Other Triangles with Integer Sides Satisfying the Hypothesis

All of these in the table at the right are non-similar triangles with integer sides where the lengths of   a  and   b   are less that 40.   The list is not exhaustive.

Can you find other non-similar triangles where one of sides of length a or b is less than 40?
How can you search for more such triangles?
These were found using a spreadsheet.
Can we prove there is an infinite set of non-similar triangles satisfying the hypotheses?