A geometric proof for the irrationality of √2

Tom Apostol in 2000 gave a geometric proof of the irrationality of 2, which for a bright student can even be considered as a “proof without words”:

Image

(The figure has been taken from wikipedia)

The explanation is as follows: Suppose \sqrt{2} is rational and equals m/n where m,n\in\mathbb{N} with (m,n)=1. Consider an isoceles right \vartriangle ABC with legs of length n and hypotenuse m. Draw the blue arcs with center A. Observe that \vartriangle ABC is congruent to \vartriangle ADE by SAS, and furthermore note the various lengths given in the figure follow from this fact. Hence we have an even smaller right isosceles triangle \vartriangle CDF, with hypotenuse length 2n-m and legs m-n. The fact these values are integers even smaller than m and n and in the same ratio is clear from the figure and thus we contradict the hypothesis that (m,n)=1.

The above proof may be aptly summarized in one line:  If \sqrt{2}=m/n is in lowest terms then \sqrt{2}=(2n-m)/(m-n) is in lower terms. In this case however the definition of the square root is implicitly used to show that 2n-m<m and m-n<n.

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