Prove that the angles of a triangle all have rational cosines if and only if the triangle is similar to one with rational sides.
The law of cosines states
a^2 = b^2 + c^2  2*b*c*cos A
Rearranging:
cos A = (b^2 + c^2  a^2) / (2*b*c)
so if a, b and c are rational, so will be cos A, and similarly for the other angles of the triangle.
The other way around is harder: to find the rational ratios of the sides of the triangle based on rational cosines.

Posted by Charlie
on 20050302 16:22:03 