Prove that
`|{:(b+c, c+a, a+b),(c+a, a+b,b+c),(a+b, b+c, c+a):}| =2(a+b+c)(ab+bc+ca-a^(2)-b^(2)-c^(2)).`

Explanation

Let the given determinant be `Delta`. Then,
`Delta =|{:(b+c, c+a, a+b),(c+a, a+b,b+c),(a+b, b+c, c+a):}| `
`=|{:(2(a+b+c), 2(a+b+c), 2(a+b+c)),(c+a, a+b,b+c),(a+b, b+c, c+a):}| [R_(1) to (R_(1) + R_(2) + R_(3))] `
`=2(a+b+c)*|{:(" "1, " "1, " "1),(c+a, a+b,b+c),(a+b, b+c, c+a):}| ["taking(a+b+c)common from"R_(1)]`
`=2(a+b+c)*|{:(" "1, " "0, " "0),(c+a, b-c,b-a),(a+b, c-a, c-b):}| [{:(C_(2) to (C_(2)-C_(1))" and"),(C_(3) to (C_(3) -C_(1))):}]`
`=2(a+b+c)*1*|{:(b-c, b-a), (c-a, c-b):}| ["expanded by"R_(1)]`
`2(a+b+c) *[(b-c)(c-b)-(c-a)(b-a)]`
`=2(a+b+c)(ab+bc+ca-a^(2)-b^(2)-c^(2))`.
Hence, `Delta =2(a+b+c)(ab+bc+ca-a^(2)-b^(2)-c^(2))`.