell space space straight b squared end cell cell space space 1 end cell row straight c cell space space straight c squared end cell cell space space 1 end cell end table close vertical bar space equals space open vertical bar table row 1 cell space space straight a end cell cell space space space straight a squared end cell row 1 cell space space straight b end cell cell space space space straight b squared end cell row 1 cell space space straight c end cell cell space space space straight c squared end cell end table close vertical bar comma space by space passing space straight R subscript 3 space over space two space rows. equals space open vertical bar table row 1 cell space space space straight a end cell cell space space space space straight a squared end cell row 0 cell space space space space straight b minus straight a end cell cell space space space space space straight b squared minus straight a squared end cell row 0 cell space space space space straight c minus straight a end cell cell space space space straight c squared minus straight a squared end cell end table close vertical bar comma space by space straight R subscript 2 minus straight R subscript 1 comma space space straight R subscript 3 minus straight R subscript 1 equals space open vertical bar table row cell straight b minus straight a end cell cell space space space space space space straight b squared minus straight a squared end cell row cell straight c minus straight a end cell cell space space space space space straight c squared minus straight a squared end cell end table close vertical bar comma space by space expanding space with space straight C subscript 1 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 1 cell space space space straight b plus straight a end cell row 1 cell space space space straight c plus straight a end cell end table close vertical bar space equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b minus straight a right parenthesis equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c minus straight b right parenthesis space equals 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.

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