AoPS - Problem

For each natural $i$, let $gcd(a_i,b_i,c_i)=1$ and $a_i^2+b_i^2=c_i^2$, i.e. Consider infinitely many distinct pythogarean triplets. Now consider many points on a semicircle with diameter $AB=1$ and the distance from the points to $A$ is $\frac{a_i}{c_i}\in \mathbb{Q}$ and distance to $B$ is $\frac{b_i}{c_i}\in \mathbb{Q}$. Then it is easy to check that all the distance between 2 points are rational by Ptolemy and the area of any triangle is rational. Now we scale the whole figure by $\sqrt2$ and the lengths are now all irrational but the area is now 2 times which is still rational.