Does there exist a scalene triangle with integer side-lengths and an integer height.

Question

I recently got fascinated by this question:

Does there exist a scalene triangle $\triangle ABC$ where $AB, AC, $ and $BC$ are integers and $\triangle ABC$ has an integer height?

There is some ambiguity around where the 'height' is but I assume that it is perpendicular from any of the sides.

Cheers!

EDIT: Right-angled triangles are not permitted, I think the question wants a generalisation, if it exists...

Answer 1

A nice example is

• a $13,14,15$ triangle, with height $12$ perpendicular to the $14$ side

and if you multiply these numbers by $195$ you get the more impressive example with all three heights integers:

• a $2535, 2730,2925$ triangle with respective heights $2520,2340,2184$