Question

If a(y + z) = b(z + x) = c(x + y) and out of a, b, c no two of them are equal then show that,

`(y - z)/[a ( b - c )] = ( z - x)/[ b ( c - a)] = ( x - y)/[c ( a - b )]`

Answer 1

Let `a( y + z ) = b( z + x ) = c( x + y) = k`

⇒ `(a( y + z))/(abc) = (b( z + x))/(abc) = (c( x+ y))/(abc)`

`(y + z)/(bc) =(z + x)/(ac) = ( x+ y)/(ab)` = k

`k = (( x + y) - ( z + x))/(ab - ac) = (y - z)/(a(b - c))`    ...(1)

`k = (( y + z) - (x + y))/(bc - ab) = (z - x)/(b(c - a))`     ...(2)

`k = (( z + x) - (y + z))/(ac - bc)  = (x - y)/(c(a - b))`    ...(3)

`(therefore a/b = c/d = k = (a - c)/ (b - d))`

`therefore (y - z) /(a(b - c)) = (z - x) /(b(c - a)) = (x - y) /(c(a - b))=k/(abc)`