3 Proper Integer Heronian Triangles whose Area and Perimeter are Equal

Theorem
There are exactly $3$ proper integer Heronian triangles whose area and perimeter are equal.

These are the triangles whose sides are:


 * $\tuple {6, 25, 29}$
 * $\tuple {7, 15, 20}$
 * $\tuple {9, 10, 17}$

Proof
First, using Pythagoras's Theorem, we establish that these integer Heronian triangles are indeed proper:

Now we show they have area equal to perimeter.

We use Heron's Formula throughout:


 * $\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$

where:
 * $\AA$ denotes the area of the triangle
 * $a$, $b$ and $c$ denote the lengths of the sides of the triangle
 * $s = \dfrac {a + b + c} 2$ denotes the semiperimeter of the triangle.

Thus we take the $3$ triangles in turn:

It remains to be demonstrated that these are indeed the only such proper integer Heronian triangles which match the criterion.