Posted: Thu Sep 30, 2004 5:37 am    Post subject: Looking for the clearest or easiest solution (construction)

It’s well-known problem of which I know solution, but I need to present it to ordinary high school students which seems to be a little difficult to me… Prove that it’s impossible to construct (following only the rules of Euclidean construction) a triangle given by lengths of its two sides and its inradius (=radius of its inscribed circle). I use this well-known formula:

SQRT

where is inradius, are lengths of sides of our triangle and
We can adjust this formula to the cubic equation with variable or (after substitution ) (it’s an easier form of demanded cubic equation). After a little counting we’ll state that this cubic polynom has no rational root which will imply its irreducibility in rational numbers. And as a final step we’ll use Wantzel Theorem which will show that no root of considered polynomal is a constructible number (following Eucleides). But is there anything clearer and easier…

Thanks for any help.