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 2005-03-02 16:22:03