if a:b = c:d then prove that ab/(a^2+b^2) = cd/(c^2+d^2)

Given that:

a : b = c : d

This means:

a/b = c/d

We need to prove that:

ab / (a² + b²) = cd / (c² + d²)

Step 1: Expressing a in terms of b and c in terms of d
Since a/b = c/d, we can write:

a = (b * c) / d
c = (d * a) / b

Similarly, we can express b and d in terms of a and c:

b = (a * d) / c
d = (c * b) / a

Step 2: Compute LHS
LHS = ab / (a² + b²)

Substituting a = (b * c) / d:

LHS = [(b * c) / d] * b / [(b² * c²) / d² + b²]

Simplify the denominator:

= [(b² * c) / d] / [(b² * c² + b² * d²) / d²]

Multiply numerator and denominator by d²:

= (b² * c * d²) / [d * (b² * c² + b² * d²)]

Factor out b² in the denominator:

= (b² * c * d²) / [b² * d * (c² + d²)]

Cancel b² from numerator and denominator:

= (c * d²) / [d * (c² + d²)]

= cd / (c² + d²)

Step 3: Compare with RHS
RHS = cd / (c² + d²)

Since LHS = RHS, the given expression is proven.

Thus, ab / (a² + b²) = cd / (c² + d²).