I was doodling on some piece of paper a problem that sprung into my mind. After a few minutes of resultless tries, I advanced to try to solve the problem using computer based means.
The problem stated is
Does a right angle triangle with integer sides such that $P = A\cdot k \quad \quad \text{or} \quad \quad k\cdot P = A$ where $(P , \ A , \ k )\in \mathbb{N}$ exist?
(Here P is the perimeter of the triangle. Sum of the sides. And A is the area of the triangle. a*b/2 in a triangle with a b c and c is the hypotenuse)
Obviously for the simple cases such triangles exists. For an example when
$ A=2P \qquad $ the triangle $12,16,20$ works
$ A=P \qquad \; \; $ the triangle $6,8,10$ works
$2A=P \qquad $ the triangle $3,4,5$ works
I tried solving this by hand, first for the special case where $2P=A$. This ended up giving me
$ \frac{a \cdot b}{2} = A $
and
$ P = \frac{A}{2} = \frac{a \cdot b}{4} \qquad \text{also} \qquad P = a + b + c $
So
$ \frac{ab}{4} = a + b + c $
By knowing that this is a right angle this leads to the equation (Using the Pythagorean theorem)
$ a^2 + b^2 = c^2 $
Now we have two equations and three unknowns, which also needs to be integers! Sadly I was not able to continue from here. I have only learned how to solve linear Diophantine equations. Not a system of nonlinear Diophantine equations.
Just to restate my question below =)
Is there a right angle triangle with integer sides such that $P = A\cdot k \quad \quad \text{or} \quad \quad k\cdot P = A$ where $(P , \ A , \ k )\in \mathbb{N}$ ?
Regards, Werner