Let a , b , c , d and p be non-zero distinct real numbers such that a 2 + b 2 + c 2 p 2 - 2 ( a b + b c + c d ) p + b 2 + c 2 + d 2 = 0 . Then

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Rearranging the expression, we get ( a p - b ) 2 + ( b p - c ) 2 + ( c p - d ) 2 = 0 ⇒ a p - b = 0 , b p - c = 0 , c p - d = 0 ⇒ p = b a = c b = d c Which is condition of GP.

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