Let *a*, *b*, and *c* be the sides of a triangle. Let *r* be the radius of an inscribed circle and *R* the radius of a circumscribed circle. Finally, let *p* be the perimeter. Then the previous post said that

2*prR* = *abc.*

We could rewrite this as

2*rR* = *abc* / (*a* + *b* + *c*)

The right hand side is maximized when *a* = *b* = *c*. To prove this, maximize *abc* subject to the constraint *a + b* + *c* = *p* using Lagrange multipliers. This says

[*bc*, *ac*, *ab*] = λ[1, 1, 1]

and so *ab* = *bc* = *ac*, and from there we conclude *a* = *b* = *c*. This means among triangles with any given perimeter, the product of the inner and outer radii is maximized for an equilateral triangle.

The inner radius for an equilateral triangle is (√3 / 6)*a* and the outer radius is *a*/√3, so the maximum product is *a*²/6.