If a,b and c are real numbers, and
`Delta ={:[( b+c,C+a,a+b),( c+a,a+b,b+c),( a+b,b+c,c+a) ]:}=0`
Show that either `a+b+c=0 "or" a= b= c . `

Explanation

` Delta=|{:(b+c,,c+a,,a+b),(c+a,,a+b,,b+c),(a+b,,b+c,,c+a):}|`
Applying `R_(1) to R_(1)+R_(2)+R_(3)` we have
` Delta=|{:(2(a+b+c),,2(a+b+c),,2(a+b+c)),(c+a,,a+b,,b+c),(a+b,,b+c,,c+a):}|`
`=2 (a+b+c) |{:(1,,1,,1),(c+a,,b-c,,b-a),(a+b,,c-a,,c-b):}|`

Applying `C_(2) to C_(2)-C_(1) " and "C_(3) to C_(3)-C_(1)` we have
`Delta =2(a+b+c) |{:(1,,0,,0),(c+a,,b-c,,b-a),(a+b,,c-a,,c-b),(,,,,):}|`
Expanding along `R_(1)` we have
`Delta =2 (a+b+c) (1) [(b-c)(c-b)-(b-a)(c-a)]`
`=2 (a+b+c) [-b^(2)-c^(2)+2bc-bc+ba+ac-a^(2)]`
`=2 (a+b+c) [ab+bc+ca-a^(2)-b^(2)-c^(2)]`
It is given that `Delta =0` Therefore,
`(a+b+c) [ab+bc+ca-a^(2)-b^(2)-c^(2)]=0`
`" or " (1//2)(a+b+c) [(a-b)^(2)+(b-c)^(2)+(c-a)^(2)=0`
`" or Either " a+b+c =0`
` " or " (a-b)^(2) +(b-c)^(2) +(c-a)^(2)=0`
` rArr " Either " a+b+c =0 " or " a=b=c`