Smallest Scalene Obtuse Triangle with Integer Sides and Area

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Theorem

The smallest scalene obtuse triangle with integer sides and area has sides of length $4, 13, 15$.


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 $\triangle ABC$.

Here we have:

\(\ds s\) \(=\) \(\ds \dfrac {4 + 13 + 15} 2\)
\(\ds \) \(=\) \(\ds 16\)


Thus:

\(\ds A\) \(=\) \(\ds \sqrt {16 \paren {16 - 4} \paren {16 - 13} \paren {16 - 15} }\)
\(\ds \) \(=\) \(\ds \sqrt {16 \times 12 \times 3 \times 1}\)
\(\ds \) \(=\) \(\ds \sqrt {2^4 \times 2^2 \times 3 \times 3 \times 1}\)
\(\ds \) \(=\) \(\ds \sqrt {2^6 \times 3^2}\)
\(\ds \) \(=\) \(\ds 2^3 \times 3\)
\(\ds \) \(=\) \(\ds 24\)


This theorem requires a proof.
In particular: It remains to be shown that it is the smallest. This can be done by exhaustion: the number of triples defining an obtuse scalene triangle are not that many.
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Sources

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