If b is the mean proportion between a and c, show that (a⁴ + a²b² + b⁴) / (b⁴ + b²c² + c⁴) = a² / c²

Since b is the mean proportional between a and c:

b² = ac

Therefore: b⁴ = a²c²

Consider the left-hand side:

(a⁴ + a²b² + b⁴)/(b⁴ + b²c² + c⁴)

Substitute b² = ac and b⁴ = a²c²:

= [a⁴ + a²(ac) + a²c²] / [a²c² + (ac)c² + c⁴]

= [a²(a² + ac + c²)] / [c²(a² + ac + c²)]

= a²/c²

= Right-hand side.

Hence proved.

The key fact is b² = ac because b is the mean proportional between a and c. Replacing b² and b⁴ by expressions in a and c gives a common factor that cancels.