Using properties of determinants Prove that
`|{:(a+b+c,,-c,,-b),(-c,,a+b+c,,-a),( -b,,-a,,a+b+c):}| = 2 (a+b) (b+c) (c+a)`

Explanation

`Delta =- |{:(a+b+c,,-c,,-b),(-c,,a+b+c,,-a),(-b,,-a,,a+b+c):}|`
Applying `R_(3) to R_(3) +R_(2),R_(1) to R_(1)+R_(2)`
`Delta= |{:(a+b,,a+b,,-(a+b)),(-c,,a+b+c,,-a),(-(b+c),,b+c,,b+c):}|`
`=(a+b)(b+c) |{:(1,,1,,-1),(-c,,a+b+c,,-a),(-1,,1,,1):}|`
Applying `R_(1) to R_(1)+R_(3)`
`Delta =(a+b)(b+c) |{:(0,,2,,0),(-c,,a+b+c,,-a),(-1,,1,,1):}|`
`= (a+b)(b+c) (-2) |{:(-c,,-a),(-a,,1):}|`
`=2(a+b)(b+c)(c+a)`