Lemma: Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them

(Lemma to Proposition 59 from Book 10 of Euclid's “Elements”)

If a straight line is cut unequally then (the sum of) the squares on the unequal (parts) is greater than twice the rectangle contained by the unequal (parts). * Let $AB$ be a straight line, and let it have been cut unequally at $C$, and let $AC$ be greater (than $CB$). * I say that (the sum of) the (squares) on $AC$ and $CB$ is greater than twice the (rectangle contained) by $AC$ and $CB$.


Modern Formulation

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Proofs: 1

Proofs: 1 2

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Adapted from (subject to copyright, with kind permission)

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