# 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
5 years ago

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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.

- 5 years ago