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