Using the property of determinants and without expanding, prove that:`|b+c q+r y+z c+a r+p z+x a+b p+q x+y|=2|a p x b q y c r z|`

Explanation

`|{:(b+c,q+r,y+z),(c+a,r+p,z+x),(a+b,p+q,x+y):}|=2|{:(2(a+b+c),2(p+q+r),2(x+y+z)),(c+a,r+p,z+x),(a+b,p+q,x+y):}|`
`(R_(1)toR_(1)+R_(2)+R_(3))`
`=2|{:(a+b+c,p+q+r,x+y+z),(c+a,r+p,z+x),(a+b,p+q,x+y):}|`
`=2|{:(a+b+c,p+q+r,x+y+z),(-b,-q,-y),(-c,-r,-z):}|`
`(R_(2)toR_(2)-R_(1),R_(3)toR_(3)-R_(1))`
`=2|{:(a,p,x),(-b,-q,-y),(-c,-r,-z):}|" "(R_(1)toR_(1)+R_(2)+R_(3))`
`=2(-1)(-1)|{:(a,p,x),(b,q,y),(c,r,z):}|=2|{:(a,p,x),(b,q,y),(c,r,z):}|`
= R.H.S