Hello. I thought up the following problem this morning:
Is it possible to construct an isosceles triangle with integer side length and area?
Mathematica was unable to find any solutions, but it did not say that there were none. I tried to work with the associated Diophantine equation, but I couldn't see immediately how to show that there weren't any solutions.
I'd appreciate any thoughts on the matter.
Regarding Mathematica.
Heron's formula:
Let . Then
Screenshot from Mathematica:
Edit: Of course it's also possible to avoid Heron's formula by dividing the isosceles triangle into two right triangles to begin with.
Label the two congruent sides and the other side . Treat as the base and draw an altitude from the base to the opposite vertex.
and
Substitute
etc.