Question

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

Answer 1

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

x = k (b + c- a) 

Y = k (c+a-b)

z = k (a+b-c) 

Now ,

`("x + y+z")/("a + b + c")`

`= ("k"("b + c - a") + "k" ("c + a - b")+ "k" ("a + b - c"))/("a + b + c")`

`= ("k" ("b + c - a + c + a - b + a + b - c"))/("a + b + c")`

`= ("k"("a + b + c"))/("a + b + c") = "k"`

Hence , 

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

Proved.