Determining all triples $(a,b,c)$ of positive integers that are sides of a triangle inscribed in a circle of diameter $6.25$

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An Olympiad Geometry question

Determine all triples $(a, b, c)$ of positive integers which are the lengths of the sides of a triangle inscribed in a circle of diameter $6.25$ units.

So, in this, basically diameter = 2R = $\frac{a}{sinA}$ and so on. Or we also have R = $\frac{abc}{4\Delta}$.

But I don't understand how this may help in this problem.

Any hint/ solution is appreciated,

Thanks!

Best Answer

Observe that, since $6.25$ is the length of the longest chord(diameter) of the circle, $a$, $b$ and $c$ will belong to $\{1,2,3,4,5,6\}$.

Also, notice that, $25$ will divide $abc$. Then, at least $2$ of these terms will be equal to $5$($\Delta $ will be rational ).Let $a=5$ and $b=5$ without loss of generality. Now, just plug in those values into the equation and you will get the value of $c$.

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