Question

if `x/a = y/b = z/c` prove that `(ax - by)/((a + b)(x - y)) + (by - cz)/((b + c)(y - z)) + (cz - az)/((c + a)(z - x)) = 3`

Answer 1

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

=> x = ak, y = bk, z = dk

L.H.S

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

`= (a(ak) - b(bk))/((a + b)(ak - bk)) + (b(bk) - c(ck))/((b + c)(bk - dk)) + (c(ck) - a(ak))/((c + a)(dk - ak))`

`= (k(a^2 - b^2))/(k(a + b)(a- b)) + (k(b^2 - c^2))/(k(b + c)(b - c)) + (k(c^2 - a^2))/(k(c + a)(c - a))`

`= (k(a^2 - b^2))/(k(a^2 - b^2)) + (k(b^2 - c^2))/(k(b^2 - c^2)) + (k(c^2 - a^2))/(k(c^2 - a^2))`

= 1 + 1 + 1 = 3 = R.H.S