If a, b and c are real numbers, and
$increment space equals space open vertical bar table row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space straight a plus straight b end cell row cell straight c plus straight a end cell cell space straight a plus straight b end cell cell space straight b plus straight c end cell row cell straight a plus straight b end cell cell space straight b plus straight c end cell cell space straight c plus straight a end cell end table close vertical bar space equals space 0 comma$ show that either $straight a plus straight b plus straight c space equals space 0 space or space space straight a space equals space straight b space equals space straight c.$

Solution

increment space equals space open vertical bar table row cell straight b plus straight c end cell cell straight c plus straight a end cell cell straight a plus straight b end cell row cell straight c plus straight a end cell cell straight a plus straight b end cell cell straight b plus straight c end cell row cell straight a plus straight b end cell cell straight b plus straight c end cell cell straight c plus straight a end cell end table close vertical bar space equals space open vertical bar table row cell 2 left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell straight c plus straight a end cell cell straight a plus straight b end cell row cell 2 left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell straight a plus straight b end cell cell straight b plus straight c end cell row cell 2 left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell straight b plus straight c end cell cell straight c plus straight a end cell end table close vertical bar comma space by space straight C subscript 1 space rightwards arrow space straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3

equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row 1 cell space straight c plus straight a end cell cell space space straight a plus straight b end cell row 1 cell space straight a plus straight b end cell cell space space straight b plus straight c end cell row 1 cell space straight b plus straight c end cell cell space space straight c plus straight a end cell end table close vertical bar equals 2 left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row 1 cell space straight c plus straight a end cell cell space straight a plus straight b end cell row 1 cell space straight a plus straight b end cell cell space straight b plus straight c end cell row 1 cell space straight b plus straight c end cell cell space straight c plus straight a end cell end table close vertical bar comma space by space straight R subscript 2 space rightwards arrow space straight R subscript 2 space minus straight R subscript 1 comma space straight R subscript 3 space rightwards arrow space straight R subscript 3 space minus straight R subscript 1 equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space open vertical bar table row cell straight b minus straight c end cell cell space space straight c minus straight a end cell row cell straight b minus straight a end cell cell space space straight c minus straight b end cell end table close vertical bar equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space open square brackets left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight b right parenthesis space minus space left parenthesis straight b minus straight a right parenthesis space left parenthesis straight c minus straight a right parenthesis close square brackets equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis space open square brackets negative straight b squared minus straight c squared plus 2 bc minus bc plus ab plus ca minus straight a squared close square brackets equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space left parenthesis ab plus bc plus ca minus straight a squared minus straight b squared minus straight c squared right parenthesis equals space minus left parenthesis straight a plus straight b plus straight c right parenthesis thin space open square brackets 2 straight a squared plus 2 straight b squared plus 2 straight c squared minus 2 ab minus 2 bc minus 2 ca close square brackets equals negative left parenthesis straight a plus straight b plus straight c right parenthesis space left square bracket left parenthesis straight a squared plus straight b squared minus 2 ab right parenthesis plus left parenthesis straight b squared plus straight c squared minus 2 bc right parenthesis plus left parenthesis straight c squared plus straight a squared minus 2 ca right parenthesis right square bracket equals negative left parenthesis straight a plus straight b plus straight c right parenthesis space left square bracket left parenthesis straight a minus straight b right parenthesis squared plus left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared right square bracket Now space increment space equals space 0 space give space us space space space space space space minus left parenthesis straight a plus straight b plus straight c right parenthesis thin space left square bracket left parenthesis straight a minus straight b right parenthesis squared plus left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared space equals space 0 rightwards double arrow space space space space space left parenthesis straight a plus straight b plus straight c right parenthesis thin space left square bracket left parenthesis straight a minus straight b right parenthesis squared plus left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared right square bracket space equals space 0 therefore space space space space space space space space space space space space space either space space straight a plus straight b plus straight c space equals space 0 or space space space space space space space space left parenthesis straight a minus straight b right parenthesis squared plus left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared rightwards double arrow space space space space space space space space space space space space space left parenthesis straight a minus straight b right parenthesis squared space equals space 0 comma space space space space space left parenthesis straight b minus straight c right parenthesis squared space equals space 0 comma space space space left parenthesis straight c minus straight a right parenthesis squared space equals space 0 rightwards double arrow space space space space space space space space space space space space space space space space space space straight a equals straight b comma space space straight b equals space straight c comma space space straight c equals straight a rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space straight a equals straight b equals straight c therefore space space space space space increment space equals space 0 space space space space rightwards double arrow space space either space straight a plus straight b plus straight c space equals space 0 space or space space space straight a equals straight b equals straight c. space space space space space space space space space space

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