Question

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

Answer 1

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

x = k(b + c – a),

y = k(c + a – b),

z = k(a + b – c)

`(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 proved.

Answer 2

`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)

x + y + c = k[(b + c – a) + (c + a − b) + (a + b − c)]

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

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

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

Hence proved.