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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")`
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Solution
`"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.
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