Question

If `(b + c - a)/(y + z - x) = (c + a - b)/(z + x - y) = (a + b - c)/(x + y - z)` then prove that each ratio is equal to `a/x = b/y = c/z`

Answer 1

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

Apply componendo,

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

⇒ `(2c)/(2z) = (2a)/(2x)`

⇒ `c/z = a/x`   ...(1)

Apply componendo again,

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

⇒ `(2b)/(2y) = (2c)/(2z)`

⇒ `b/y = c/z`   ...(2)

From (1) and (2):

`a/z = b/y = c/z`

Hence proved.