Are there any known right triangles that have integer side lengths and rational angles? If not, has it been proven that none exist?
Do you mean rational in radians, or in degrees?
I was thinking of degrees, or fractions of a whole (2π radians), though either would be interesting. I would be quite surprised if any such triangles existed with rational angles in radians, given that π is irrational.
I bit late but i i think it is proven there is no solutions, except for the special case 0° and side lengths 1, 1 and 0. Let us consider the triangle with a²+b²=c² and a = c sin(pi q) where q is the angle as a fraction of half a circle. So you are looking for a solution where a, b, c are integer and q is rational. So we first need to find a rational value for q where sin(pi q) is rational. According to https://math.stackexchange.com/questions/87756/when-is-sinx-rational#87768 this happens only for the well known case of 30°, so q=1/6 and a/c=1/2. However, in this case b=c/2 × sqrt(3) which is irrational, so with this angle we can never create integer side length.
It looks like there aren’t any, by Niven’s Theorem https://www.proofwiki.org/wiki/Niven’s_Theorem
Thanks!
