Question

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

Answer 1

∵ b is the mean proportional between a and c, then,
b² = a × c ⇒ b² = ac …(i)
Now a, c, a² + b² and b² + c² are in proportion

if `a/c = (a^2 + b^2)/(b^2 + c^2)`
if a(b² + c²) = (a² + b²)
if a(ac + c²) = c(a² + ac)  ...[from (i)]
if ac(a + c) = a²c + ac²
if ac(a + c) = ac(a + c)
which is true.
Hence proved.