The dominions of a certain Eastern monarch formed a perfectly square
tract of country. It happened that the king one day discovered that his
four sons were not only plotting against each other, but were in secret
rebellion against himself. After consulting with his advisers he decided
not to exile the princes, but to confine them to the four corners of the
country, where each should be given a triangular territory of equal area,
beyond the boundaries of which they would pass at the cost of their
lives. Now, the royal surveyor found himself confronted by great natural
difficulties, owing to the wild character of the country. The result was
that while each was given exactly the same area, the four triangular
districts were all of different shapes, somewhat in the manner shown in
the illustration. The puzzle is to give the three measurements for each
of the four districts in the smallest possible numbers--all whole
furlongs. In other words, it is required to find (in the smallest
possible numbers) four rational right-angled triangles of equal area.
[Illustration]