(FB/Math/BM – Problem proposed by Souradeep Purkayastha)
How many unique families of triangles exist with same circumradius and same inradius? I mean, in how many ways can side lengths (a,b,c) be ‘combinated’, for fixed r and R?
There are infinitely many such lengths (a,b,c).
If the area is T then r = 2T/(a + b + c), and R = abc/(4T), so
2rR = abc/(a + b + c). The a, b, c satifying this and the conditions for a, b, c to be sides of a triangle, i e a + b > c > 0 &cycl… , are possible for given r and R. The equation gives c = 2rR(a + b)/(ab – 2rR) which describes a surface in ℝ³. Points (a,b,c) on this surface where also a + b > c > 0 etc are possible.