Prove: `|[b+c,a,a],[b,c+a,b],[c,c,a+b]|=4a b c`

Prove: |[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]|=(a+b+c)^3

Prove: |[b+c ,a-b ,a], [c+a, b-c ,b], [a+b, c-a, c]|=3a b c-a^3-b^3-c^3

Prove that |[b+c ,a-b, a] ,[c+a, b-c ,b],[ a+b, c-a, c]|=3abc-a^3-b^3-c^3dot

Prove: |[a+b+2c, a, b], [c, b+c+2a, b], [c ,a ,c+a+2b]|=2(a+b+c)^3

Prove the identities: |[a, b, c],[ a-b,b-c,c-a],[ b+c,c+a, a+b]|=a^3+b^3+c^3-3a b c

Prove that |a ,b+c, a^2,b, c+a, b^2,c, a+b, c^2|=-(a+b+c)xx(a-b)(b-c)(c- a)

Prove: \ |(a, b, c),( a-b,b-c,c-a),( b+c,c+a, a+b)|=a^3+b^3+c^3-3a b c

Prove that : |{:(a,b,c),(b,c,a),(c,a,b):}|=3 a b c-a^(3)-b^(3)-c^(3)

Prove that: |[b, c-a^2,c] ,[a-b^2,a b-c^2,c ],[a-b^2,a ,b-c^2b c-a^2a b-c^2b c-a^2c a-b^2]|=|[a, b, c],[ b ,c ,a],[ c, a ,b]|^2 .

Using the property of determinants and without expanding, prove that: |[1,b c, a(b+c)],[1,c a, b(c+a)],[1,a b, c(a+b)]|=0