Without expanding, prove that each of the following determinant vanishes : `|(1,a,b+c),(1,b,c+a),(1,c,a+b)|`

Without expanding, prove that the following determinants vanish: {:|(a,b,a+b),(b,c,b+c),(c,a,c+a)|

Without actual expansion, prove that the following determinants vanish: {:|(1,a,a^2-bc),(1,b,b^2-ca),(1,c,c^2-ab)|

Without expanding, prove that the following determinants vanish : |{:(a,b,a+b),(b,c,b+c),(c,a,c+a):}|

Without actual expansion, prove that the following determinants vanish : |{:(a,b,c),(a-b,b-c,c-a),(b,c,a):}|

Without actual expansion, prove that the following determinants vanish: {:|(a,b,c),(a-b,b-c,c-a),(b,c,a)|

Without actual expansion, prove that the following determinants vanish: {:|(0,a,-b),(-a,0,-c),(b,c,0)|

Without expanding show that the following determinants vanish |(4,a,b+c),(4,b,c+a),(4,c,a+b)|

Without expanding,show that the following determinant vanishes. abs ((1,bc,a(b+c)),(1,ca,b(c+a)),(1,ab,c(a+b)))

Without expanding, prove that the following determinants vanish: {:|(b^2c^2,bc,b+c),(c^2a^2,ca,c+a),(a^2b^2,ab,a+b)|

Without actual expansion, prove that the following determinants vanish: {:|(x,a,x+a),(y,b,y+b),(z,c,z+c)|