I am making this post to thank the Project Euler problem development team and specifically every one who was involved in developing problem 914.
I had a really wonderful time solving problem 914. First off, some coordinate geometry to figure out some formulae about the inradius. If you are attempting problem 914, then you must have some knowledge about primitive pythagorean triples from earlier problems. Then, programming a naive solution in JavaScript that scales up to R = 10 ^ 8. At this point theoretical abstract analysis may or may not get you futher, so we did a concrete data analysis. This leads to the 'a-ha' moment that takes you all the way to R = 10 ^ 18, easier to program this part in Python 3 with PyPy. I am assuming something like this is what the problem posers had in mind: (i) high-school maths, (ii) maths learned by solving earlier problems, (iii) naive solution to give insight into the real solution. All in all a good Project Euler problem-solving experience!!