How would you be able to find all triangles with a given inradius and all three sides of the triangle have integer lengths? More specifically how would you do it without randomly guessing at numbers?
For example, if you were given an inradius of 2, you could find a triangle with sides of 5, 12, 13 or 7, 15, 20.
If you wanted to go the work out the inradius from a triangle with known sides it would be fairly simple using the following formula. $$R = \frac{\sqrt{s(s-a)(s-b)(s-c)}}{s}$$ I tried a range of methods, such as rearranging Heron's formula to find specific sides. However, that left me with equations such as the following. $$a = \frac{A^2}{s(s-b)(s-c)} + s$$ This equation still has an equation full of unknown values and showed me I was attempting to solve the problem incorrectly. The problem now is that I don't know what else I can do to solve the initial question.