Triangles with Integer Area and Integer Sides in Arithmetical Sequence

Theorem
The triangles with the following sides in arithmetic progression have integer areas:
 * $3, 4, 5$
 * $13, 14, 15$
 * $15, 28, 41$
 * $15, 26, 37$

Their areas are:
 * $6, 84, 126, 156$

Proof
From Heron's Formula, the area $A$ of $\triangle ABC$ is given by:
 * $A = \sqrt{s \left({s - a}\right) \left({s - b}\right) \left({s - c}\right)}$

where $s = \dfrac{a + b + c} 2$ is the semiperimeter of $\triangle ABC$.

For $3, 4, 5$:

For $13, 14, 15$:

For $15, 28, 41$:

For $15, 26, 37$: