Without expanding prove that,
`|{:(1,bc,b+c),(1,ca,c+a),(1,ab,a+b):}|=|{:(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2)):}|`

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|{:(1,a,bc),(1,b,ca),(1,c,ab):}|=|{:(1,a,a^2),(1,b,b^2),(1,c,c^2):}|

|{:(1,a,bc),(1,b,ca),(1,c,ab):}|

|{:(a,a^2,bc),(b,b^2,ca),(c,c^2,ab):}|=|{:(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3):}|

Without expanding prove that, |{:(b^2c^2,bc,b+c),(c^2a^2,ca,c+a),(a^2b^2,ab,a+b):}|=0

Without expanding prove that |{:(1,a,a^(2)-bc),(1,b,b^(2)-ca),(1,c,c^(2)-ab):}|=0

|{:(1,bc,a(b+c)),(1,ca,b(c+a)),(1,ab,c(a+b)):}|=0

Without expanding prove that , |{:(1,ab,1/a+1/b),(1,bc,1/b+1/c),(1,ca,1/c+1/a):}|=0

Without expanding show that |[1/a,1,bc],[1/b,1,ca],[1/c,1,ab]|=0

|{:(1,bc,bc(b+c)),(1,ca,ca(c+a)),(1,ab,ab(a+b)):}|=0

Without expanding prove that, abs((1,ab,1/a+1/b),(1,bc,1/b+1/c),(1,ca,1/c+1/a)) = 0