The following proof was discovered by Tom M. Apostol, and was published as “Irrationality of the Square Root of Two – A Geometric Proof” in the American Mathematical Monthly, November 2000, pp. 841–842.
In short, if 
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. Therefore no such triangle could exist in the first place, and is irrational.
In oher words, suppose that is rational. Then by scaling up the isosceles right triangle with sides 1, 1, and appropriately, we obtain the smallest possible isosceles right triangle whose sides are all integers.
Note: If 
, where 
is in lowest terms, then the desired tringle has catheti with length b and hypotenuse a. Indeed, for the pythagorean theorem, we have:
but, for hypothesis, 
. So: 
.
This is 
in the figure below:
By hypothesis, OA, OB, and AB are all integers.
Now construct arc BC, whose center is at A.
Note: 
, and thus AC is an integer. Since 
, OC is also an integer.
Let CD be the perpendicular to OA at point C. Then 
is also an isosceles right triangle, so OC = CD, and CD is an integer.
and 
are tangents to the same arc from the same point D, so 
, and BD is an integer. Since OB and BD are both integers, so is OD.
Since OC, CD, and OD are all integers, is another isosceles right triangle with integer sides, which contradicts the assumption that was the smallest such.
The logical contradiction we have arrived at shows that is a irrational number.
