The following diagram provides a neat and not well known proof that the square root of 2 is irrational. (This proof is by Tom Apostol, NOT ME.  It is here because I like it.  The wording is mine.)

 

root2

 

 

Here are the details. Suppose is rational then =m/n for some integers m and n. Let n be the smallest integer for which there is an m with =m/n.

 

Let ABC be an isosceles right triangle with side n. In the above diagram, length(AC)=length(BC)=n . And length(AB)=m= n. Now consider a circle, O, with center at A and radius equal to length(AC) or n. At point D, the intersection of O with AB, we erect a perpendicular which intersects BC at E. Note that since DE and EC are tangents to circle O, they are equal. But DE is equal to BD and BD is an integer (namely m-n). But then EC is also the integer m-n and then BE is an integer (namely 2n-m) . Since BED is a 45-45-90 triangle with integer sides we now have a contradiction to the choice of n as the smallest integer for which there is an m with =m/n. (length(DE)<length(BE)<length(BC)=n)