Triangles with Integer Area and Integer Sides in Arithmetical Sequence

Theorem

The triangles with the following sides in arithmetic sequence 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:

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

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

For $3, 4, 5$:

$\ds s$

$=$

$\ds \frac {3 + 4 + 5} 2$

$\ds $

$=$

$\ds 6$

$\ds \leadsto \ \ $

$\ds A$

$=$

$\ds \sqrt {6 \paren {6 - 3} \paren {6 - 4} \paren {6 - 5} }$

$\ds $

$=$

$\ds \sqrt {6 \times 3 \times 2 \times 1}$

$\ds $

$=$

$\ds \sqrt {6 \times 6}$

$\ds $

$=$

$\ds 6$

For $13, 14, 15$:

$\ds s$

$=$

$\ds \frac {13 + 14 + 15} 2$

$\ds $

$=$

$\ds 21$

$\ds \leadsto \ \ $

$\ds A$

$=$

$\ds \sqrt {21 \paren {21 - 13} \paren {21 - 14} \paren {21 - 15} }$

$\ds $

$=$

$\ds \sqrt {21 \times 8 \times 7 \times 6}$

$\ds $

$=$

$\ds \sqrt {\paren {3 \times 7} \times 2^3 \times 7 \times \paren {2 \times 3} }$

$\ds $

$=$

$\ds 3 \times 7 \times 2^2$

$\ds $

$=$

$\ds 84$

For $15, 28, 41$:

$\ds s$

$=$

$\ds \frac {15 + 28 + 41} 2$

$\ds $

$=$

$\ds 42$

$\ds \leadsto \ \ $

$\ds A$

$=$

$\ds \sqrt {42 \paren {42 - 15} \paren {42 - 28} \paren {42 - 41} }$

$\ds $

$=$

$\ds \sqrt {42 \times 27 \times 14 \times 1}$

$\ds $

$=$

$\ds \sqrt {\paren {2 \times 3 \times 7} \times 3^3 \times \paren {2 \times 7} }$

$\ds $

$=$

$\ds 2 \times 3^2 \times 7$

$\ds $

$=$

$\ds 126$

For $15, 26, 37$:

$\ds s$

$=$

$\ds \frac {15 + 26 + 37} 2$

$\ds $

$=$

$\ds 39$

$\ds \leadsto \ \ $

$\ds A$

$=$

$\ds \sqrt {39 \paren {39 - 15} \paren {39 - 26} \paren {39 - 37} }$

$\ds $

$=$

$\ds \sqrt {39 \times 24 \times 13 \times 2}$

$\ds $

$=$

$\ds \sqrt {\paren {3 \times 13} \times \paren {2^3 \times 3} \times 13 \times 2}$

$\ds $

$=$

$\ds 2^2 \times 3 \times 13$

$\ds $

$=$

$\ds 156$

This article is complete as far as it goes, but it could do with expansion.
The citation below generalises this result, and this page could be turned into it.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.