# Proposition: 2.04: Square of Sum

### (Proposition 4 from Book 2 of Euclid's “Elements”)

If a straight line is cut at random then the square on the whole (straight line) is equal to the (sum of the) squares on the pieces (of the straight line), and twice the rectangle contained by the pieces. * For let the straight line $AB$ have been cut, at random, at (point) $C$. * I say that the square on $AB$ is equal to the (sum of the) squares on $AC$ and $CB$, and twice the rectangle contained by $AC$ and $CB$.

### Modern Formulation

With $$a=AC$$ and $$b=CB$$, this proposition is a geometric version of the algebraic identity: $(a+b)^2 = a^2+2\,a\,b+b^2.$

See also binomial theorem for $n=2.$

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13
Sections: 14

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### References

#### Adapted from CC BY-SA 3.0 Sources:

1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
2. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016