(xy)z = xyz
similarly
xyxz = x(y+z)
we have a case of
(xu)v (xy)z (xp)q
=x(uv + yz + pq)
u = 1/(a-b)
v= 1/(a-c)
y =1/(b-a)
z=1/(b-c)
p=1/(c-a)
q=1/(c-b)
uv + yz + pq =1/{(a-b)(a-c)} + 1/{(b-a)(b-c)} +1/{(c-a)(c-b)}
= 1/{(a-b)(a-c)} -1/{(a-b)(b-c)} +1/{(a-c)(b-c)} # (b-a)=-(a-b), (c-a)=-(a-c) (c-b) = -(b-c)
LCM of denominators above fractions (a-b)(a-c) , (b-c)(a-c), (a-c)(b-c) is (a-b)(b-c)(a-c)
1/{(a-b)(a-c)} =(b-c)/{(a-b)(b-c)(a-c)}
1/{(a-b)(b-c)} =(a-c)/{(a-b)(b-c)(a-c)}
1/{(a-c)(b-c)} =(a-b)/{(a-b)(b-c)(a-c)}
1/{(a-b)(a-c)} -1/{(a-b)(b-c)} +1/{(a-c)(b-c)}
=(b-c)/{(a-b)(b-c)(a-c)} - (a-c)/{(a-b)(b-c)(a-c)} + (a-b)/{(a-b)(b-c)(a-c)}
= { (b-c) -(a-c) + (a-b)}/{(a-b)(b-c)(a-c)}
=( b - c -a +c +a -b )/{(a-b)(b-c)(a-c)}
=(b -b -c +c -a +a)/{(a-b)(b-c)(a-c)}
=0/{(a-b)(b-c)(a-c)}
=0
uv + yz + pq = 0
Thus
x(uv + yz + pq) = x0 = 1