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I read in book $6$, proposition $19$ of the elements the statement "Similar triangles are to one another in the squared ratio of (their) corresponding sides." Firstly, what does it mean by squared ratio? I thought similar triangles were to one another the ratio of their corresponding sides, not their square.

Here is the proposition and its proof:

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Jacob Willis
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When Euclid says "similar triangles are to one another..." he is referring to the ratio of their areas. If all of the sides of triangle $T_1$ are $k$ times longer than the corresponding sides of triangle $T_2$, then the area of $T_1$ is $k^2$ times the area of $T_2$.

mweiss
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  • Why does Euclid prove it all complicated like this? Isn't is true because $\dfrac{1/2(kb)(kh)}{1/2bh} = k^2$? – Jacob Willis Dec 25 '15 at 18:40
  • Euclid worked without formulas or measurements. He did not regard "area" as a numerical quantity to be computed by measuring lengths and then operating on them arithmetically. His concept of "equal areas" is a purely geometric one. – mweiss Dec 25 '15 at 19:55