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Triangles with equal perimeter and area

Interesting problem that I'd like to share:

Find all triangles whose side lengths are integers such that the area equals perimeter.

[ Note: you can find the answer on google if you so desire. ]

Note by C Lim
4 years, 4 months ago

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4 votes

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One approach is the heron's formula by which we get a diophantine equation.

Another approach I have thought of is that for scalene triangles with side lengths \(a,b,c\), we must have that :

\(abx=2y(a+b+c)\)

Where \(sin \theta=\frac{x}{y}\) and \(x,y\) and are co-prime positive integers such that \(x<y\), and \(\theta\) is the angle between \(a\) and \(b\).

If we consider any isosceles triangle with integer side lengths \(a,b,b\) after doing some algebra we get that:

\(b=\frac{a(a^2+1)}{2(a^2-1)}\)

But this is not possible because of the triangle inequality:

We can use that since \(2b>a\)

By which we get a contradiction that \(-1>1\)

Obviously there will be no equilateral triangles that satisfy this.

Aditya Parson - 4 years, 4 months ago

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