If ` x/(y+z) = y/(z+x) = z/(x + y)`, then prove that the value of each ratio is equal to `1/2 " or " (-1)`.

Explanation

` x/(y + z) = y/(z +x) = z/(x+y) = k " (let)" [kne 0]`
` :. X = k (y + z), y = k (z + x) , z = k ( x + y)`.
Now, ` x + y + z = k (y + z) + k (z + x) + k(x + y)`
` = k (y + z + z + x + x + y)`
` = k (2x + 2y + 2z)`
` = 2k (x + y +z)`
or, ` x + y + z - 2k(x + y +z) = 0`
or, ` (x + y+z) (1 - 2k) = 0`.
` :. " either" x + y + z = 0 " or, " 1 - 2k = 0`
` rArr 2k = 1 rArr k = 1/2`.
` :. ` each ratio ` = 1/2`
Again , ` x + y + z rArr y + z = - x`
` rArr x/(y+z) = x/(-x) = - 1` ltbr. Similarly , ` z + x = - y rArr y/(z + x) = y/(-y) = - 1`
and ` x + y = - z rArr z/(x + y) = z/(-z) = - 1 `.
` :. x/(y+z) = y/(z+x) = z/(x+y)` implies that the value of each ratio is equal to ` 1/2 " or " (-1)`.