Simone Muselli

A fabled pair of triangles

Once upon a time, in the kingdom of Integrity, there were two peasant farmers, named Rightness and Equality. They went to King Fairness to ask ​for land to farm. Rightness wanted it to have the form of a right angled triangle, while Equality requested his land be shaped like an isosceles triangle. Then the righteous king Fairness decided to fulfil their requests, but with some extra conditions. The two fields had to have ​the same area and perimeter; moreover, the lengths of all edges had to be integral, otherwise it would not have been possible to measure them. Considering the then state of economic turmoil in the Kingdom of Integrity, what were the sizes of the smallest possible fields satisfying the previous requirements?