Question

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

Answer 1

It is given that b is the mean proportional between a and c.

Thus,
$\implies \dfrac{a}{b} = \dfrac{b}{c} \\ \\ \implies ac = b {}^{2} \: \: \: \: \: \:$


Now, multiply by ( a - c ) on both sides,

$\implies ac(a - c) = b {}^{2} (a - c) \\ \\ \implies a {}^{2} c - a {c}^{2} = b {}^{2} a - {b}^{2} c \\ \\ \implies {a}^{2} c + b {}^{2} c = {b}^{2} a + a {c}^{2} \\ \\ \implies c( {a}^{2} + {b}^{2} ) = a( {b}^{2} + {c}^{2} ) \\ \\ \implies \dfrac{ {a}^{2} + {b}^{2} }{ {b}^{2} + {c}^{2} } = \dfrac{a}{c}$


As the ratio of a and c is equal to ratio of a² + b² and b² + c², a , c , a² + b² and b² + c² are in proportional.

Proved .