Maths question about ratio

(b² + bx + x²) : (a² + ay + y²) = (b² - bx + x²) : (a² - ay + y²)

(a² - ay + y²)(b² + bx + x²) = (a² + ay + y²)(b² - bx + x²)

[(a² + ay + y²) - 2ay] (b² + bx + x²) = (a² + ay + y²) [(b² + bx + x²) - 2bx]

(a² + ay + y²)(b² + bx + x²) - 2ay(b² + bx + x²) = (a² + ay + y²)(b² + bx + x²)- 2bx(a² + ay + y²)

ay(b² + bx + x²) = bx(a² + ay + y²)

ab²y + abxy + ax²y = a²bx + abxy + bxy²

ab²y - a²bx - bxy² + ax²y = 0

(ab²y - a²bx) - (bxy² - ax²y) = 0

ab(by - ax) - xy(by - ax) = 0

(ab - xy)(by - ax) = 0

ab - xy = 0 or by - ax = 0

ab = xy or by = ax

x/a = b/y or x/b = y/a

x : a = b : y or x : b = y : a