Question

Prove that:
$open vertical bar table row cell straight a squared end cell bc cell ac plus straight c squared end cell row cell straight a squared plus ab end cell cell straight b squared end cell ac row ab cell straight b squared plus bc end cell cell straight c squared end cell end table close vertical bar space equals space 4 straight a squared straight b squared straight c squared.$

Solution

Let $increment space equals space open vertical bar table row cell straight a squared end cell bc cell ac plus straight c squared end cell row cell straight a squared plus ab end cell cell straight b squared end cell ac row ab cell straight b squared plus bc end cell cell straight c squared end cell end table close vertical bar$
$equals space abc open vertical bar table row straight a straight c cell straight a plus straight c end cell row cell straight a plus straight b end cell straight b straight a row straight b cell space space straight b plus straight c end cell straight c end table close vertical bar equals space abc open vertical bar table row straight a cell space straight c end cell cell space space straight a plus straight c end cell row 0 cell space minus 2 straight c end cell cell space minus 2 straight c end cell row straight b cell space straight b plus straight c end cell cell space space space space straight c end cell end table close vertical bar comma space space space by space straight R subscript 2 space minus space straight R subscript 1 space minus space straight R subscript 3 space equals space minus 2 abc squared space open vertical bar table row straight a straight c cell straight a plus straight c end cell row 0 1 cell space 1 end cell row cell straight b space end cell cell space straight b plus straight c end cell cell space straight c end cell end table close vertical bar equals space minus 2 abc squared space open vertical bar table row straight a cell space space minus straight a end cell cell space space straight a plus straight c end cell row 0 cell space space space space 0 end cell cell space 1 end cell row straight b cell space space space space straight b end cell cell space space straight c end cell end table close vertical bar comma space by space straight C subscript 2 space minus straight C subscript 3 equals negative 2 abc squared space open vertical bar table row straight a cell negative straight a end cell row straight b cell space space space space straight b end cell end table close vertical bar comma space expanding space with space straight R subscript 2 space equals space 2 abc squared left parenthesis ab plus ab right parenthesis space equals space 2 abc squared left parenthesis 2 space straight a space straight b right parenthesis space equals space 4 space straight a squared space straight b squared space straight c squared.$

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