Relationships between heronian Triangles

[Mikestone8]

On listing Heronian triangles with shortest side 3 in order of size, I find that the ratio of the height (and hence the area) of each successive triangle to the previous one seems to converge toward 3+ 2√2. On experimenting with Heronian triangles of base 4, I seem to detect a similar pattern with convergence toward a ratio of 2+ √3.

However, so far I have totally failed to discover any similar pattern in cases where the shortest side is 5. Is there none or am I missing something obvious?

Or is there any general solution to this sort of question?

 

[Alonzo]

No idea, but I had never heard the term Heronian triangle and just looked it up, so thanks for that info. Did you learn about these in a class?

 

[Mikestone8]

No. It was accidental.

I knew of course that the 3,4,5 triangle (and all Pythagorean) ones) had a rational area. However, in a moment of idle curiosity I tried the 13,14,15 triangle and found that it also had one. I went on to try 23,23,25 but that one didn't.

I contacted a maths website (iirc it was Stack Exchange) and enquired if there were others and if so how to fin them. Like you I had never heard the term" but got a lot of information about them. I found the two relationships which I mentioned in my first message, but haven't been able to find one for a shortest side of 5 - hence my enquiry.