Question

Prove that
$open vertical bar table row 1 cell space space space space straight a end cell cell space space space space space straight a squared minus bc end cell row 1 cell space space space space straight b end cell cell space space space space straight b squared minus ca end cell row 1 cell space space space straight c end cell cell space space space space straight c squared minus ab end cell end table close vertical bar space equals 0$

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Solution

Let space space increment space equals space open vertical bar table row 1 cell space space space space straight a end cell cell space space space space space straight a squared minus bc end cell row 1 cell space space space space straight b end cell cell space space space space straight b squared minus ca end cell row 1 cell space space space straight c end cell cell space space space space straight c squared minus ab end cell end table close vertical bar space equals space open vertical bar table row 1 cell space space space space straight a end cell cell space space space straight a squared end cell row 1 cell space space space space straight b end cell cell space space space space straight b squared end cell row 1 cell space space space space straight c end cell cell space space space space straight c squared end cell end table close vertical bar space plus space space open vertical bar table row 1 cell space space space space space space straight a end cell cell space space minus bc end cell row 1 cell space space space space space straight b end cell cell space space minus ca end cell row 1 cell space space space space space straight c end cell cell space space minus ab end cell end table close vertical bar space space space space space space space space space space space space equals space space open vertical bar table row 1 cell space space space space straight a end cell cell space space space straight a squared end cell row 1 cell space space space space straight b end cell cell space space space space straight b squared end cell row 1 cell space space space space straight c end cell cell space space space space straight c squared end cell end table close vertical bar space minus space begin inline style 1 over abc end style space open vertical bar table row straight a cell space space space space space straight a squared end cell cell space space space space abc end cell row straight b cell space space space space space straight b squared end cell cell space space space space abc end cell row straight c cell space space space space straight c squared end cell cell space space space abc end cell end table close vertical bar space space space space space space space space space space space space equals space space space open vertical bar table row 1 cell space space space space straight a end cell cell space space space straight a squared end cell row 1 cell space space space space straight b end cell cell space space space space straight b squared end cell row 1 cell space space space space straight c end cell cell space space space space straight c squared end cell end table close vertical bar space minus space begin inline style abc over abc end style space open vertical bar table row straight a cell space space space space space space space straight a squared end cell cell space space space space space space space space 1 end cell row straight b cell space space space space space space straight b squared end cell cell space space space space space space 1 end cell row straight c cell space space space space space space straight c squared end cell cell space space space space space space 1 end cell end table close vertical bar space space space space space space space space space space space space equals space space space open vertical bar table row 1 cell space space space space straight a end cell cell space space space straight a squared end cell row 1 cell space space space space straight b end cell cell space space space space straight b squared end cell row 1 cell space space space space straight c end cell cell space space space space straight c squared end cell end table close vertical bar space minus space open vertical bar table row 1 cell space space space space straight a end cell cell space space space space space space space straight a squared end cell row 1 cell space space space space straight b end cell cell space space space space space space straight b squared end cell row 1 cell space space space space straight c end cell cell space space space space space space straight c squared end cell end table close vertical bar space space space space space space space space space space space equals space space 0 space space space space space space space space space space space space

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