About An Inscribed Circle

I’m not precisely sure how to embed this, so, let me just point over to the Five Triangles blog (on blogspot) where there’s a neat little puzzle. It starts with Pythagorean triplets, the sets of numbers (a, b, c) so that a2 plus b2 equals c2. Pretty much anyone who knows the term “Pythagorean triplet” knows the set (3, 4, 5), and knows the set (5, 12, 13), and after that knows that there’s more if you really have to dig them up but who can be bothered?

Anyway, the problem at Five Triangle’s “Inscribed Circle” here draws that second Pythagorean triplet triangle, the one with sides of length 5, 12, and 13, and inscribes a circle within it. The problem: find the radius of the circle?

I’m embarrassed to say how much time I took to work it out, but that’s because I was looking for purely geometric approaches, when casting it over to algebra turns this into a pretty quick problem. I do feel like there should be an obvious geometric solution, though, and I’m sure I’ll wake in the middle of the night feeling like an idiot for not having that before I talked about this.