The Square Root of 2 is Irrational (Geometric Proof)

Problem: Show that $ \sqrt{2}$ is an irrational number (can’t be expressed as a fraction of integers). Solution: Suppose to the contrary that $ \sqrt{2} = a/b$ for integers $ a,b$, and that this representation is fully reduced, so that $ \textup{gcd}(a,b) = 1$. Consider the isosceles right triangle with side length $ b$ and hypotenuse length $ a$, as in the picture on the left. Indeed, by the Pythagorean theorem, the length of the hypotenuse is $ \sqrt{b^2 + b^2} = b \sqrt{2} = a$, since $ \sqrt...