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Proposition: 2.10: Sum of Squares (Half)

(Proposition 10 from Book 2 of Euclid's “Elements”)

If a straight line is cut in half, and any straight line added to it straight-on, then the sum of the square on the whole (straight line) with the (straight line) having been added, and the (square) on the (straight line) having been added, is double the (sum of the square) on half (the straight line), and the square described on the sum of half (the straight line) and (straight line) having been added, as on one (complete straight line).

• For let any straight line $AB$ have been cut in half at (point) $C$, and let any straight line $BD$ have been added to it straight-on.
• 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 \(a:=AD\) and \(b:=BD\), we have \(AC=(a-b)/2\) and \(CD=(a+b)/2\), and the proposition states that

\[a^2+b^2=2\left(\left(\frac{a-b}2\right)^2+\left(\frac{a+b}2\right)^2\right).\]

Table of Contents

Proofs: 1

Thank you to the contributors under CC BY-SA 4.0!

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References

Adapted from CC BY-SA 3.0 Sources:

1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014

Adapted from (Public Domain)

1. Casey, John: "The First Six Books of the Elements of Euclid"

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"