Question

If `(x + y)/(ax + by) = (y + z)/(ay + bz) = (z + x)/(az + bx)`, prove that each of these ratio is equal to `(2)/(a + b)` unless x + y + z = 0

Answer 1

`(x + y)/(ax + by) = (y + z)/(ay + bz) = (z + x)/(az + bx)`

= `(x+ y + y + z + z + x)/(ax + by + ay + bx + az + bx)`

= `(2(x  y + z))/(x(a + b) + y(a + b) + z(a + b)`

= `(2(x + y + z))/((a + b)(x + y + z)`

= `(2)/(a + b)`
if x + y + z ≠ 0.
Hence proved.