If b is the mean proportional between a and c , prove that a, c, a² + b² and b² + c² are proportional.

Given, b is the mean proportional between a and c then,

b² = ac.    [….Eq 1]

If a, c, a² + b² and b² + c² are proportional then,

$\Rightarrow \dfrac{a}{c} = \dfrac{a^2 + b^2}{b^2 + c^2} \\[0.5em] \Rightarrow a(b^2 + c^2) = c(a^2 + b^2) \\[0.5em]$

Solving L.H.S first

   a(b² + c²)
= a(ac + c²)     [Putting value of b² from Eq 1]
= ac(a + c)

Solving R.H.S

   c(a² + b²)
= c(a² + ac) [Putting value of b² from Eq 1]
= ac(a + c)

Since, L.H.S. = R.H.S = ac(a + c), hence the numbers,
a, c, a² + b² and b² + c² are in proportion.