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Using the property of determinants and w...

Using the property of determinants and without expanding ` {:|( a-b,b-c, c-a),( b-c,c-a,a-b),( c-a,a-b,b-c)|:} =0`

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|{:(b+c,a,b),(c+a,c,a),(a+b,b,c):}|

|{:(a,b,c),(b,c,a),(c,a,b):}|