Welcome to Math Help Forum!
With the usual notation, the radius of the incircle of a triangle, , is given by the formulaand the radius of the circumcircle, , by the formulaThis second formula - the Sine Rule, of course - gives
and so if (i.e. ):I denoted the lengths of the line segments as follows:
and, as usual,and I also denoted the angle:Then, using the Sine Rule on :So the area of the triangle is given by:and hence the radius, , of its incircle (when simplified) by:In the same way, the radius, , of the incircle of is:
So the problem can now be re-stated as:Given that , prove that, for all valid values of ,It looks promising, but it just ain't so! The LHS can be manipulated as follows:
and, in a similar way, the RHS
So, in order for these to be equal, we would need
i.e. , for all values of , given . But this simply is not so.
I have put some calculations together into an Excel spreadsheet to confirm this - that we can fulfill all the conditions and yet produce different radii. I attach the Excel file.