Heron of Alexandria/Problems/Rectangles

Problem

 * Find two rectangles with integral sides, such that:
 * the area of the first is $3$ times the area of the second
 * the perimeter of the second is $3$ times the perimeter of the first.

Solution
The rectangles are:
 * $53 \times 54$

and:
 * $318 \times 3$

Proof
As can be seen:

and:

and it is seen that the two rectangles fulfil the given conditions.

Let the given ratio be made general, that is, $n$ rather than $3$.

Let $u, v$ and $x, y$ be the sides of the rectangles.

Then we can write:

Thus the general solution is:

Setting $n = 3$ gives the solution.