Proof That √2 Is Irrational
In short, if √2 were rational, we could construct an isosceles right triangle with integer sides. Given one such triangle, it is possible to construct another that is smaller. Repeating the construction, we could construct arbitrarily small integer triangles. But this is impossible since there is a lower limit on how small a triangle can be and still have integer sides.
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The big triangle and the little triangle share one angle, and both have a right angle. Since the angles must sum to 180, the triangles must have the same third angle. Therefore they are similar. Since the legs of the big triangle have equal lengths (it was isosceles), the legs of the small triangle must also be equal (by similarity).
