Brahmagupta-Fibonacci Identity/Proof 1

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Let $a, b, c, d$ be numbers.


$\paren {a^2 + b^2} \paren {c^2 + d^2} = \paren {a c + b d}^2 + \paren {a d - b c}^2$


\(\ds \) \(\) \(\ds \paren {a c + b d}^2 + \paren {a d - b c}^2\)
\(\ds \) \(=\) \(\ds \paren {\paren {a c}^2 + 2 \paren {a c} \paren {b d} + \paren {b d}^2} + \paren {\paren {a d}^2 - 2 \paren {a d} \paren {b c} + \paren {b c}^2}\) Square of Sum, Square of Difference
\(\ds \) \(=\) \(\ds a^2 c^2 + 2 a b c d + b^2 d^2 + a^2 d^2 - 2 a b c d + b^2 c^2\) multiplying out
\(\ds \) \(=\) \(\ds a^2 c^2 + a^2 d^2 + b^2 c^2 + b^2 d^2\) simplifying
\(\ds \) \(=\) \(\ds \paren {a^2 + b^2} \paren {c^2 + d^2}\)


Source of Name

This entry was named for Brahmagupta‎ and Leonardo Fibonacci‎.

Historical Note

Both Brahmagupta‎ and Leonardo Fibonacci‎ described what is now known as the Brahmagupta-Fibonacci Identity in their writings:

  • 628: Brahmagupta: Brahmasphutasiddhanta (The Opening of the Universe)
  • 1225: Fibonacci: Liber quadratorum (The Book of Squares)

However, it appeared earlier than either of those in Diophantus of Alexandria's Arithmetica of the third century C.E.