Badly approximable triangles and Delone sets

We investigate the linear equation \(x+y+z=1\) where \(\pi x\), \(\pi y\), \(\pi z\) are three angles of a triangle and the numbers \(x\), \(y\), \(z\) are badly approximable. We show that there are exactly two solutions which have the smallest partial quotients by lexicographical ordering. Lastly, we give a construction of associated Delone sets which are expected to have rotationally invariant diffraction.