Proposition: 2.09: Sum of Squares of Sum and Difference
(Proposition 9 from Book 2 of Euclid's “Elements”)
If a straight line is cut into equal and unequal (pieces) then the (sum of the) squares on the unequal pieces of the whole (straight line) is double the (sum of the) square on half (the straight line) and (the square) on the (difference) between the (equal and unequal) pieces.
- For let any straight line $AB$ have been cut - equally at $C$, and unequally at $D$.
- I say that the (sum of the) squares on $AD$ and $DB$ is double the (sum of the squares) on $AC$ and $CD$.
Modern Formulation
Algebraically, with \(a=AC\) and \(b=CD\), this proposition states that
\[(a+b)^{2}+(a-b)^{2}=2(a^2+b^2).\]
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1
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References
Adapted from CC BY-SA 3.0 Sources:
- Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
Adapted from (Public Domain)
- Casey, John: "The First Six Books of the Elements of Euclid"
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
